honours project - magetic activity in sun-like stars - emily walsh bsc mathematics and physics

The University of Dundee, School of Engineering, Physics andMathematics

BSc Mathematics and Physics Honours ProjectMagnetic Activity in Sun-Like Stars

Emily Walsh: [email protected]

April 28, 2016

Contents

1 The Need for a Dynamo 31.1 Introduction of Magnetic Activity on the Sun & other Sun-Like Stars . . . . . . 31.2 Estimate of Decay Time of a Fossil Field on the Sun . . . . . . . . . . . . . . . 5

2 Discrete Maps as Dynamo Models 82.1 Motivation Behind the iterative map . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.1 Description of ‘erf’ function . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Use of Matlab to find Solutions to the one-dimensional Iterative Map . . . . . . 112.3 Bifurcation Diagram Summarising Previous Solutions . . . . . . . . . . . . . . . 18

2.3.1 Periodic Windows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.4 Chaotic Solution Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4.1 Sensitivity To Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . 192.4.2 Lyapunov Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.4.3 Feigenbaum Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4.4 Sharkovskii’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.5 Replacement of The Function Within The Map . . . . . . . . . . . . . . . . . . 242.5.1 Finding Solutions of Iterative Map With The Function Replaced . . . . . 262.5.2 Bifurcation Diagram of The Iterative Map With The Function Replaced 32

2.6 Introducing Stochastic Forcing To The Map . . . . . . . . . . . . . . . . . . . . 33

3 Mathematics of Dynamo Theory 353.1 Anti-Dynamo Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.1.1 Cowling’s Anti-dynamo Theorem: . . . . . . . . . . . . . . . . . . . . . . 353.2 Dynamo Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2.1 Parker’s 1955 Model: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.2.2 Tacholine Dynamos: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4 Conclusion 404.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Appendix 44

A 44A.1 MATLAB code used for finding solutions to the one-dimensional iterative map

(section 2.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44A.2 MATLAB code used for generating cobweb maps for the one-dimensional itera-

tive map (section 2.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44A.3 MATLAB code used to produce the bifurcation diagram for the one-dimensional

iterative map (section 2.3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45A.4 MATLAB code used to display both the original function f(pn) and the new

function which it was replaced by (section 2.5) . . . . . . . . . . . . . . . . . . . 46

1

Magnetic Activity in Sun-Like Stars

A.5 MATLAB code used for finding solutions to the one-dimensional iterative mapwith function replaced (section 2.5.1) . . . . . . . . . . . . . . . . . . . . . . . . 46

A.6 MATLAB code used for generating cobweb maps for the one-dimensional itera-tive map with function replaced (section 2.5.1) . . . . . . . . . . . . . . . . . . . 46

A.7 MATLAB code used to produce the bifurcation diagram for the one-dimensionaliterative map with function replaced (section 2.5.2) . . . . . . . . . . . . . . . . 47

A.8 MATLAB code used to introduce stochastic forcing to the one-dimensional it-erative map (section 2.6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2

1

The Need for a Dynamo

1.1 Introduction of Magnetic Activity on the Sun &

other Sun-Like Stars

The Sun is the impressive star positioned in the centre of our solar system and the closeststar to us here on Earth. It is vital to everyday life on Earth, Earth’s climate and on spaceweather, therefore investigating the changing magnetic activity on the Sun plays an importantrole in life on earth. Even though the magnetic field lines on the Sun’s surface can’t be seen,using models these fascinating magnetic fields which are constantly changing over time can bevisualised.

The Sun is essentially a massive ball of plasma compressed by its own gravitational attrac-tion and is extremely hot, with the surface reaching temperatures of up to 5,785 Kelvin [1].It has been shown that the Sun is not believed to be a solid body and is in fact a gaseousbody by the knowledge that the plasma of the sun rotates around the Sun’s axis at a higherspeed at the equator of the Sun than at the poles. Plasma in space includes many free chargedparticles and is therefore highly energetic. In the convection zone of the Sun giant convectioncurrents take place, and these flows of hot plasma containing the moving free particles createsa magnetic field. This process is similar to generating a magnetic field from current flowingin a wire but the convection current is driven by the heat from the Sun’s fusion rather thana battery. The Sun generates a magnetic field that extends out into space, and using a modelto show this, both the open and closed magnetic field lines can be seen. Some of the magneticfield lines are open and extend to far off distances in the solar system unlike the closed oneswhich come out from the surface of the Sun and loop around and close on the solar surfaceagain [2], [3]. This can be seen in Figure 1.1.1 where the closed field lines are white and theopen field lines are the green and purple ones which have north and south polarity. From thesediagrams it is shown that the magnetic field lines have changed dramatically, from being soclose together and concentrated at the poles in 2011 to then becoming very messy and chaotic.This could be the perfect conditions for solar events to happen like coronal mass ejections orsolar flares.

In areas of the Sun where there is a strong magnetic field a phenomena occurs producingdark observable spots which are known as sunspots. As previously stated, in the convectionzone of the Sun giant convection currents take place, within these convection flows the hotmatter rises and the cooler matter falls back down again. It is a similar process to boilingliquid, the hot bubbles of liquid rise quickly reaching higher levels than before and then whenthey become cooler than their surroundings they fall back down before rising again and the

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Figure 1.1.1: Changing Magnetic Activity on the Sun between January 2011 (left) and July2014 (right) [4]

process repeats. So in context to the Sun, as these convective currents are constantly providingmore and more hot matter to the top this maintains the particular temperature at the surface.However in these regions of strong magnetic fields, the fields manage to somehow inhibit theconvection and stop the convective flow so there is not enough hot matter reaching the top inthese regions and this results in a cool spot, observed as a dark sunspot. In areas of the Sunwith intense magnetic activity there is therefore a greater number of sunspots [5].

Sunspots tend to appear in groups and tend to appear in pairs within the groupings, thepairs are usually associated with a north and south pole of the magnetic field, having oppo-site polarity. From one year to the next sunspots do not appear on the same position on theSun, and as time progresses they move closer to the equator. The sunspots have an orderedform of polarity, such that, in the northern hemisphere the leader sunspots in the group wouldhave north polarity and be closest to the equator and the same for the south pole [6]. As thesunspots move closer to the equator each pair of sunspots is migrating and it is believed thatthe positive and negative leader sunspots which are coming towards each other start diffusinginto each other and cancelling out. Once a cancellation of the positive and negative leadersunspots takes place only the ones following behind are left. These ones which are left areof the opposite polarity of each of the poles and will affect the North and South poles of theglobal magnetic field . In the process of an 11 year cycle it is then the case that the Suns poleswill flip where the North will become South and the South become North, this then flips back11 years later meaning the cycle of the Sun is essentially a 22 year cycle. Even though activityon the Sun seems fairly regular, the sunspots do not always appear in the same quantities.There was a inconsistent period starting in about 1645 and occuring until 1715 when sunspotnumbers became dramatically reduced and even slightly rare, this period of time is called theMaunder Minimum, [7].

Similar to what was discussed for the Sun, stellar rotation and the convective motions ofthe stars generate strong magnetic fields and productive several magnetic phenomena includingstarspots in the photosphere. Sun-like stars, or solar-type stars are stars on the lower mainsequence which are known to show chromospheric activity similar to that on the Sun, thisactivity can be detected by measuring the profiles of the Ca II H & K chromospheric emissionlines. Ca II H & K emissions, is the emission of two lines of singly charged calcium emittedfrom the areas of concentrated magnetic field and can be measured in both the Sun and sun-like stars. Using the individual measurements from sun-like stars and the Sun itself, it is thenpossible to make comparisons to solar activity on the Sun. Ca II H & K emission has beenmeasured in lower main sequence stars since 1957 and most of the information currently usedis from a project known as the ‘HK project’. From this project it is believed that young stars

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are more active and therefore rotate faster than the less active and slower older stars [8].

Like the Sun, rotation in sun-like stars is an important part in the analysis and observationsof the properties of the stars. In sun-like stars the rotation is much faster than that of the Sun,which only rotates once every 25 days at its equator. Unlike the Sun’s 11 year solar cycle, thecycle of these sun-like stars varies as they are all rotating at different speeds.

Direct observation of star spots is not possible, unlike being able to observe the sunspotson the Sun. It was possible to spatially resolve the sunspots but on these other stars it is not.Starspots are however observed in different ways; they can be detected by direct imaging, butthis has limited application where only very large and very near objects are observable. Theycan also be detected in another way, through photometric light curves, throughout the starsrotation cycle the starspots come in and out of view by causing changes in the stars brightness,or periodic light variations. This can then be displayed on a curve graph and the pattern inthe stars light curve will repeat once per rotation of the star, however only simple spot con-figurations can be retrieved. And finally the most used method of detecting the starspots iscalled Doppler imaging, which is a technique using a series of spectral line profiles, of a rapidlyrotating star, to compute the stellar temperature distribution. As the spot moves across thesurface of the star the line profile changes, an image of the surface of the star can then beproduced from an observed line profile [9].

Magnetic activity in sun-like stars declines with age and this is related to the loss in angularmomentum throughout the main sequence lifetime. Therefore young stars will exhibit highlevels of activity and rapid rotation and older stars will be less active with slower rotationspeeds as was proven from the HK project before.

1.2 Estimate of Decay Time of a Fossil Field on the Sun

It has previously been shown that a magnetic dynamo on the Sun does infact exist due tothe knowledge that a fossil field on the Sun, such as a sunspot, decays in a shorter time thanthe lifetime of the Sun itself, but yet can still be seen on the Sun today. In this section anestimate of the decay time of a fossil field will be calculated and used in this assumption thatthe magnetic dynamo is necessary on the Sun. The Magnetohydrodynamic (MHD) equationswhich will be used in this section are:

Amperes Law:

~j =1

µ0

∇× ~B

Ohms Law:

~E = −~v × ~B +~j

σFaradays Law:

∂ ~B

∂t= −∇× ~E

The Solenoidal Constraint:

∇ · ~B = 0

To calculate the decay time the Induction Equation, another MHD equation, is used. Itcan be obtained from the following derivation using the equations above:

5


In Solar MHD the electric field ~E and the current density ~j are eliminated and it is theprimary variable ~B which is worked with [10]. As Ohm’s Law gives ~E from ~v, ~B, ~j and ~σ, and

the only other place ~E appears is in Faraday’s Law, therefore ~E can be easily eliminated:

To eliminate ~E use Amperes Law:

~j =1

µ0

∇× ~B (1.2.1)

and the generalised Ohms Law,

~E = −~v × ~B +~j

σ(1.2.2)

Rearranging (1.2.2) and then substituting in equation (1.2.1) an equation for ~E is obtained:

~E =1

σ~j − ~v × ~B

~E =1

σ

(1

µ0

(∇× ~B)

)− ~v × ~B

~E = −~v × ~B +1

µ0σ(∇× ~B)

This is then substituted into Faraday’s Law, to eliminate ~E:

∂ ~B

∂t= −∇× ~E

= −∇×(−~v × ~B +

1

µ0σ(∇× ~B)

)= ∇× (~v × ~B)−∇×

(1

µ0σ(∇× ~B)

)= ∇× (~v × ~B)−∇× (η(∇× ~B))

Since the magnetic diffusivity, η =1

µ0σ, and assuming η is constant this gives:

∂ ~B

∂t= ∇× (~v × ~B)− η∇× ~B

= ∇× (~v × ~B)− η(∇× (∇× ~B))

The last term in this equation can be expanded out using the vector identity:

∇(∇ · ~B)−∇× (∇× ~B) = ∇2 ~BThe equation then becomes:

∂ ~B

∂t= ∇× (~v × ~B)− η(−∇2 ~B +∇(∇ · ~B)

= ∇× (~v × ~B) + η(∇2 ~B −∇(∇ · ~B))

∇ · ~B = 0 , leaving the following equation, known as the Induction Equation:

∂ ~B

∂t= ∇× (~v × ~B) + η∇2 ~B (1.2.3)

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To calculate an estimate for the decay time of a fossil field on the Sun the ∇× (~v× ~B) termcan be neglected from the induction equation, this is because an estimation of the decay timeif the plasma were stationary is calculated to avoid any complication, so ~v = 0.

The induction equation then simplifies to:

∂ ~B

∂t=

1

µ0σ∇2 ~B (1.2.4)

In cartesian coordinates this equation for the x coordiate is:

∂Bx

∂t=

1

µ0σ

(∂2Bx

∂x2+∂2By

∂y2+∂2Bz

∂xz

)

The solution of these equations shows that magnetic fields decay together with the currentproducing them and therefore it is possible to derive an appropriate decay time. Assuming thatthe currents vary significantly in radius R, then from equation (1.2.4) the decay time becomes:[11]

τD = µ0σR2

τD =R2

η(1.2.5)

To find the simple estimate for the decay time of a fossil field on the Sun values for R2 andη are substituted in:

An estimate for η as a constant in the solar corona is η = 1× 109 T−32 m2s−1

Treating R as an estimate of the radius of a typical sunspot, R is substitued as R = 25750 km =2.575× 10−7 m. (Sunspots range from about 1,500 km to 50,000 km, [12].)

τD =R2

η

τD =(2.575× 107)2

1× 109

τD = 663062.5 s

τD = 7.67433 days

τD ≈ 1 week (1.2.6)

This value is an estimate of the decay time of a typical sunspot, comparing this to thelifetime of the Sun can explain why a magnetic dynamo is invoked.

The lifetime of the Sun is approximately 10 billion years [1]. The decay time for this typicalsunspot is only 1 week which is much smaller and only an extremely small fraction of 10 billionyears. From this discovery it can then be said that the sunspots clearly have not been therefrom the start of the Sun’s lifetime or they would have all decayed by now and there wouldbe none left. So if it is known that the sunspots must be decaying and more being created,then there has to be something else happening to explain why this is the case. Hence, why amagnetic dynamo is invoked to explain stellar magnetic activity.

7

2

Discrete Maps as Dynamo Models

2.1 Motivation Behind the iterative map

From before it was said that there had to be something else explaining the creation anddecaying of the short lived sunspots, which is why a magnetic dynamo was invoked. One ofthe models that exists to explain the magnetic dynamo and develop a model of the solar cycleis the Babcock-Leighton model, a flux-transport dynamo model, a detailed description of thismodel is found in Babcock’s 1961 paper, [13], [14].

Figure 2.1.1: “ Babcock’s qualitative scenario for dynamo action. Stage 1: the initialpoloidal dipolar field has magnetic field lines closing in the interior at shallow depths. Stage2: submerged flux is stretched out by differential rotation to produce toroidal flux. Stage 3:magnetic buoyancy causes flux to emerge at sunspot groups. Stage 4: following flux of activeregions migrates to the poles and reverses the polar field, creating a new poloidal dipolar fieldof opposite polarity [13], [15].”

Following the Charbonneau paper, [14], in 1961 Babcock made his discoveries on how thetoroidal and poloidal field components operate within the Sun. He came to the conclusion thatthe toroidal field, which is the azimuthal field, can be generated by differential rotation andthis field exists just below the surface of the Sun. It rises by magnetic buoyancy and the stagesof this process can be seen in Figure 2.1.1 [15]. The poloidal field component is the field inthe opposite direction to toroidal, the field stretching between the two poles. Earlier it wasexplained that the sunspots on the Sun are in a certain formation with the leading sunspot

8


having the same polarity as the hemisphere they appear in and that they are moving towardsthe equator and migrate. The pairs that the sunspots appear in with one sunspot having northand the other south polarity can sometimes be described as bipolar sunspots, with each endbeing one pole of a localised magnetic field called a flux tube. The Babcock-Leighton modeluses this information and Babcock suggested that when these bipolar sunspots eventually de-cay, they release a fraction of their magnetic flux into the photosphere. Under several actionsincluding the differential rotation discussed, this flux accumulates in the polar regions untilit overwhelms the “old” poloidal flux and causes a reversal of the poloidal field, hence a newpoloidal field component of opposite sign is produced [16]. In 1969 Leighton further describedthe migration of the flux, he suggested that it is due to the supergranular eddy diffusion that thepoloidal flux migration takes place, he then necessarily incorporated an α-effect. The α-effectis described in later sections, when talking about how cyclonic twisting Parker loops are created.

The Babcock-Leighton model operates in the surface layer of the Sun and the region wherethe toroidal fields are produced is much deeper within, so for this dynamo model to work thetwo different source regions must be able to somehow transfer information to one another. Thiscommunication of the two regions has been suggested to have taken place via advection by aquadrupolar meridional flow penetrating the convective envelope. This meridional circulationwhich takes place does control the cycle period but it also introduces a long time delay in thedynamo mechanism. Due to the Babcock-Leighton model having this long time delay of thesolar cycle it means that it is then possible to use a one dimensional iterative map to computethe cycle to cycle amplitude variations [16].

The strength of the toroidal field at cycle n+1 is linearly proportional to the surface poloidalfield strength of cycle n, it is proportional to the preceeding cycle:

Tn+1 = aPn

The strength of the poloidal field at cycle n+1 is non-linearly proportional to the toroidalfield strength of the same cycle:

Pn+1 = f(Tn+1)Tn+1

Here f(Tn+1) is an unspecified function which is measuring the efficiency of the Babcock-Leighton mechanism as a function of the toroidal field strength.

Converting the above two equations into non dimensional form provides the following non-dimensionalised equations, where P and T are representative values for the poloidal and toroidalfield amplitudes:

tn+1 = apn

pn+1 = (T /P )f(tn+1)tn+1

where pn = Pn/P , tn = Tn/T , and a=(P /T )∇Ω∇t. The last term here, a, is dimen-sionless and can be set to 1, being absorbed into the maps parameter α. Substituting thenon-dimensionalised equation for toroidal field strength into the one for poloidal field strengthresults in the following equation:

9


pn+1 = (T /P )f(pn)pn

In the Charbonneau paper, [17], the function is chosen as:

f(tn+1;β) = (P /T )(1 + β(1− tn+1)), β > 0

β in this equation is a measure of the efficiency of the Babcock-Leighton mechanism whenpoloidal fields are regenerated. Substituting this then back into the previous equation leads to:

pn+1 = pn(1 + β(1− pn), [≡ g(pn;β)], β > 0

The one-dimensional iterative map then used in the later calculations will be:

Pn+1 = αf(Pn)Pn

This map is one dimensional as it only involves a single dynamical variable, Pn, and it isdescribed as a map as it allows a cycle amplitude in terms of the amplitudes of preceding cyclesto be calculated.

2.1.1 Description of ‘erf’ function

Figure 2.1.2: Plot of the error function [18]

The function erf(x) is a function that can often be found when using software such asMaple or Matlab and is known as the “error function”. The error function is encountered whenintegrating the normal distribution (which is normalized from the Gaussian function).

The error function is used in measurement theory (using probability and statistics), and itsuse in other branches of mathematics is typically unrelated to the characterization of measure-ment errors.

The property erf (-z) = -erf (z) means that the error function is an odd function. Thisdirectly results from the fact that the integrand e−t

2is an even function.

10


It can be defined as:

erf(x) =2√π

∫ x

0

e−t2

dt

The complementary error function is defined by:

erfc(x) = 1− erf(x)

Erf can also be defined as a Maclaurin Series:

erf(x) =2√π

∞∑n=0

(−1)nz(2n+ 1)

n!(2n+ 1)

The Imaginary error function is defined by:

erfi(x) = − Ierf(Ix) =2√π

∫ x

0

et2

dt [19]

2.2 Use of Matlab to find Solutions to the one-dimensional

Iterative Map

The one dimensional model for a simple dynamo in the form of an iterative map previouslyshown, is used here to find solutions. The previously undefined function is now replaced withthe function below, from following page 615/6 on the Charbonneau 2005 paper [14]:

pn+1 = αf(pn)pn, f(pn) =α

4

(1 + erf

(pn −B1

ω1

))(1− erf

(pn −B2

ω2

))(2.2.1)

In studying this model the following values for the constants are substituted into the map,B1 = 0.6, B2 = 1.0, ω1 = 0.2, ω2 = 0.8:


4

(1 + erf

(pn − 0.6

0.2

))(1− erf

(pn − 1.0

0.8

))(2.2.2)

Using Matlab, treating α as a control parameter and starting with the same initial valuep(1) = 1.1, solutions for this iterative map were found. The solutions are displayed for severaldifferent values of α where noticeable abrupt changes occured and these can be seen in Figures2.2.1 - 2.2.6.

Another way of displaying the solutions to this mapping is the use of cobweb diagrams (alsoseen in Figures 2.2.1 - 2.2.6). These cobweb diagrams/maps provide another visual represen-tation of what is happening as the value of α is altered.

11


The method of producing a cobweb map is as follows:

• Firstly plot the curve pn+1 = f(pn) and the line pn+1 = pn in the (pn, pn+1) plane.

• Mark the initial condition p(1) on the x-axis.

• Draw a line vertically upwards till it meets the curve pn+1 = f(pn).

• Now reflect this point in the line pn+1 = pn onto the pn-axis.

• Repeat this procedure with p(1) getting replaced by p(2) and so on.

• As a result the sequence of points p(1), p(2), p(3), ... is obtained on the pn-axis that iscalled the solution of the one-dimensional iterative map.

0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1

1.2

1.4Graph Displaying Solutions of One−Dimensional Iterative Map For α=0.2

n

p n

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.005

0.01

0.015

0.02

0.025

0.03Cobweb Diagram for α = 0.2 Showing a Solution Dropping to Zero

pn

p n+1

Figure 2.2.1: Solution Jumping Quickly To Zero For the One Dimensional Iterative Mapwhen α=1.5 and Corresponding Cobweb Map (Where the axis of the graphs correspond ton=number of iterations, pn= strength of the poloidal magnetic field component at cycle n andpn+1= strength of the poloidal magnetic field component at cycle n+1. The pink curve on thecobweb map is the function from the one-dimensional map and the green line is pn+1 = pn.)

When α is chosen to be a very small value of 0.2, seen in Figure 2.2.1, this is clearly outsidethe α range and pn reached a constant value of zero very quickly after only 2 iterations.

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The corresponding cobweb map for this value of α=0.2 can also be seen in this figure, the y-axisis scaled much smaller for this graph so it can be seen how quickly pn reached this constantzero value from the map.

0 20 40 60 80 100 1201.06

1.07

1.08

1.09

1.1

1.11

1.12


n

p n

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Cobweb Diagram for α = 1.5 Showing a Solution Approaching a Fixed Point

pn

p n+1

Figure 2.2.2: Solution Converging To a Fixed Point For the One Dimensional Iterative Mapwhen α=1.5 and Corresponding Cobweb Map (where n=number of iterations, pn= strength ofthe poloidal magnetic field component at cycle n and pn+1= strength of the poloidal magneticfield component at cycle n+1)

Increasing α to 1.5 a change in the solution is seen, instead of pn abruptly dropping to zeroit converges to a single value, this is shown in Figure 2.2.2. After 13 iterations pn reached thefixed point of 1.0788. As long as α=1.5 then pn will always eventually converge to this valueof 1.0788 and stay there, this value is called a fixed point, or a steady state.

As pn converges to the fixed point of 1.0788 this can also be seen on the cobweb map inthis figure. This fixed point is in the centre of the diagram, where the cobweb appears to becircling round and round with lines extending between the pink function curve and the greeny=x curve, and finishing at the point in the centre, 1.0788.

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0 20 40 60 80 100 1200.9

1

1.1

1.2

1.3

1.4


n

p n

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Cobweb Diagram for α = 1.67 Showing a Period Two Solution

pn

p n+1

Figure 2.2.3: Period Two Solution For the One Dimensional Iterative Map when α=1.67 andCorresponding Cobweb Map (where n=number of iterations, pn= strength of the poloidal mag-netic field component at cycle n and pn+1= strength of the poloidal magnetic field componentat cycle n+1)

More abrupt changes begin to appear as α is increased from this point, pn starts to appearto reach a point where it is oscillating between two values. At α=1.67, seen in Figure 2.2.3, aperiod 2 solution is occurring, pn never settles to a fixed point and instead oscillates betweenthese two points (1.4101 & 0.9208).

The cobweb map for α=1.67 shows the two values which pn is oscillating between, these areshown on the plot as the noth-west and south-east corners on the pink curve from the square.As pn has reached this point where it just oscillates between the two, it follows the same pathover and over again. This is why the cobweb displayed appears as a sqaure.

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0 20 40 60 80 100 1200.8

1

1.2

1.4


n

p n

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Cobweb Diagram for α = 1.73 Showing a Period Four Solution

pn

p n+1

Figure 2.2.4: Period Four Solution For the One Dimensional Iterative Map when α=1.73 andCorresponding Cobweb Map (where n=number of iterations, pn= strength of the poloidal mag-netic field component at cycle n and pn+1= strength of the poloidal magnetic field componentat cycle n+1)

Similarly as α is increased again a period 4 solution is found. A period 4 solution at α=1.73is displayed in Figure 2.2.4, where pn settles to oscillating between four points (1.4475, 0.9291,1.5138 & 0.8240).

The period 4 cobweb map for α=1.73 shows the oscillation between the four points. Simi-larly for the period 2 solution the same path is followed over and over again, this time betweenfour instead of two points, and displays what looks like two squares from the cobweb mapping.

15


0 20 40 60 80 100 1200.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Graph Displaying Solutions of One−Dimensional Iterative Map For α=1.855

n

p n

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Cobweb Diagram for α = 1.855 Showing a Chaotic Solution

pn

p n+1

Figure 2.2.5: Chaotic Solution For the One Dimensional Iterative Map when α=1.855 andCorresponding Cobweb Map (where n=number of iterations, pn= strength of the poloidal mag-netic field component at cycle n and pn+1= strength of the poloidal magnetic field componentat cycle n+1)

This regular final behaviour which pn eventually displays, either as a fixed point solution oran oscillation between n points, is called an ‘attractor’, since the initial conditions will eventu-ally be attracted to it [20].

As α is increased a pattern is established where pn eventually reaches a point where it staysat a single value, oscillates between two, oscillates between 4, and this pattern continues witheight values, sixteen, and so on. However as the intervals between the α values get smallerand smaller this becomes difficult to detect and eventually a chaotic solution is reached. Thechaotic value was found in this report to be around α = 1.855 and this can be seen on Figure2.2.5, it is clear from this that there is no consistent pattern reached at the end like in all otherprevious solutions.

The cobweb map for α = 1.855 clearly shows the chaotic solution with no resemblance of arecurring pattern as the lines appear to have taken independent paths all over the mapping.

16


0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1

1.2

1.4Graph Displaying Solutions of One−Dimensional Iterative Map For α =1.3

n

p n

0 20 40 60 80 100 1200

0.5

1

1.5

2Graph Displaying Solutions of One−Dimensional Iterative Map For α=1.89

n

p n

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Cobweb Diagram For α = 1.3 Showing a Solution Outside the α Range

pn

p n+1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Cobweb Diagram For α = 1.89 Showing a Solution Outside the α Range

pn

p n+1

Figure 2.2.6: Solution For the One Dimensional Iterative Map when α is outside the range,1.3< α <1.89 and the Corresponding Cobweb Maps (where n=number of iterations, pn=strength of the poloidal magnetic field component at cycle n and pn+1= strength of the poloidalmagnetic field component at cycle n+1)

If α is chosen outside the range 1.3< α <1.89, the top graphs in Figure 2.2.6 show that thesolution drops out to zero. This therefore provides the information needed to obtain the rangeof α values.

17


2.3 Bifurcation Diagram Summarising Previous Solu-

tions

In the previous section to find solutions where α was being increased, pn first converged toa fixed point, then settled at a period 2 oscillation, then a period 4 oscillation, and so on untilchaos was reached. In dynamical systems, each of these distinguishable changes which are seenas period doubling are called bifurcations.

The previous solutions can be summarised together in one diagram called a bifurcation di-agram. This succession of bifurcations eventually reaching chaos (seen below in Figure 2.3.1),is called the period doubling route to chaos.

When pn reaches a period 2 oscillation this is seen on the diagram as the first branch, againwhen pn undergoes another period doubling and period 4 oscillations occur, this is seen as thenext branching where the two branches then become four. This bifurcation diagram clearlyshows how difficult it gets to tell when more period doubling occurs with the intervals becomingso small and the route to chaos can be demonstrated well in this graph.

1.3 1.4 1.5 1.6 1.7 1.80

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8Bifurcation Diagram For the One−Dimensional Iterative Map

α

Pn

Figure 2.3.1: Bifurcation Diagram Summarising the Previous Solutions (where pn= strengthof the poloidal magnetic field component at cycle n and α is the increasing values of α)

18


2.3.1 Periodic Windows

In the bifurcation diagram (Figure 2.3.1) there is clearly what looks to be a few white stripsvertically on the plot. From the bifurcation diagram it can be seen that the chaotic regionsuddenly ends when α exceeds a critical value, this value is about 1.89. Once this critical valueis reached the solution just drops to zero and the motion appears to revert back to period-1motion seen as the solid black line at the bottom. As the parameter α is increased there areregions where the solution is not chaotic but is insead periodic. These narrow windows ofperiodicity are known as periodic windows.

There are a few periodic windows, with some being much more visible than others, someof the widest windows can be seen in Figure 2.3.2. The widest window is the period 3 windowwhich is shown in the zoomed in figure. As α is further increased however it is clear that theperiodic window breaks down and eventually disappears, this is due to the action of a cascadeof period-doubling bifurcations. This is the same mechanism as in the original period doubling,the only difference being that the orbits are now of period 3·2n, instead of 2·2n.

1.74 1.76 1.78 1.8 1.82 1.84 1.86 1.88 1.9

0.6

0.8

1

1.2

1.4

1.6

1.8Section of Bifurcation Diagram Showing Periodic Windows

α

Pn

1.82 1.825 1.83 1.835

0.6

0.8

1

1.2

1.4

1.6

Period 3 Periodic Window

α

Pn

Figure 2.3.2: Sections of the bifurcation diagram zoomed in to show the periodic windows.On the figure on left period 6, 5 & 3 windows are most visible. The figure on the right showsa close up of the period 3 window where period doubling occurs again as the window breaksdown.

2.4 Chaotic Solution Analysis

2.4.1 Sensitivity To Initial Conditions

Assuming the series of values p0, p1, p2 and so on to be called the trajectory of p for thisexplanation. At the values of α where chaos occurred (around α=1.855), two trajectoriesstarting from very similar initial conditions, will diverge from each other instead of convergingto a fixed point or oscillation [20]. This property is sometimes called the butterfly effect.

19


0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1

1.2

1.4Sensitivity To Initial Conditions For Solution Dropping to Zero When α=0.2

n

p n

0 20 40 60 80 100 1201.06

1.08

1.1

1.12

1.14Sensitivity To Initial Conditions For Solution Approaching Fixed Point When α=1.5

n

p n

0 20 40 60 80 100 1200.9

1

1.1

1.2

1.3

1.4

1.5Sensitivity To Initial Conditions For Period Two Solution When α=1.67

n

p n

0 20 40 60 80 100 1200.8

1

1.2

1.4

1.6

Sensitivity To Initial Conditions For Period Four Solution When α=1.73

n

p n

0 20 40 60 80 100 1200.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Sensitivity To Initial Conditions For Chaotic Solution When α=1.855

n

p n

Figure 2.4.1: Graphs showing the sensitivity to initial conditions for the different cases ofα (where n=number of iterations, pn= strength of the poloidal magnetic field component atcycle n)

20


For each of the values of α the graph was iterated as done previously, to obtain a tra-jectory, with the initial condition of p(1)=1.1 (this is shown as the black line in Figure 2.4.1)The initial conditions were then changed very slightly by increasing 1.1 to 1.000001, and themap was then iterated again, plotting it on top of the first (this is shown as the pink line inFigure 2.4.1). For the first four cases (α = 0.2, α = 1.5, α = 1.67, α = 1.73) the plots appear asjust a pink line, this is because the divergence occurs so slowly, the change in initial conditionsdoesn’t make much difference to the previous one if any. The second iteration follows the exactsame pattern and it just looks like one line.

However for the chaotic case, α = 1.855 there is a much more noticeable change. The twotrajectories start off very close together and appear as one pink line, as in the other cases, butjust after 20 iterations or so they start to diverge significantly. In the previous α cases bothtrajectories had the same pattern or correlation but in this chaotic case, after the 20 iterationsthere is soon no similarities in the two trajectories and it can be seen in Figure 2.4.1 to be verychaotic. This is described as sensitive dependence on initial conditions.

2.4.2 Lyapunov Exponents

This sensitivity to initial conditions can be quantified by the Lyapunov exponent. The Lya-punov exponent, L, computed using the derivative method is defined by:

L =1

n(ln|f ′u(x1)|+ ln|f ′u(x2)|+ ...+ ln|f ′u(xn)|) ,

where f ′u represents differentiation with respect to x and x0, x1, x2, ..., xn are successive it-erates. The Lyapunov exponent may be computed for a sample of points near the attractor toobtain an average Lyapunov exponent. [21]

The theorem states that:‘If at least one of the average Lyapunov exponents is positive, then the system is chaotic; ifthe average Lyapunov exponent is negative, then the orbit is periodic and when the averageLyapunov exponent is zero, a bifurcation occurs.’

Due to time limitations of this project it was not possible to fully complete the plot for theLyapunov exponent, in Figure 2.4.2 the beginning of a plot for the Lyapunov exponent can beseen. If there was time to fully correct this plot the first bifurcation point should corresond tothe value found on the bifurcaton diagram and touch the x axis, then the blue curve shouldmove into the positive y axis region where chaotic solutions occur. In completing this againthe differentiated function would be required to be corrected and hopefully put right this error.Once corrected it would then be possible to change the number of iterates and have a varietyof larger iterates included like 1000 and 10,000.

21


1.3 1.4 1.5 1.6 1.7 1.8 1.9 2−12

−10

−8

−6

−4

−2

0

α

Lyap

unov

Exp

onen

t

Lyapunov Exponent for 100 Iterates

Figure 2.4.2: Lyapunov exponent graph, corrections required to be made to this.

2.4.3 Feigenbaum Number

The Feigenbaum constant δ is a universal constant for functions approaching chaos viaperiod doubling. It has already been shown that the function used in the analysis of this mapapproaches chaos via period doubling. Another way of proving that this is infact a chaoticsolution that is reached is the analysis of the bifurcation diagram to obtain the Feigenbaumconstant. Feigenbaum discovered that if dk is defined by dk=bk+1 - bk, then:

δ = limk→∞

dkdk+1

= 4.669202....

The first four bifurcations found on the bifurcation diagram were:b1 = 1.5808,b2 = 1.7160,b3 = 1.7415,b4 = 1.7472.

Therefore the Feigenbaum constant was able to be calculated as:

δ = limk→∞

dkdk+1

= limk→∞

bk+1 − bkbk+2 − bk+1

δ1 = limk→∞

b2 − b1b3 − b2

= limk→∞

1.7160− 1.5805

1.7415− 1.7160= 5.31372

δ2 = limk→∞

b3 − b2b4 − b3

= limk→∞

1.7415− 1.7160

1.7472− 1.7415= 4.47368

(2.4.1)

The values calculated from this bifurcation provided Feigenbaum constants of 5.31372 and4.47368, these are not exactly the universal constant 4.669092... but as this value is only alimit of the process as k tends to infinity, it is difficult to get the exact value.

22


2.4.4 Sharkovskii’s Theorem

The theorem is stated as follows:

‘The Sharkovskii Theorem involves the following ordering of the set N of positive integers,which is known as the Sharkovskii ordering:

3 < 5 < 7 < 9 < ... < 2 ·3 < 2 ·5 < 2 ·7 < ... < 22 ·3 < 22 ·5 < 22 ·7 < ... < 23 < 22 < 2 < 1

Now let F be a continuous function from the reals to the reals and suppose p < q in theabove ordering. Then if F has a point of least period p, then F also has a point of least periodq. A special case of this general result, also known as Sharkovskii’s theorem, states that if acontinuous real function has a periodic point with period 3, then there is a periodic point ofperiod n for every integer n.

A converse to Sharkovskii’s theorem says that if p < q in the above ordering, then we canfind a continuous function which has a point of least period q, but does not have any points ofleast period p. For example, there is a continuous function with no points of least period 3 buthaving points of all other least periods. Sharkovskii’s theorem includes the period three theoremas a special case [22].’

As described in section 2.3.1 there is clearly periodic windows within the bifurcation dia-gram. The period 3 periodic window is shown in Figure 2.3.2, and as Sharkovkii’s theoremstates if a continuous real function has a periodic point with period 3, then there is a periodicpoint of every period n for every integer n. So from this theorem it can be said that there isa periodic point for every integer, periodic windows of 6, 5 & 3 are the most visible ones inFigure 2.3.2.

23


2.5 Replacement of The Function Within The Map

The map used throughout this report as seen earlier is:


4

(1 + erf

(pn −B1

ω1

))(1− erf

(pn −B2

ω2

))Since this map with the given function produces a universal single-hump map, it may be

the case that one might wonder if choosing a different function with qualitatively similar prop-erties could represent a better or similar approximation for this map. In this section it will beexplained how changing the function f(pn) to some other qualitatively similar function wouldmake changes to the map.

Plotting f(pn) appears as below in Figure 2.5.1:

−1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

1.2

1.4

Original Functinon f (pn )

pn

f (p n )

Figure 2.5.1: Plot of the original function f(pn)

The function which could be used as a replacement to the original function by producing asuitable curve was considered. Several different solutions and combinations were experimentedwith, one of them being f(pn) = exp(t − c)2. However, when plotting this function with theoriginal it wasn’t a suitable enough curve. It was an exponential increase and decrease, howeveras seen in Figure 2.5.2 the new one chosen is much more symmetrical than the original and itis not 0 when pn = 0.

24


−1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

1.2

1.4

Original Functinon f (pn ) and Unsuccessful Replacement Function

pn

f (p n )

Figure 2.5.2: Plot of the original function f(pn) (black curve) with the unsuccessful replace-ment function f(pn) = exp(t− c)2 (green curve).

It was then considered that tanh and erf have qualitatively similar properties, so the er-ror function in the original function could be replaced by tanh to get a qualitatively similarreplacement function:

f(pn) =α

4

(1 + tanh

(pn −B1

ω1

))(1− tanh

(pn −B2

ω2

))This was a suitable choice and in Figure 2.5.3 the differences in these two functions are shown.They follow much more of the same pattern and this new function meets the necessary require-ments, appearing similar enough to the first.

−1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

1.2

1.4

Original Functinon f (pn ) and Replacement Function With Qualitatively Similar Properties

pn

f (p n )

Figure 2.5.3: Plot of the original function f(pn) (black curve) with the successful replacement

function f(pn) =α

4

(1 + tanh

(pn −B1

ω1

))(1− tanh

(pn −B2

ω2

))(pink curve).

25


2.5.1 Finding Solutions of Iterative Map With The Function Re-placed

To describe the maps change in behaviour due to the change of function, the next stepwas to repeat the process of finding solutions to the original map again, with the new functionsubstituted into the map and the differences observed.


4

(1 + tanh

(pn −B1

ω1

))(1− tanh

(pn −B2

ω2

))

0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1

Graph Displaying Solutions of Iterative Map With Replaced f(pn) when α=0.2

n

p n

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Cobweb Diagram When α = 0.2 For Iterative Map With Replaced Function f(pn)

pn

p n+1

Figure 2.5.4: Solution dropping to zero for the One Dimensional Iterative Map with thefunction f(pn) replaced, when α=0.2 and Corresponding Cobweb Map (where n=number ofiterations, pn= strength of the poloidal magnetic field component at cycle n and pn+1= strengthof the poloidal magnetic field component at cycle n+1). The pink curve on the cobweb map isthe new function being used for the one-dimensional map and the green line is pn+1 = pn.)

Choosing α again to be a very small value of 0.2 the solution drops out to zero very quickly,as it did in the original map. Therefore the altered map results in the same behaviour beingdisplayed as before.

26


0 20 40 60 80 100 1201

1.1

1.2

1.3

1.4


n

p n

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2


pn

p n+1

Figure 2.5.5: Solution converging to a fixed point for the One Dimensional Iterative Map withthe function f(pn) replaced, when α=1.6 and Corresponding Cobweb Map (where n=numberof iterations, pn= strength of the poloidal magnetic field component at cycle n and pn+1=strength of the poloidal magnetic field component at cycle n+1)

The behaviour of the map follows the same form as α is increased, a solution which convergesto a fixed point can again be seen using this map. The value of alpha at which this occuredwas very similar to before and the solution reaching a fixed point at α=1.6 is shown in Figure2.5.5 where a fixed point of 1.1758 is reached.

27


0 20 40 60 80 100 1201

1.1

1.2

1.3

1.4

1.5


n

p n

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2


pn

p n+1

Figure 2.5.6: Period 2 solution for the One Dimensional Iterative Map with the functionf(pn) replaced, when α=1.734 and Corresponding Cobweb Map (where n=number of iterations,pn= strength of the poloidal magnetic field component at cycle n and pn+1= strength of thepoloidal magnetic field component at cycle n+1)

Again the map’s behaviour seems to be duplicating the previous one as α is increased, withsolutions of period doubling clearly being shown. The period two solution is shown in Figure2.5.6. The value of α at which this solution took place was again very similar to before, onlyhaving to increase slightly to 1.734.

28


0 20 40 60 80 100 1200.8

1

1.2

1.4

1.6

1.8


n

p n

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2


pn

p n+1

Figure 2.5.7: Period 4 solution for the One Dimensional Iterative Map with the functionf(pn) replaced, when α=1.895 and Corresponding Cobweb Map (where n=number of iterations,pn= strength of the poloidal magnetic field component at cycle n and pn+1= strength of thepoloidal magnetic field component at cycle n+1)

The period four solution was also reached with the new mapping, at the again slightly largervalue of α=1.895. In Figure 2.5.7 it can be seen that the maps solution once again follows thesame characteristic pattern as before displaying a solution which reaches a point of oscillatingbetween four different values.

29


0 20 40 60 80 100 120

0.8

1

1.2

1.4

1.6

1.8

2


n

p n

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2


pn

p n+1

Figure 2.5.8: Chaotic solution for the One Dimensional Iterative Map with the function f(pn)replaced, when α=1.997 and Corresponding Cobweb Map (where n=number of iterations, pn=strength of the poloidal magnetic field component at cycle n and pn+1= strength of the poloidalmagnetic field component at cycle n+1)

Chaotic solutions of the new map are able to be found also and are displayed in Figure2.5.8 at α=1.997. This map with the new function chosen therefore replicates the solutionsfrom before displaying the solutions of different periods and reaching a chaotic solution.

30


0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1


n

p n

0 20 40 60 80 100 1200

0.5

1

1.5

2

2.5


n

p n

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2


pn

p n+1

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3


pn

p n+1

Figure 2.5.9: Solution For the Altered One Dimensional Iterative Map when α is outside therange, 1.348< α <2.1 and the Corresponding Cobweb Maps (where n=number of iterations,pn= strength of the poloidal magnetic field component at cycle n and pn+1= strength of thepoloidal magnetic field component at cycle n+1)

The range of α values for the new mapping is again very similar to before, (in the originalmap the range was 1.3< α <1.89) and comparing this to the map including the new functionthe range of alpha values becomes 1.348< α <2.1.

31


2.5.2 Bifurcation Diagram of The Iterative Map With The FunctionReplaced

1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Bifurcation Diagram For the Iterative Map With Replaced Function

α

Pn

Figure 2.5.10: Bifurcation Diagram Summarising the Previous Solutions for the New MapIncluding the Changed Function (where pn= strength of the poloidal magnetic field componentat cycle n and α is the increasing values of α)

This bifurcation diagram in Figure 2.5.10 displays again the solutions for the changed mapclearly indicating that the range of α values now is 1.348< α <2.1.

Therefore, having found solutions to the map again with the new qualitatively similarfunction, the behaviour displayed was exactly the same as before and hence it can be concludedthat if f(pn) is replaced by a differenct function the map behaves in exactly the same manner,a period doubling route to chaos.

32


2.6 Introducing Stochastic Forcing To The Map

The Even-Odd Effect, or the Gnevyshev-Ohl Rule is a rule proposed by Gnevyshev andOhl in 1948. They discovered that if the arrangement of solar cycles was such that they werearranged in pairs with a cycle of even number and a following odd numbered cycle then thesum of the sunspot numbers in the odd cycle is higher than in the even cycle [23].

The dynamo process was reduced to the one-dimensional map and it is evident that the ex-planation then for the odd-even effect is predicted on some very small changes to the parameter1.3< α <1.89. This clearly then provides enough knowledge to believe that the sunspot cycle’sodd-even effect is not particularly sturdy. The parameter α may vary stochastically over asignificant range of numerical values. This could result in the dynamo being taken outside therange of α where period 2 cycle behaviour holds, to test this theory a stochastic forced mapcan be used. This can be done by adding noise onto the map where a small random number isadded at each step of results in the equation:

pn+1 = αf(pn)pn + εn, f(pn) =α

4

(1 + erf

(pn −B1

ω1

))(1− erf

(pn −B2

ω2

))

Adding the noise, εn, to the graph as εn ∈ [0, 0.09] the graph for the map with varying αbecomes the plot seen in Figure 2.6.1. It is clearly shown from this graph that disturbancesoccur. At values where previously a solution of a steady state or periodic solution occured, nowresults in a graph with disruptions not previously there. There is a persistence of the odd-eveneffect in certain regions of the graph including between iterations of approximately 24-28 and50-55.

0 10 20 30 40 50 60 70 80 90 1000.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6Stochastically Forced Map

n

p n

Figure 2.6.1: Graph showing the stochastically forced map. There is now disruptions oc-curing within the map. (where n=number of iterations, pn= strength of the poloidal magneticfield component at cycle n)

33


Intermittency is an effect of a stochastically forced map which can be seen by looking at alonger sequence of amplitude iterates. In Figure 2.6.2 the plot is displayed over a much longersequence of iterates, between 1 and 2000. From this graph periods of intermittency can beseen (as shown in Charbonneau 2000 paper) with the most noticable periods being around650, 1050 and 1750 iterates.

0 200 400 600 800 1000 1200 1400 1600 1800 20000.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6Stochastically Forced Map

n

p n

Figure 2.6.2: Graph showing the stochastically forced map over a longer sequence of iteratesshowing intermittency. (where n=number of iterations, pn= strength of the poloidal magneticfield component at cycle n)

34

3

Mathematics of Dynamo Theory

Primordial Magnetic fields are prevelent through the universe forming at the very beginningof time. The formation of a star is enveloped in the contraction of gas clouds which form solarmasses like our Sun. This gas cloud enables the joining of the magnetic field with a solar massand creates light throughout our solar system. There have been many theories as to where theprimordial field is located, namely, it was thought that it was located within radiative zone ofthe Sun. This theory then advocated the idea that the surface magnetic field was created dueto the location of the primodial magnetic field in the inner core of the sun. It was then discov-ered that this may not be true as the diffusion time for the decay of a global solar magneticfield is about 1010 years.

How the Sun’s magnetic field operates is not fully understood, however observations haveallowed solar scientists to rule out possible concepts. The theory of an MHD operator hasbeen omitted due to the fact that there does not seem to be any oscillation in velocity. Theunderstanding of how the fields are generated is by a mechanism for the frequent regenerationof the magnetic fields from the motion of the charged particles in the plasma, which makes upthe convection zone. This is known as the dynamo mechanism which was briefly explained inthe earlier sections. Dynamo theory is a theory that proposes the mechanism by which theSun-like-stars, generate a magnetic field. This may simply be by the conversion of mechanicalto electrical energy in the mechanical dynamo but it is not as simple in solar dynamo. Severaldynamo and anti-dynamo theorems exist, including the Babcock-Leighton model which is aflux-transport dynamo and was discussed earlier. In this section some more of the theoremsare further explained.

3.1 Anti-Dynamo Theorems

3.1.1 Cowling’s Anti-dynamo Theorem:

Cowlings theorem states that:

‘A steady axisymmetric magnetic field cannot be maintained by dynamo action [15].’

Cowlings theorem can thereby be mathematically described as follows:

In cylindrical polar coordinates (R,φ,z), consider a steady, axisymmetric field ~B(R,z) thatdepends on R and z only. This magnetic field can be written as the sum of a toroidal componentBφ and a poloidal component Bp:

~B = Bφ~iφ + ~Bp

35


Because of the axisymmetry, the magnetic configuration in all meridional (along the merid-ian/in the north-south direction) planes is the same and the projections of field lines onto suchplanes must be closed curves. In each meridional plane therefore at least one O-type neutralpoint must exist, where Bp vanishes so the field is purely azimuthal and the azimuthal currentis non-zero.

Ohm’s Law from section 1.2, ~E = −~v× ~B+~j

σcan be integrated around the closed azimuthal

field line, C, through the neutral points, resulting in:∫C

~j

σ· d~s =

∫C

~E · d~s+

∫C

~v × ~B · d~s

Following this, Stoke’s Theorem can be used for the transformation of the first term on theright hand side,

Stoke’s Theorem: ∫δS

~F · d~s =

∫S

(∇× ~F ) · d~a

=⇒∫C

jφds

σ=

∫S

∇× ~E · d~S +

∫C

~v × ~B · d~s

As Faraday’s Law is,∂ ~B

∂t= −∇× ~E and it is assumed that the magnetic field is steady, it

is possible to state that this ∇× ~E term vanishes.

Along this azimuthal field line C, ~B is parallel to the path element d~s, so the second termon the right hand side also vanishes and the above equation reduces down to:

∫C

jφds = 0

However because jφ does not vanish along the field line C, the equation cannot equal 0,therefore it cannot be satisfied. It can be concluded, given the evidence presented above thata steady magnetic field cannot be axisymmetric. [15]

3.2 Dynamo Theorems

3.2.1 Parker’s 1955 Model:

Parker’s model is a turbulent dynamo model discovered in 1955 in which he proposes thatturbulent motions inside the convection zone of the Sun may have the ability to sustain theSun’s poloidal field. In previous sections it was explained that the stretching and twisting ofmagnetic field lines may be due to differential rotation. This differential rotation stretches outthe poloidal flux, creating a strong toroidal component.

36


Figure 3.2.1: Diagram showing the magnetic dynamo effects. ω-effect shown in images (a)-(c), where the pre-existing magnetic field line (small arrow) is twisted and stretched out bydifferential rotation of the plasma (thick grey arrows) into the stong toroidal component. Inimages (e)-(g) the α-effect is shown, the radial motions cause twisting of the newly generatedtoroidal field component under the Coriolis force, producing a new poloidal field component[24].

Parker’s discovery determined that the particular rotation of the Sun results in parts ofthe plasma from the convection zone emerging from the Sun acting like cyclones, rising loopsof magnetic flux. The Coriolis effect causes these loops to twist. The action of the loopsdue to the Coriolis effect twists the toroidal magnetic field producing a component of mag-netic field in the poloidal direction. This movement can be seen in Figure 3.2.1. In the twohemispheres the toroidal field has opposite signs, the helical motions however are also oppo-site in the sense that they are clock-wise in the northern hemisphere and anti-clockwise inthe southern hemisphere. Therefore Parker stated that the resulting poloidal loops having thesame sense in both hemispheres, could lead to the production of a large-scale poloidal field [15].

Considering the axisymmetric field ~B = Bφ~iφ + ~Bp from the previous section if it is acted

on by a flow ~v = vφ~iφ + ~vp. The φ-component of the induction equation becomes:

δBφ

δt+R(~vp · ∇)

(Bφ

R

)= R ~Bp · ∇

(vφR

)+ η

(∇2 − 1

R2

)Bφ

The first term in this equation on the left is the rate of change of the toroidal field and thesecond represents its advection with a flow. The first term on the right-hand side is showinghow differential rotation acting on a poloidal field can enhance the toroidal flux.

Integrating the poloidal component of the Induction equation gives:

δApδt

+~vpR· ∇(RAp) = η

(∇2 − 1

R2

)Ap

37


The integration of the poloidal component does not allow the poloidal component of mag-netic flux to be generated from the toroidal component, because this equation implies that ~Bp

decays away in time.

An electric field Eφ = αBφ can be added to the equation, as both the rate of the poloidalflux and toroidal flux are proportional to each other. Adding this electric field results in theequation becoming:

δApδt

+~vpR· ∇(RAp) = αBφ + η

(∇2 − 1

R2

)Ap

Given the evidence presented above it can be assumed that dynamo action is a possibilityunder the Parker dynamo model [15].

3.2.2 Tacholine Dynamos:

Another type of dynamo is a tachocline dynamo. These dynamos are not fully understood, likemany dynamo theorems, however there has been an incredible amount of discussion on howthese are believed to operate.

Figure 3.2.2: Interior structure of the Sun, showing the tachocline in between the radiativeand convection layers [25].

The tachocline, also known as the interface layer is present as a thin layer between theradiative zone and the convective zone, as can be seen on Figure 3.2.2.

It is believed that the magnetic dynamo may have been generated in this layer, as thechanging speeds of fluid flow across the tachocline can stretch the magnetic field lines and re-sult in the creation of stronger ones. There are different tachocline dynamo models, includingthe ‘overshoot dynamo’, ‘interface dynamo’ and the ‘solar tachocline’.

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The Interface Dynamo:

The interface dynamo theory states that if the toroidal field at the bottom of the convec-tion zone is 105 G , the turbulent flow present in this zone would not possess enough strengthto twist the magnetic field. The α-effect related to convective turbulence would then end upgetting switched off due to the flow not having enough strength.

The tachocline was discovered in earlier years and with the knowledge of this and thensome more information coming to light about the turbulent flow, this prompted the ‘interface’dynamo to be proposed. Within the tachocline the mean toroidal field is located, which isgenerated by differential rotation. The other field, known as the poloidal is found in the lowerregion of the convection zone. This field is generated by an α effect rather than differentialrotation like previously, due to the convective motions in the convection zone it ends up beingpumped down into the tachocline where the toroidal field is stored. The stability of the fieldsresults in them exhibiting different field strengths, these strengths can vary between 104 G and105 G in both poloidal and toroidal components.

When Parker analysed this ‘interface dynamo’ in 1993 he observed that the dynamo seemsto exhibit certain behaviour acting like a surface wave tied to the lower surface of the convec-tion zone. Parker displayed these findings by extending a simple planar dynamo model into atwo-layer model with an α-effect in the first region, z > 0 and a uniform shear in z < 0. Hefurther explained his observations mathematically giving a full understanding of this proposi-tion.

After his numerical explanation Parker came to the conclusion that the benefit of thisinterface model is that it includes methods explaining how the shear creates a large toroidalfield, and this field is then capable of being correctly stored due to its stable nature. On theother hand, Parker came across some disadvantages of the model, he knew that the poloidalflux ends up being pumped down to the tachocline but the exact details of how that flux istransported back and forth is still a bit of a mystery [15].

Overshoot Dynamo:

The overshoot dynamo theory states that the dynamo instead of taking place within theconvection zone, actually operates in a thin region called the overshoot region. The overshootregion is located just below the bottom of the convection zone, seen in Figure 3.2.2.

This dynamo theory works on the assumption that the overshoot region is a more favourableone. The reason behind this being that the magnetic field is capable of being held in this regiondue to magnetic buoyancy, restricting the time it has to be amplified by a magnetic dynamo,as it is in the convection zone. This region is much more stable than the convection zone andthe problem of the magnetic field being removed to quickly would not occur here as magneticbuoyancy would be suppressed. Convective plumes from the preferable but more unstablelayers would overshoot and then end up infiltrating into this overshoot layer, supplying theturbulent motions that could be what powers the magnetic dynamo [15].

39

4

Conclusion

4.1 Conclusion

The Sun is the most vital source of energy for life on Earth, and is the largest star in thecentre of our solar system. Investigations of the Sun’s invisible and changing magnetic fieldby using models therefore provides fundamental information for life on Earth. In this reportsunspots are used as the main indicators for the Sun’s changing magnetic field, they are visiblein regions of the Sun where there is a strong magnetic field. This changing magnetic field pro-vides enough information to understand the way in which a dynamo operates. The estimationof the decay time of a fossil field lead to the assumption that the magnetic dynamo must exist,as it is needed to explain the creation and decaying of the short lived sunspots.

The dynamo has several physical properties which are still to this day not fully understoodbut it is shown in this report that it is possible to study the dynamo’s behaviour by the use ofa simple mathematical model in the form of a one-dimensional map, given in Charbonneau’s2005 paper. This map was used throughout the report and has been shown to display severaldifferent behaviours, including a steady state solution converging to a fixed point, periodicsolutions and reaching a chaotic solution. All of these behaviours are found in stellar dynamoswhich are explained in Baliunas et al, 1995. This simple model therefore clearly demonstratesthe dynamo’s several different types of behaviour sufficiently without having to make use ofany other more complex models.

The model was then extended by substituting the function, f(pn), for a different term withqualitatively similar properties. Solutions of identical behaviour were achieved in using theextended version of the map, agreeing with the predictions from the known properties of theone-dimensional map. Due to this model being simple it allowed for much more analysis thana more intricate model would have resulting in being able to get a lot more from the model.One of the findings in the analysis of the map was that it was shown that the model only workswith a certain range of α.

The solutions which reached chaos in the map were further analysed using different math-ematical techniques. One of the techniques was the maps sensitivity to initial conditions, inall of the solutions were the map displayed a steady state or periodic behaviour it made nochange in changing the initial conditions. For the chaotic behaviour it did however, differentinitial conditions, no matter how small they were, resulted in the two trajectories divergingsignificantly. This can be quantified by the Lyapunov exponent.

In conclusion, section 3 then moved onto discussing several different known dynamo and

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anti-dynamo theorems. The results of these theorems were examined and the difference be-tween a dynamo and anti-dynamo theorem was shown. In the dynamo theorems they presentedthe evidence that toroidal flux can be generated by differential rotation from the initial poloidalfield, flux emerges as sunspot groups and a new poloidal dipolar field of opposite polarity isproduced. The difference being that in an anti-dynamo theorem an axisymmetric dynamo isnot possible, a steady axisymmetric magnetic field cannot be maintained by dynamo action.

Finally, having thoroughly analysed the model provided in the Charbonneau paper and theseveral types of behaviour the dynamo exhibits, the outcome of this report has been successfulin producing a suitable hypothesis for the Sun’s dynamo.

41

Bibliography

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[2] Vocativ. Youtube: Watch the sun’s mesmerizing magnetic fields [online].available: https://www.youtube.com/watch?v=xodxtxo4c5i, 2016.

[3] NASA Goddard. Youtube: Understanding the magnetic sun [online].available: https://www.youtube.com/watch?v=2g1eppppiom, 2016.

[4] Brian Dunbar. Nasa: Understanding the magnetic sun [online]. available:http://www.nasa.gov/content/goddard/nasa-funded-sounding-rocket-to-catch-aurora-in-the-act, 2016.

[5] Science Channel. Youtube: What are sunspots? [online].available: https://www.youtube.com/watch?v=zc2dfds8g0q, 2015.

[6] Dept. Physics & Astronomy University of Tennessee. The magnetic field of the sun [online].available: http://csep10.phys.utk.edu/astr162/lect/sun/magnetic.html, 2016.

[7] John A Eddy. The maunder minimum. p.1189, 1976.

[8] SL Baliunas, RA Donahue, WH Soon, JH Horne, J Frazer, L Woodard-Eklund, M Brad-ford, LM Rao, OC Wilson, Q Zhang, et al. Chromospheric variations in main-sequencestars. The Astrophysical Journal, 438:269–287, 1995.

[9] Maria Massi. Stellar magnetic activity [online].available: http://www3.mpifr-bonn.mpg.de/staff/mmassi/c6-stellaractivity.pdf, 2016.

[10] Alan Hood. The induction equation [online]. available:http://www-solar.mcs.st-andrews.ac.uk/ alan/sun course/chapter2/node8.html, 2010.

[11] Arnold Hanslmeier. The Sun and Space Weather. p.99. Springer, 2010.

[12] Jennifer Bergman. Size of sunspots [online].available: http://www.windows2universe.org/sun/atmosphere/sunspot size.html, 2005.

[13] HW Babcock. The topology of the sun’s magnetic field and the 22-year cycle. TheAstrophysical Journal, 133:572–587, 1961.

[14] Paul Charbonneau, Cedric St-Jean, and Pia Zacharias. Fluctuations in babcock-leightondynamos. i. period doubling and transition to chaos. The Astrophysical Journal, 619:613–622, 2005.

[15] Eric Priest. Magnetohydrodynamics of the Sun: Chapter 8 Dynamo Theory. CambridgeUniversity Press, 2014.

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[16] Paul Charbonneau, Cedric St-Jean, and Pia Zacharias. Fluctuations in babcock-leightondynamos. i. period doubling and transition to chaos. The Astrophysical Journal, 619:613,2005.

[17] Paul Charbonneau. Multiperiodicity, chaos, and intermittency in a reduced model of thesolar cycle. Solar Physics, 199:385–404, 2001.

[18] Prof J R Culham. Error and complementary error functions, engineering courses [online].available: http://www.mhtlab.uwaterloo.ca/courses/me755/web chap2.pdf, 2004.

[19] Eric W Weisstein. “ erf” from mathworld [online].available: http://mathworld.wolfram.com/erf.html, 2016.

[20] Melanie Mitchell. Complexity: A guided tour. p.30. Oxford University Press, 2009.

[21] Stephen Lynch. Dynamical systems with applications using MATLAB: Chapter 3 - Non-linear Discrete Dynamical Systems. Springer, 2004.

[22] Eric W Weisstein. “ sharkovsky’s theorem” from mathworld [online].available: http://mathworld.wolfram.com/sharkovskystheorem.html, 2016.

[23] David H Hathaway. Gnevyshev–ohl rule (even–odd effect) [online]. available:http://solarphysics.livingreviews.org/open?pubno=lrsp-2010-1&page=articlesu26.html, 2010.

[24] Joanna Dorothy Haigh, Michael Lockwood, and Mark S Giampapa. The Sun, SolarAnalogs and the Climate: Saas-Fee Advanced Course 34, 2004. Swiss Society for As-trophysics and Astronomy, volume 34. p.122. Springer Science & Business Media, 2005.

[25] P Gosselin. Earth changes, sott [online]. available:https://www.sott.net/article/271874-solar-cycle-24-remains-the-weakest-in-170-years-svaalgard-none-of-us-alive-have-ever-seen-such-a-weak-cycle, 2014.

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Appendix A

A.1 MATLAB code used for finding solutions to the

one-dimensional iterative map (section 2.2)

1 % Wanting to compute p(n+1)=a ∗ ( (1/4) ∗(1+ e r f ( ( p(n) −0.6) /0 . 2 ) )∗(1− e r f ( ( p(n) −1.0)/0 . 8 ) ) ) ∗p(n)

2

3 % Denote alpha as a4

5 a=1.5 ;6 p=ze ro s (100 ,1 ) ;7 p (1) =1.1 ;8 f o r n=1:100; % s t a r t the loop , running from n=1, to n=1009 p(n+1)=a ∗ ( ( a /4)∗(1+ e r f ( ( p(n) −0.6) /0 . 2 ) )∗(1− e r f ( ( p(n)−1) /0 . 8 ) ) ) ∗p(n) ;

10 end11

12 p % computing p n13

14 p lo t (p , ’−ko ’ , ’ LineWidth ’ ,1 , ’ MarkerEdgeColor ’ , ’ k ’ , ’ MarkerFaceColor ’ , ’ k ’ , ’MarkerSize ’ , 2 ) % window w i l l pop up to show the p l o t o f the sequence

15 t i t l e ( ’Graph Disp lay ing So lu t i on s o f One−Dimensional I t e r a t i v e Map For \ alpha=1.5 ’ , ’ f o n t s i z e ’ , 14)

16 x l ab e l ( ’n ’ , ’ f o n t s i z e ’ , 14) ;17 y l ab e l ( ’ p n ’ , ’ f o n t s i z e ’ , 14) ;

A.2 MATLAB code used for generating cobweb maps

for the one-dimensional iterative map (section 2.2)

1 % wr i t i ng out i t e r a t i v e map and de f i n i n g alpha as a :2 % p(n+1)=a ∗ ( ( a /4)∗(1+ e r f ( ( p(n) −0.6) /0 . 2 ) )∗(1− e r f ( ( p(n)−1) /0 . 8 ) ) ) ∗p(n) ;3

4 % se t t i n g alpha to be the value we ca l c u l a t ed f o r each graph o f s o l u t i o n s5 % prev i ou s l y :6

7 f s i z e =16;8 nmax=100;9 halfm=nmax/2 ;

10 p=ze ro s (1 ,nmax) ; p1=ze ro s (1 ,nmax) ; p2=ze ro s (1 ,nmax) ;11 p (1) =0.9 ;12 a=1.67;13 ax i s ( [ 0 2 0 2 ] )14 f o r n=1:nmax15 p(n+1)=a ∗ ( ( a / 4 . ) ∗(1+ e r f ( ( p(n) −0.6) /0 . 2 ) )∗(1− e r f ( ( p(n) −1.) /0 . 8 ) ) ) ∗p(n) ;16 end17

44


18 f o r n=1: halfm19 p1 (2∗n−1)=p(n) ;20 p1 (2∗n)=p(n) ;21 end22

23 p2 (1 ) =0;p2 (2 )=p (2) ;24 f o r n=2: halfm25 p2 (2∗n−1)=p(n) ;26 p2 (2∗n)=p(n+1) ;27 end28 hold on29 p lo t (p1 , p2 , ’ k ’ , ’ LineWidth ’ , 1 . 2 ) ;30

31 k=l i n s p a c e (0 ,2 , 100) ;32 p lo t (k , a ∗( a /4)∗(1+ e r f ( ( k−0.6) /0 . 2 ) ) .∗(1− e r f ( ( k−1) /0 . 8 ) ) .∗ k , ’m’ , ’ LineWidth ’ , 1 . 2 )

;33

34 x=[0 2 ] ; y=[0 2 ] ;35 p lo t (x , y , ’ g ’ , ’ LineWidth ’ , 1 . 2 ) ;36 hold o f f37 t i t l e ( ’Cobweb Diagram For \ alpha = 1.67 Showing a Period Two So lu t i on ’ , ’ f o n t s i z e

’ , 14)38

39 x l ab e l ( ’ p n ’ , ’ Fonts i z e ’ ,16)40 y l ab e l ( ’ p n+1 ’ , ’ Fonts i z e ’ ,16)

A.3 MATLAB code used to produce the bifurcation dia-

gram for the one-dimensional iterative map (section

2.3)

1 Npre = 200 ; Nplot = 100 ;2 p = ze ro s ( Nplot , 1 ) ;3

4 f o r a = 0 . 2 : 0 . 0 0 0 5 : 2 . 0 ,5 p (1) = 0 . 8 ;6 f o r n = 1 : Npre ,7 p (1) = a ∗ ( ( a /4)∗(1+ e r f ( ( p (1 ) −0.6) /0 . 2 ) )∗(1− e r f ( ( p (1 )−1) /0 . 8 ) ) ) ∗p (1) ;8 end ,9 f o r n = 1 : Nplot−1,

10 p(n+1)= a ∗ ( ( a /4)∗(1+ e r f ( ( p(n) −0.6) /0 . 2 ) )∗(1− e r f ( ( p(n)−1) /0 . 8 ) ) ) ∗p(n) ;11 end ,12 p lo t ( a∗ ones ( Nplot , 1 ) , p , ’ k . ’ , ’ markers i ze ’ , 4 . 5 ) ;13 hold on ;14 end15

16 t i t l e ( ’ B i f u r c a t i on Diagram For the One−Dimensional I t e r a t i v e Map ’ , ’ f o n t s i z e ’ , 14);

17 x l ab e l ( ’ \ alpha ’ , ’ Fonts i z e ’ ,16) ; y l ab e l ( ’ P n ’ , ’ Fonts i ze ’ ,16) ;18 s e t ( gca , ’ xl im ’ , [ 1 . 3 1 . 9 ] ) ;19

20 hold o f f ;

45


A.4 MATLAB code used to display both the original

function f (pn) and the new function which it was

replaced by (section 2.5)

1 f s i z e =16;2 nmax=100;3

4 a=2;5 t=l i n s p a c e (−1 ,4 ,100) ;6 hold on7 p lo t ( t , ( a /4)∗(1+ e r f ( ( t−0.6) /0 . 2 ) ) .∗(1− e r f ( ( t−1) /0 . 8 ) ) , ’ k ’ , ’ LineWidth ’ , 1 . 5 ) ;8 p lo t ( t , ( a /4)∗(1+tanh ( ( t−0.6) /0 . 2 ) ) .∗(1− tanh ( ( t−1) /0 . 8 ) ) , ’m’ , ’ LineWidth ’ , 1 . 5 ) ;9 hold o f f

10

11 t i t l e ( ’ Or i g i na l Functinon \ i t f ( p n ) ’ , ’ f o n t s i z e ’ , 13)12 x l ab e l ( ’ p n ’ , ’ Fonts i z e ’ ,16)13 y l ab e l ( ’ f ( p n ) ’ , ’ Fonts i z e ’ , 16)

A.5 MATLAB code used for finding solutions to the

one-dimensional iterative map with function replaced

(section 2.5.1)

1 % Denoting alpha as a2

3 a=1.6 ;4 p=ze ro s (100 ,1 ) ;5 p (1) =1;6

7 f o r n=1:100; % s t a r t the loop , running from n=1, to n=1008 p(n+1)=a ∗ ( ( a /4)∗(1+tanh ( ( p(n) −0.6) /0 . 2 ) )∗(1− tanh ( ( p(n)−1) /0 . 8 ) ) ) ∗p(n) ;9 end

10


13 p lo t (p , ’−ko ’ , ’ LineWidth ’ ,1 , ’ MarkerEdgeColor ’ , ’ k ’ , ’ MarkerFaceColor ’ , ’ k ’ , ’MarkerSize ’ , 2 )

14

15 t i t l e ( ’Graph Disp lay ing So lu t i on s o f I t e r a t i v e Map With Replaced \ i t f ( p n ) \rmwhen \ alpha=1.6 ’ , ’ f o n t s i z e ’ , 14)

16 x l ab e l ( ’n ’ , ’ f o n t s i z e ’ , 14) ;17 y l ab e l ( ’ p n ’ , ’ f o n t s i z e ’ , 14) ;

A.6 MATLAB code used for generating cobweb maps

for the one-dimensional iterative map with function

replaced (section 2.5.1)

1 f s i z e =16;2 nmax=100;3 halfm=nmax/2 ;4 p=ze ro s (1 ,nmax) ; p1=ze ro s (1 ,nmax) ; p2=ze ro s (1 ,nmax) ;5 p (1) =1;6 a=0.2 ;7 ax i s ( [ 0 2 0 0 . 0 5 ] )

46


8 f o r n=1:nmax9 p(n+1)=a ∗ ( ( a / 4 . ) ∗(1+tanh ( ( p(n) −0.6) /0 . 2 ) )∗(1− tanh ( ( p(n) −1.) /0 . 8 ) ) ) ∗p(n) ;

10 end11

12 f o r n=1: halfm13 p1 (2∗n−1)=p(n) ;14 p1 (2∗n)=p(n) ;15 end16

17 p2 (1 ) =0;p2 (2 )=p (2) ;18 f o r n=2: halfm19 p2 (2∗n−1)=p(n) ;20 p2 (2∗n)=p(n+1) ;21 end22 hold on23 p lo t (p1 , p2 , ’ k ’ , ’ LineWidth ’ , 1 . 2 ) ;24

25 k=l i n s p a c e (0 ,3 , 100) ;26 p lo t (k , a ∗( a /4)∗(1+tanh ( ( k−0.6) /0 . 2 ) ) .∗(1− tanh ( ( k−1) /0 . 8 ) ) .∗ k , ’m’ , ’ LineWidth ’

, 1 . 2 ) ;27

28 x=[0 3 ] ; y=[0 3 ] ;29 p lo t (x , y , ’ g ’ , ’ LineWidth ’ , 1 . 2 ) ;30 hold o f f31 t i t l e ( ’Cobweb Diagram When \ alpha = 0 .2 For I t e r a t i v e Map With Replaced Function

\ i t f ( p n ) ’ , ’ f o n t s i z e ’ , 14)32

33 x l ab e l ( ’ p n ’ , ’ Fonts i z e ’ ,16)34 y l ab e l ( ’ p n+1 ’ , ’ Fonts i z e ’ ,16)

A.7 MATLAB code used to produce the bifurcation di-

agram for the one-dimensional iterative map with

function replaced (section 2.5.2)

1 Npre = 200 ; Nplot = 100 ;2 p = ze ro s ( Nplot , 1 ) ;3 f o r a = 0 . 2 : 0 . 0 0 0 5 : 2 . 0 ,4 p (1) = 0 . 8 ;5 f o r n = 1 : Npre ,6 p (1) = a ∗ ( ( a /4)∗(1+tanh ( ( p (1 ) −0.6) /0 . 2 ) )∗(1− tanh ( ( p (1 )−1) /0 . 8 ) ) ) ∗p (1) ;7 end ,8 f o r n = 1 : Nplot−1,9 p(n+1)= a ∗ ( ( a /4)∗(1+tanh ( ( p(n) −0.6) /0 . 2 ) )∗(1− tanh ( ( p(n)−1) /0 . 8 ) ) ) ∗p(n) ;

10 end ,11 p lo t ( a∗ ones ( Nplot , 1 ) , p , ’ k . ’ , ’ markers i ze ’ , 4 . 5 ) ;12 hold on ;13 end ,14 t i t l e ( ’ B i f u r c a t i on Diagram For the I t e r a t i v e Map With Replaced Function ’ , ’

f o n t s i z e ’ , 14) ;15 x l ab e l ( ’ \ alpha ’ , ’ Fonts i z e ’ ,16) ; y l ab e l ( ’ P n ’ , ’ Fonts i ze ’ ,16) ;16 s e t ( gca , ’ xl im ’ , [ 1 . 3 2 . 1 ] ) ;17 %’ ylim ’ , [ 0 . 7 1 . 6 ]18 hold o f f ;

47


A.8 MATLAB code used to introduce stochastic forcing

to the one-dimensional iterative map (section 2.6)

1 % Denote alpha as a2

3 % Adding range o f alpha to the map :4 rng (0 , ’ tw i s t e r ’ ) ;5 d = 1 . 6 ;6 e = 1 . 7 ;7 a = ( e−d) .∗ rand (500 ,1 ) + d ;8

9 p=ze ro s (500 ,1 ) ;10 p (1) =1.1 ;11

12 % Adding no i s e ( r ) to the map :13 rng (0 , ’ tw i s t e r ’ ) ;14 c = 0 ;15 b = 0 . 0 9 ;16 r = (b−c ) .∗ rand (500 ,1 ) + c ;17

18 f o r n=1:100; % s t a r t the loop , running from n=1, to n=10019 p(n+1)=(a (n) ∗ ( ( a (n) /4)∗(1+ e r f ( ( p(n) −0.6) /0 . 2 ) )∗(1− e r f ( ( p(n)−1) /0 . 8 ) ) ) ∗p(n) )+

r (n) ;20 end21


24 p lo t (p , ’−ko ’ , ’ LineWidth ’ ,1 , ’ MarkerEdgeColor ’ , ’ k ’ , ’ MarkerFaceColor ’ , ’ k ’ , ’MarkerSize ’ , 2 )

25 t i t l e ( ’ S t o c h a s t i c a l l y Forced Map ’ , ’ f o n t s i z e ’ , 14)26 x l ab e l ( ’n ’ , ’ f o n t s i z e ’ , 14) ;27 y l ab e l ( ’ p n ’ , ’ f o n t s i z e ’ , 14) ;28 s e t ( gca , ’ xl im ’ , [ 0 100 ] , ’ yl im ’ , [ 0 . 6 , 1 . 6 ] ) ;

48

honours project - magetic activity in sun-like stars - emily walsh bsc mathematics and physics

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