6 Stability

There are two different ways to think about the stability for a system of nonlin-ear equations. Firstly, we could ask whether if a point is in a particular region ofphase space whether it stays near this region. Secondly we can consider makinga small change to the system and ask whether this results in a significant changein the solutions. The latter in called structural stability. We will consider bothtypes in turn.

6.1 Stability of a Point in Phase Space

There are two main definitions of stability in phase space x. One is whentrajectories starting at nearby points flow to a point xf . This is asymptoticstability. The other is when trajectories starting near a point xf stay near thispoint. This is called Liapunov stability.

We can be more precise in these definitions. Suppose x(t) is some trajectoryand x(t = 0) = x0 is near xf :

Liapunov stability. A fixed point xf is Liapunov stable if for all possible > 0 there exists a such that for all x0 satisfying |x0 xf | < we have|x(t) xf | < for all t.

Asymptotic stability. A fixed point xf is asymptotically stable if itis Liapunov stable and in addition there exists a such that for all x0satisfying |x0 xf | < we have x(t) xf for all t.

The important point to grasp about Liapunov stability is that can be madeas small as one likes, i.e. stability only occurs if by an appropriate choice of one can remain arbitrarily close to xf . Asymptotic stability is clearly a subclassof Liapunov stability. It is also possible that trajectories may flow ultimatelyto a point but without Liapunov stability. This is known as quasi-asymptoticstability. An illustration of all three cases are illustrated in fig. 13.

In two dimensions a stable star, node or focus is asymptotically stable,whereas a centre is Liapunov stable only. The region where all trajectories flowto a point is the domain of asymptotic stability. The largest region is the basinof attraction. We have already seen an example of this when considering thephase portrait for rabbits and sheep, where the phase space separated into abasin of attraction for rabbits only and for sheep only separated by the lineknown as the stable manifold.

Quasi-asymptotic stability is demonstrated by the set of equations

r = r(1 r2), = sin2(/2), (256)

where r and represent polar coordinates. This system of equations has a fixedpoint at

r = 1, = 0 (x = 1, y = 0). (257)

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Figure 13: An illustration of (a) Liapunov stability, (b) asymptotic stability,(c) quasi-asymptotic stability.

This is a stable fixed point. This is clear by considering perturbations aboutr = 1. If we let = we are clearly attracted directly back to the fixed point.However, if = 2 we only get back to the fixed point by travelling in acircuit encompassing the origin and reaching a distance up to 2 from the fixedpoint no matter how close we start.

6.2 Limit Cycles

We can also sometimes find asymptotic solutions corresponding to a continuousregion rather than a single point. Periodic orbits sometimes attract (or repel)nearby points. We can define the stability of such orbits in a similar manneras for fixed points. Define the set of points on the orbital trajectory as , anddefine a neighbourhood of by N(, ) which contains points y where we canfind points x in such that |y x| < . We define

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Liapunov orbital stability. For all > 0 there exists a such that forall y0 N(, ) then for all t we have y(t) N(, ).

Asymptotic stability. We require that is Liapunov stable and thatthere is a such that for all y0 N(, ) then we have y(t) approaches as t.

Asymptotically stable orbits are called limit cycles. These are important inpopulations, economics and in physiology. They do not arise in purely linearsystems. Consider the linear equations x = Gx, where G is a constant matrix.If x(t) is a solution to this equation then so is cx(t), where c is a constant.Hence, any orbital solution has indeterminate size.

A simple example of a limit cycle is a van der Pol oscillator, described by

x+ (x2 1)x+ x = 0, (258)

where is positive. If the oscillation amplitude exceeds 1 there is damping ofthe oscillation, while if the amplitude is less than 1 there is negative dampingor enhancement. Another simple example is given in circular polars by theequations

r = r(1 r2), = 1. (259)

The stable solution for r is clearly r = 1, while a constant rotation about theorigin occurs for any value of r.

6.3 Structural Stability of Solutions

This considers changes to the system of equations. Consider the equations

x = g(x). (260)

This system of equations is said to be structurally stable if for small we can addh(x) to the right-hand side such that solutions remain qualitatively equivalentto the original solutions, i.e. there is a one-to-one mapping between the twosets of solutions. This is a complicated topic, and we can only treat it in aqualitative manner. We can consider the stability by looking at the form ofsolutions in the linear case.

6.4 The Linearisation Theorem

This states that for any simple fixed point of a system of nonlinear differentialequations the phase space portrait close to the fixed point remains qualitativelythe same (one-to-one mapping) as for the linearised form of the equations pro-vided the fixed point is not a centre. Hence, an attracting or repelling fixedpoint retains its form, but a star may become a focus (or vice versa), for exam-ple. Centres are unstable in this sense because they rely on a precise fine-tuningsuch that there is no growth or decay at all in the solutions (trG = 0 precisely)which can easily be disturbed by nonlinear terms.

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Figure 14: (a) The phase portrait for a non-simple fixed point obtained inthe linear limit. (b) the distortion of this phase portrait by the addition of anon-linear term.

A simple fixed point means that there is not a zero eigenvalue for the lin-ear system. A zero eigenvalue again relies on a very precisely defined systemand small perturbations can remove a line along which there is no evolution.Consider the linear equations

x = x, y = 0. (261)

This results in the phase portrait in fig. 14(a). These can have the nonlinearextension

x = x, y = y2. (262)

Even for small the nonlinear term alters the evolution in the y directioncompletely and results in the much-altered phase portrait in fig. 14(b).

Centres and non-simple fixed points are called borderline cases, becausethey only exist due to very precise eigenvalues, whereas small effective changesin eigenvalues for other fixed points do not alter the fact that there is growth, ordamping. This suggests that it is unlikely that we will find centres as genuinesolutions to systems of nonlinear equations since their existence is so fragile.This is generally true. However, they can be protected in special circumstanceswhere there are conserved quantities, as is often the case for physical systems.We will consider these conservative systems next.

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7 Conservative Systems

Conservative systems form an important class of dynamical systems. Theyoccur very frequently since physical systems often have one, or more conservedquantities, e.g. energy, and angular momentum. The existence of this conservedquantity then often makes it easier to find the form of solutions.

7.1 First Integrals and Conservative Systems

A first integral of a system of differential equations is a function Q(x) that isconstant on a given trajectory. It is called first integral because it often arisesfrom integration of the equation for the slope of a trajectory, e.g.

dy

dx=

f(x)

g(y)

dy g(y) =

dx f(x) + c, (263)

Therefore the first integral is

Q(x, y) = y

y0

dy g(y) x

x0

dx f(x), (264)

where we could add a constant.We do not allow the function Q(x) to have the same value for a continuous

region of phase space, i.e. for all solutions, since this tends to lead to trivialresults. A conservative system is then defined to be one which contains a firstintegral for all possible solutions in phase space.

In D dimensions Q(x) = c gives a manifold of dimension D1 within whicha solutions trajectory lies. For example, in 3 dimensions the solutions lie on asurface of constant first integral.

7.2 Examples

There are various examples of conservative systems. The obvious one is aHamiltonian System. A simple example of this for a particle moving ina potential in 1 dimension is

x = p/m, p = dV (x)

dx. (265)

The trajectory for this isdp

dx=dV/dx

p/m, (266)

and so dp p/m =

dx

dV (x)

dx+ c, (267)

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and so

c =p2

2m+ V (x) E, (268)

i.e. the energy E is the first integral. In fact we know from our results in section1 that this is a general result, with the Hamiltonian always being a conservedquantity unless the potential has explicit time dependence, and in most casesthe Hamiltonian is equal to the energy of the system. In this case we have thegeneral first integral

E =p2

2m+ V (r). (269)

Another example of a conservative system is the Linear Saddle. This hasthe equations

x = x, y = y, (270)

which gives a saddle point at the origin. The trajectory is

dy

dx= y/x, (271)

which integrates to giveln y = ln x+ c. (272)

Therefore the first integral is

c Q(x, y) = xy (273)

is a constant on all trajectories and the system is conservative.

7.3 First Integral and Trajectories

Knowing the first integral gives us information about the form of solutions.In 2 dimensions Q(x, y) = c reveals a trajectory in phase space. However,different trajectories (with different solutions) can have the same first integral,e.g. consider the system

x = x(2 y), y = y(2 y). (274)

This has the same first integral as the linear saddle. This means that thetrajectories are exactly the same lines as for the linear saddle. However, therate of movement along the trajectories is different in this case. In particularthere is a fixed line at y = 2, and for y > 2 the direction of the arrows on thetrajectories is reversed.

7.4 Consequences for Conservative Systems

There are various consequences for and features of conservative systems.

1. As we have seen above, Q(x) = c gives a useful method for finding tra-jectories.

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2. There are no purely attracting or repelling fixed points for a conservativesystem, i.e. there are no nodes, stars, foci or limit cycles. This is becauseif a conserved quantity exists then it has some value at the attracting orrepelling fixed point. But all trajectories are linked by this fixed point andQ(x) is a constant throughout a continuous region of phase space. Thisis ruled out by the definition of a first integral. For example, consider thestable star:

x = x, y = y. (275)

This leads to the trajectory

dy

dx=

y

x ln y = ln x+ c

Q(x, y) = x/y = c. (276)

This is ill-defined at the origin and is not valid everywhere in phase space.It is true that away from the origin a trajectory satisfies x = cy, i.e. astraight line, but c cannot be interpreted as a conserved quantity.

3. Trajectories are given by contours of constant Q(x). Since Q(x) is the firstintegral then a stationary point for Q(x) is a zero for the right-hand sideof the equations and is a fixed point. Maxima and minima are surroundedby closed orbits, as shown below.

Q(x, y)

Qmin

x

y

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Therefore we have nonlinear centres in conservative systems, even thoughwe saw that these are generally unstable for nonlinear equations. We alsofind that a saddle point in Q(x) is a saddle point in the phase portrait.An example is the linear saddle where Q(x, y) = xy has a saddle point atthe origin.

4. Trajectories that leave a saddle point in a conservative system often returnto the same point. This is because it is a point with the same first integral.These are called homoclinic trajectories. If two saddles have the sameQ(x) and trajectories leave one and enter the other we have heteroclinictrajectories. These are rare in systems that have no conserved quantity.

An important class of conservative conservative systems are Hamiltoniansystems, which nearly always have conserved quantities. We will examine someexamples of these next.

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