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Dynamical Systems: Adding lecture

Diaraf SECK

http://simons-nlaga.ucad.snLaboratoire de Mathematiques de la Decision et d’Analyse Numerique (LMDAN)

BP 16889 Dakar-Fann, Senegal ; [email protected] Mathematics School

onInsight from Mathematical Modeling into Problems in Conservation, Ecology, and

EpidemiologyUniversite Cheikh Anta Diop - Dakar.

21 mai 2018

D. SECK (Universite Cheikh Anta Diop - Dakar) Dynamical Systems: Adding lecture 21 mai 2018 1 / 29

Outline

1 Fundamental notions and Definitions

2 Stability and Global study

3 Bifurcation


Fundamental notions and Definitions

Outline



3 Bifurcation



In this part , we intend to introduce some additional notions on discretedynamical systems. It is followed by some important parts such as theconjugacy, the stability of the structure and the bifurcation theory.1- Discrete Dynamical Systems :1-1 Definitions Proprieties and Consequences :

DefinitionA discrete time invertible dynamical system is a group of transformationsdefined by a bijection map Φn : Γ −→ Γn∈Z, where Γ represents the phasespace, n is the parameter.A phase space is a structure corresponding to a set of all possibles states(”needed”) of the considered system.A discrete time non invertible dynamical system is a semi group oftransformations defined by a map Φn : Γ −→ Γn∈N,n is the parameter.

OrbitsOne calls direct orbit or forward orbit in x0 and one notee O+(x0) the set :ϕn(x0)n∈N.One calls fixed point of a dynamical system any point x0 such thatϕ(x0) = x1 = x0. In this case O+(x0) = x0.



A concept that generalizes the notion of fixed point is the periodic point notion.Let us assume that there is k ∈ N, k ≥ 1 such that xk = x0 then one calls thata point x0 is periodic of period k .If k ≥ 2 and xk = x0 but xi 6= x0, ∀i ∈ 2, · · · k − 1 then x0 is a periodicpoint with a minimal period equal to k .x2k = ϕk (xk ) = ϕk (x0) = x0.If x0 is a periodic with minimal period equals k ≥ 1 thenO+(x0) = x0, x1, · · · , xk−1.x0 is a pre-periodic point if there is j ∈ N, j ≥ 1 such that xj is a periodic pointof period k . In this case O+(x0) = x0, x1, · · · , xj , · · · , xk+j−1, xj = xk+j.



Remark

If ϕ is invertible, one can define O−(x0) = x−nn∈N = ϕ−n(x0)n∈N.

RemarkLimit setsWhen x0 is neither periodic nor pre- periodic then O+(x0) is an infinite set.And it’s description is more complicated. In general, it is impossible todescribe this set without additional hypotheses on the structure, say when thephase space is a topololigical space. And even in this case, the description isnot an easy task in many situations.

Example

Let us consider xn = (x1n , x2

n ). Let A lbe the matrix defined as follows :

A =

(1 23 0

)xn+1 = Axn is a discrete time linear system. The problem is to find xn?



Example

Let us consider the sequence of vectors defined by vn = (xn, yn, zn) wherev0 = (x0, y0, z0). Let the following system xn = xn−1 + zn−1

yn = xn−1 + yn−1zn = yn−1 + zn−1

This is equivalent to xnynzn

=

1 0 11 1 00 1 1

xn−1yn−1zn−1



2 Conjugacy :

DefinitionLet X and Y , be two sets , f : X → X and g : Y → Y , two maps. One saysthat f and g are conjugate if :∃h : X → Y a bijection such that hof = gof .

Example

1 Let A be a diagonalizable matrix, then , there exists a passage matrix Psuch that the diagonal matrix D is given by the following formulae :

D = P−1AP.

In this case on says that D and A are conjugate.

2 Let A =

(−1 −3−3 −1

)et B =

(2 00 −4

), there is a rotation matrix

such that i B = R−1AR. R =

(1√2− 1√

21√2

1√2

)should be.

And then , A and B are conjugate.D. SECK (Universite Cheikh Anta Diop - Dakar) Dynamical Systems: Adding lecture 21 mai 2018 8 / 29


Lemma

1 The conjugacy relation is an equivalence class.

2 Let f ,g,h as in the above definition, then we have :f n = (h−1ogoh)n = h−1ognoh

RemarkThe conjugacy relation is to be understood in the sense of equivalencebetween two dynamical systems.

LemmaIf f and g are two conjugate maps, then they are the same number of periodicpoints and these are mapped to each other by the conjugacy.



DefinitionLet S : Y → Y and T : X → X .One says that the dynamical system (Y ,S) is a factor of the dynamicalsystem (X ,T ) if tehre is an onto map φ : X → Y verifying

φo T = S o φ.

It is said also that (X ,T ) is an extension of (Y ,S). One says also that φ is asemi- conjugacy between S and T .


Stability and Global study

Outline



3 Bifurcation



3- Stable framework :An interesting question is the effect of perturbation of dynamical systems withrespect to this equivalence relation. In fact when there is equivalence(conjugacy relation), how, a small perturbation will behave between toequivalent systems? One way to formulate this, is to ask whether twodynamical systems which are ”close” in some sense, belong to the sameequivalence class.To formalise this idea we need to formalise the notion of a small perturbation,i.e. we need a topology on a given space of dynamical systems.



Let us attempt to say more : Let φt , ψt two equivalent dynamical systemsdefined on a phase space X , φt : X → X and ψt = h−1oφtoh with h : X → X .Let’s denote by φεt the perturbed dynamical system, with ε a very smallparameter.The question is : what is the behavior of φεt when ε goes to zero?In general there may be many such topologies and an appropriate topologywill depend on the class of systems of interest and on the kind of propertiesthat are being studied. However, assuming some space χof dynamicalsystems has been fixed, together with some equivalence relation ∼ and sometopology τ. we can make the following definition.

DefinitionWe say that a dynamical system f ∈ χ is structurally stable with respect to thetopology τ and the equivalence relation ∼ if it lies in the interior of itsequivalence class. If it is not structurally stable we say that it is (or undergoes)a bifurcation.



DefinitionStability of a point :

1 The equilibrium x0 is Lyapunov stable (or simply stable) if, for eachε > 0and t0 ≥ 0, there existsδ = δ(t0, ε) > 0 such that |x(t0)| < δ =⇒ x(t)exists for all t ≥ t0 and |x(t)| < ε.

2 The equilibrium x0is uniformly stable if δ = δ(ε).

3 The equilibrium x0 is unstable if it is not stable.

4 The equilibrium x0 is asymptotically stable if(a) x0 is stable ;and(b) ∃δ0 = δ0(t0)such that |x(t0)| < δ0 =⇒ |x(t)| → 0 at t →∞.

5 The equilibrium x0 is uniformly asymptotically stable (u.a.s ) if(a) x0 is uniformly stable ;and(b) ∃δ0 > 0 such that,for each ε > 0, there exists T = T (ε) > 0 such that|x(t0)| < δ0, t ≥ t0 + T =⇒ |(x(t))| < ε.



3- 1 Global study :

Extension theorem

Corollary

A flow locally bounded on Rn is globally defined.Let L = L(t , x) be a vector field of class C1, defined on R× Rn. Let Φ be themaximum flow associated to the ODE x = L(t , x), from t0 = 0. If it is knownthat for any (x0, t) such that Φ be defined we have,

|t | ≤ T , , |x0| ≤ R =⇒ |Φt0,t (x0)| ≤ C(T ,R) < +∞,

then the flow is defined for all the times and all the initial conditions.

Hartman-Grobman Theorem, Stable manifold theorem, Center ManifoldTheorem.



Flows near a periodic orbit : Stable Manifold Theorem for Periodic Orbits,Poincare’s Stability Condition in the case of two dimensions, Bendixson’s criterion in the case of two dimensions, Dulac’ s Criteria in the case oftwo dimensions, Poincare- Bendixson Theorem in the case of twodimensions. For additional information about these notions see forinstance the books of L. Perko, J. Hale., Hirsch-Smale book,...

Definition

A ponit p ∈ U ⊂ Rn (an open set) is a ω−limit point of the trajectory φ(., x0) ofthe system

x = L(x)(∗∗)

if there is a sequence tn → +∞ such that limn→ φ(tn, x0) = p.Similary if there is a sequence tn → −∞ such that limn→+∞ φ(tn, x0) = q andthe point q is called an α−limit point of the trajectory φ(., x0) of (**). The set ofall ω−limit points of a trajectory Γ is called the ω−limit set of Γ and it isdenoted by ω(Γ).The set of all α−limit point of a trajectory Γ is called α−limit set of Γ and it isdenoted by α(Γ).α(Γ) ∪ ω(Γ) is called the limit set of Γ.



Definition ( ω-limit sets)

One calls the set ω-limit of a trajectory (x(t)) the set of all its accumulationpoints when t →∞, i.e

Ω :=

y ; ∃(tn)→∞; x(tn)→ y.

In other words, it is the set of all the possible limits of successive positions ofthe trajectory evaluated for t tending to∞.

Theorem

α(Γ) and ω(Γ) are closed subsets of U ⊂ Rn (an open set ) and if Γ iscontained in a compact subset of Rn, then α(Γ) and ω(Γ) are non emptyconnected, compact subsets of U.



DefinitionA closed invariant set A ⊂ U, an open set of Rn is called an attracting set of(**) if there is some neighbourhood V of A such that for all x ∈ V , φt (x) ∈ Vfor all t ≥ t0 and φt (x)→ A as t →∞.An attractor of (**) is an attracting set which contains a dense orbit.

Definition (Trajectories linking equilibria)

Let (x(t))t∈R be a trajectory of an ODE linking two equilibria points :

limt→−∞

x(t) = x∗−, limt→+∞

x(t) = x∗+.

One says that x is a non homocline trajectory if x∗+ 6= x∗−, and a homoclineone if x∗+ = x∗−.



A first version of Poincare-Bendixson theorem is :

Theorem

Assume that n = 2. Let K be a nonempty compact limit set (of x = f (x) ). If Kcontains no equilibria, then K is a periodic orbit.

Another more general statement of the Poincare-Bendixson theorem :

Theorem ( Poincare–Bendixson Theorem)

Let us consider an autonomuous ODE of first order with a vector field of classC1, in an open domain O of R2. Then, in every compact subset of Ocontaining a finite number of equilibrium points, the ω− limit sets of an orbit ofthe ODE can only be one of the three following objects :- an equilibrium ;- a cycle ;- a graph composed by a finite number of unstable equilibrium pointsconnected by trajectories (homoclines or nonhomoclines) of the flow. Inadditition, two different equilibrium are connected by at most twononhomoclines, one in each direction of the time. (This graph may include anarbitrary and even a denumerable infinite of homoclines trajectories from agiven equilibrium).


Bifurcation

Outline



3 Bifurcation


Bifurcation

4-Bifurcation :This part has been prepared with helpful of the Perko ’s book and lecturenotes given by Michael Y. Li in December 7, 2004 .4- 1 Introduction :Consider a family of differential equations

x = f (x , µ), (1)

where f : Rn+1 → Rn is analytic for µ ∈ R, x ∈ Rn.Let x0(µ)be a family of equilibria of (1), namely, f (x0(µ), µ) = 0.Set z = x − x0(µ).Then,z = A(µ)z + O(|z|2),A(µ) = Df (x0(µ), µ).Let λ1(µ), · · · , λn(µ) be the eigenvalues of A(µ). If, for somei , i ∈ 1, · · · ,n;Reλi (µ) changes sign at µ = µ0, we say that µ0 is abifurcation point of (1). Sometimes, we also call (x0(µ0), µ0) atbifurcation point.


Bifurcation

Remark

1 f is analytic in x , µ implies that x0(µ) is analytic in µ , provideddetA(µ) = detDf (x0(µ), µ) 6= 0. Analyticity may fail at a bifurcation pointsince detA(µ) = λ1(µ) · · ·λn(µ).

2 Being the roots of det(λI − A(µ)) = 0, λ = λi (µ) are also analytic in µexcept possibly at bifurcation points.

4- 2 In one dimension :Since only one-dimensional eigenspace changes with µ, we may simplyassume n = 1.Therefore, f : R2 → R, and x0(µ) is a real-valued analytic function of µprovided

λ1(µ) =∂f∂x

(x0(µ), µ) = A(µ) 6= 0.


Bifurcation

Therefore, the equilibrium x0(µ) is u.a.s. if λ1(µ) < 0, and unstable ifλ1(µ) > 0.This implies that µ0 is a bifurcation point if

λ1(µ0) = 0.

Therefore, bifurcation points (x0(µ0), µ0) are solutions of

f (x , µ) = 0, and∂f∂x

(x , µ) = 0 (2)

The bifurcation diagram describes the general shape of x0(µ) for µ near thebifurcation point µ0.At µ 6= µ0, λ1(µ) = ∂f

∂x (x , µ) 6= 0. By the Implicit Function Theorem,x = x0(µ)is the unique solution of

f (x , µ) = 0,

and∂f∂x

dxdµ

+∂f∂µ

= 0.


Bifurcation

Sinceλ1(µ0) = ∂f

∂x (x0(µ0), µ0) = 0, if ∂f∂µ (x0(µ0), µ0) 6= 0, then | dx

dµ | → ∞, as µ→ µ0.

Therefore the curve x = x0(µ) has a vertical tangent line when µ = µ0 if∂f∂µ (x0(µ0), µ0) 6= 0. In the subsequent discussions, without loss of generality(w.l.o.g.), we will assume that the bifurcation point is at (0,0), namely, µ0 = 0and x0(µ0) = 0.The most common bifurcation types are illustrated by the following examples.

Example

Saddle-Node Bifurcation. Consider

x = µ− x2.

In this case, the bifurcation equations (2) becomes

x2 = µ,

2x = 0.


Bifurcation

There are two branches of equilibria :x0(µ) = ±√µ, and the bifurcation point is at (x , µ) = (0,0).

Since ∂f∂µ (0,0) 6= 0, we should expect a vertical tangent line at (0,0) for x0(µ).

Example

Transcritical Bifurcation. Consider

x = µx − x2.

The bifurcation equation (2) becomes µx − x2 = 0, µ− 2x = 0.This gives two branches of equilibria x0 = 0, and x0 = µ.For the branch x0 = 0, we have λ1 = µ and and thus the stability changesfrom stable to unstable as µ increases cross 0, and µ0 = 0 is the bifurcationpoint.For the second branch, λ1 = −µ. Therefore, this branch changes stability inthe opposite direction to the first branch, and the bifurcation point is alsoµ0 = 0.


Bifurcation

Example

Pitchfork Bifurcation. Consider

x = µx − x3.

From the bifurcation equations, we find that there are three branches ofequilibria : x0 = 0, x0 =

√µ and x0 = −√µ.

The corresponding λ1 for the three branches are λ1 = µ;−2µ, and −2µ,respectively. It is easy to see that (0,0) is the bifurcation point for all threebranches.

The subcritical (or supercritical) pitchfork bifurcation can be described as abranch of equilibria which changes stability type at the bifurcation point isintersected there by a stable (or unstable) branch.


Bifurcation

The following theorem establishes that the three type of bifurcations observedin the above examples are indeed the only generic ones there are.

TheoremLet us suppose

(i)f (x , µ) is an analytic function of (x , µ) near (0,0).

(ii) (x , µ) = (0,0) is a bifurcation point (namely 0 = f (0,0) = ∂f∂x (0,0)).

Then

(a) If ∂f∂µ (0,0) 6= 0 and ∂2f

∂x2 (0,0) 6= 0, then there exists, in aneighbourhood of (0,0), a single branch of critical points which has asaddle node bifurcation at (0,0).

(b) If ∂f∂µ (0,0) = 0 let D = det

(∂2f∂µ2

∂2f∂µ∂x

∂2f∂x∂µ

∂2f∂x2

)(0,0) = ∂2f

∂µ2∂2f∂x2 − ( ∂2f

∂x∂µ )2,

then (b1) if D > 0, then (0,0) is an isolated critical point.(b2) if D < 0, then there are two branches of critical points which intersect at(0,0). The bifurcation is either transcritical or pitchfork.


Bifurcation

RemarkIt is quite possible to do more on the bifurcation theory, let’s mention :

the one-dimensional bifurcation in two dimensions space (n = 2).

In high dimension space there is the Hopf Bifurcation (n ≥ 2).

We invite the reader to have at hands books in which this theory isdevelopped for instance the book of Perko.


Bifurcation

Thank you for your attention


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