Dynamical systems and chaos : Some theoretical ...Dynamical systems and chaos : Some theoretical preliminaries From trajectories to trajectories Laurent LAVAL [email protected]

Dynamical systems and chaos : Some theoreticalpreliminaries

From trajectories to trajectories

Laurent [email protected]

International School on Complex Dynamics Engineering (ISCDE)Batz-sur-Mer, October 2011

Laurent LAVAL ISCDE 2011 – Dynamical systems and chaos : Some theoretical preliminaries 1 / 64

OutlineIntroduction

TerminologyProblem statementObjective of the talk

Dynamical SystemsFormal definitionCharacterizing the time evolutionLaw of evolution / Evolution ruleDeterminism and predictability

Continuous-time dynamical systemsTransformations of differential equationsExistence and uniqueness of solutionsSensitivity to initial conditions and/or parameters

About TrajectoriesStationary behaviorsLimits points, sets and cyclesAttractors and basins of attraction


IntroductionTerminologyProblem statementObjective of the talk

Terminology: System – Process – PhenomenonSystem: Set of interacting or interdependent components forming an integrated

whole. (← may be a unitary or complexa whole)

System Components + Relationships

acomposed of many interconnected parts.

ExamplesAn Electronic circuit

Components: Individual electronic components connected by wires or traces.Relationships: Currents and/or voltages across the components, according, for in-stance, to Kirchoff’s laws.

A Social NetworkComponents: Individuals or organizationsInterdependencies: friendship, common interest, financial exchange, ...

A CapacitorComponents: Two electrical conductors separated by a dielectric (insulator).Relationship: The static electric field across the dielectric when a potential differ-ence (voltage) across the conductors occurs.



RemarkMost systems share common characteristics, including:

Systems have structure, defined by components and their organization

Systems have behavior, which involves inputs (or not), processing and out-puts of material, energy, information, or data

Systems have interconnectivity: the various parts of a system have func-tional as well as structural relationships to each other.

Systems may have some functions or groups of functions

Terminology: System – Process – PhenomenonProcess: A continuous action, operation, or series of changes taking place in adefinite manner, and directed to some end.

ExampleThe chemical Belousov-Zhabotinsky reaction (→ nonlinear chemical oscillator).



Terminology: System – Process – PhenomenonPhenomenon (plural phenomena): A fact, occurrence, or circumstance ob-served or observable

RemarkIn scientific usage, a phenomenon is any event that is observable, how-ever common it might be, even if it requires the use of instrumentation toobserve, record, or compile data concerning it. For example, in physics,a phenomenon may be a feature of matter, energy, or space-time, suchas Galileo Galilei’s observations of the motion of a pendulum. [Ber96]

Regardless of the research field (physics, control theory, mathematics, ...),Regardless of used terms: system, phenomenon or process,

The goal is the same:

To study (that is to define, to analyze, to understand, ...) their behaviorsso as to be able to predict or/and control their time evolution



Which systems / phenomena are the focus here ?Mainly:

Systems with nonlinear dynamics,

Chaotic phenomena,

Example: Chua’s circuit

Fig. 1: Pr. Leon O. CHUAFig. 2: The electronic circuit

Fig. 3: A “strange” (chaotic) behav-ior (Vc2 vs Vc1)



Example : The three-body problem(as originally studied by French mathematician Henri Poincaré)

Fig. 4: Henri Poincaré

“Given a system of arbitrarily many mass points that attract eachaccording to Newton’s law, under the assumption that no two pointsever collide, try to find a representation of the coordinates ofeach point as a series in a variable that is some known functionof time and for all of whose values the series converges uniformly.”

L1, L2, and L3 are saddle points: stable in the θ direction butunstable in the r direction. Objects located at these points wouldneed to use fuel to maintain their orbital positionL4 and L5 are theoretical stable points. An object placed at thesepoints, 60 degrees ahead of and behind the moon at the ra-dius of its orbit, will remain at the same point with respect to themoon.



Which fields may be concerned ?A lot !

Neuroscience (Computational neuroscience): Bursting phenomena, ...

Economics: Stock market fluctuations, ...

Chemistry: Belousov-Zhabotinsky (BZ) reaction, ...

Physics: Climate and weather prediction (e.g. E. Lorenz work), ...

Social psychology: Group dynamics, dynamics of conflicts, ...

Biology: Cardiac dynamics, spreading of epidemics, ......

Strogatz, S.H., Nonlinear Dynamics And Chaos: With Applications To Physics, Biology,Chemistry, And Engineering, Perseus Books, 1994.Izhikevich E.M., Dynamical Systems in Neuroscience: The Geometry of Excitability andBursting, The MIT press, 2007.Gondolfo, G., Economic dynamics, Springer, 2010.Guastello, S.J. and al., Chaos and Complexity in Psychology: The Theory of NonlinearDynamical Systems, Cambridge University Press, (reprint) 2011.



Problem statementRecalling the aim : To study the behaviors of systems / phenomena / processeswhich exhibit nontrivial dynamics

Basic questions

Which language to use ? Mathematics

Which mathematical tool(s) to use ?A small fiction ...Some extraterrestrials land on the earth and learn about meridi-ans. Their first question is: Are they parallel lines ?With regard to their local limited viewpoint (a small part of theearth seeming like a plan), Euclidean geometry seems to beefficient to characterize two close meridians. But, taking off theground on their flying carpet, the ET then find another viewpoint:Meridians appear as curved lines → Lobachevskian geometry(otherwise known as hyperbolic geometry) then appears moreconvenient to answer to their question.



Observation (viewpoint) as well as knowledge of existing tools may be of crucialimportance

Objective of the talkThis talk aims :

to introduce basic notions and definitions related Dynamical System Theory,

to recall some results about differential equations,

to put on the light some technical/theoretical problems,

through questions and (may be) answers


Part II

Dynamical Systems


Dynamical Systems

Formal definitionCharacterizing the time evolutionLaw of evolution / Evolution ruleDeterminism and predictability

Dynamical Systems

Let us recall the “The three-body problem”... try to find a representation of the coordinates of each point as a series in avariable that is some known function of time ...

What are dynamical systems ? (formal definition)Dynamical systems are mathematical objects used to model physical phenomenawhose behavior (the state) changes over time.

Keywords: Model, state, time, evolution

Dynamical Systems TheoryArea of applied mathematics used to describe the behavior of complex dynamicalsystems.


Dynamical Systems


How to characterize the time-evolution of physical phenomena ?Starting point: The observation (viewpoint)

An example : The arrow paradox of Zeno (for instance, see [Hug10])"if everything when it occupies an equal space is at rest, and if that which isin locomotion is always occupying such a space at any moment, the flyingarrow is therefore motionless." – Aristotle, Physics VI:9, 239b5.

The only instantaneous observation (as a snapshot) of a physical system/phe-nomenon/process may be not sufficient to discern and characterize its evolution overtime

Need of additional descriptive informations – the state variablesa– to characterizethe state of the system at a given time

aQuantities of interest whose magnitudes are expected to change with time.


Dynamical Systems


ExamplesTo define the evolution of the arrow in an Euclidean space (3D space) we canconsider its position (3 coordinates) and its velocity (3 coordinates)

Numbers of predators and prey together with their respective growth rates, in anecological system (in which two species interact),

Temperature, pressure, volume, internal energy, enthalpy, and entropy in a ther-modynamic system,

...

Remark: Some state variables of a state vector may not be physically measurable ormeasured quantities (see the concept of observability in control theory).

Concept of state estimation


Dynamical Systems


What is a state space ? (formal definition)Set of possible configurations↔ state space/phase space/configuration space

Each configuration or state (the arguments) has to be associated (by a mapping)to a number (measure) or a set (a family or vector) of numbers in the state space(that is the target set of images = the codomain)

We wish that each state (each configuration) of the system corresponds to aunique point in the state space, (← a bijective mapping)

For two close states, we wish the associated vectors (the images) to be close,

The state space could be of finite, infinite or zero dimension (Cantor set)

Definition 1 (State space)

A state space (phase space) is a separated topological space for which every (openor closed) n-dimensional topological ball (that is any subset) of the topological spaceis homeomorphic to an (open or closed) Euclidean n-ball (related to the generalizedcoordinates), involving that mappings preserve all the the topological properties ofarguments domain.


Dynamical Systems


What about time ?Time of evolution and Time of observation

(for measuring, modeling, analysis, etc ...)“Every physical phenomenon, system or process evolves with respect to continuous

time variations“

RemarkDynamical system → Reasoning with state spaces → Projection of configurationsinto the state space

Possible lost of informations about time

Possible lost of physical meaning (adimensioned mathematical descriptions)

But: Great basis for understanding the global behavior

ExampleCase of autonomous systems where mathematical modeling (descriptions) arebased on ordinary differential equations which do not explicitly depend on the inde-pendent variable (as for Time-invariant systems with time as independent variable)


Dynamical Systems


Continuous-time caseConsider the Lorenz’s mathematical model given by: x(t) = σ× y(t)−σ× x(t) red: coupling terms

y(t) = ρ× x(t)− x(t)× z(t)− y(t)z(t) = x(t)× y(t)−β× z(t) (σ,β,ρ: system parameters)

(1)

Fig. 5: Time series: x(t), y(t) andz(t)

Fig. 6: Plot of z(t) vs y(t) vs x(t)– a 3D-phase space


Dynamical Systems


Discrete-time case(s)Synchronous: quantities are evaluated at given (possibly equally) spaced times

Asynchronous : quantities are evaluated when an event occurs

Fig. 7: Time series of predators and preys Fig. 8: Plot of rate of Wolfs vs rate of Moose– 2D-Phase plane


Dynamical Systems


Section of Poincaré (discretization)

Fig. 9: Time series:x(t), y(t) and z(t)

Fig. 10: Plot of z(t) vsy(t) vs x(t) (continuous-time case)

Fig. 11: Poincare section =Phase plane – (discrete-timecase)

Remark: Use of Poincaré section leads any continuous-time periodic orbit to appearas a discrete-time orbit cycling trough a finite set of points lying on the surface of thesection (where the number of crossing points, with respect to the piercing direction,depends on the periodicity of the orbit).


Dynamical Systems


To summarize

Dynamical system (conceptual definition)A dynamical system can be thought as:

a state spacea (or phase space) that defines all possible states of the system

together with

a a set of times⊆Z (discrete-time case) or a subset of R (continuous-time case)and

a law of evolution (also called: evolution rule) which represents the ways in whichthe state variables interact over the time as well as the time-varying behavior ofthe system state (the state trajectories).

aalso called a a manifold in the continuous-time case


Dynamical Systems


Law of evolution / Evolution rule

Definition 2 (Time evolution operator)

Consider the state space X of a dynamical system, and an initial state x0 ∈ X at times0 ∈ T (set of times). The time evolution which brings the system from x0 to a statex1 ∈ X (at time s1 ∈ T ) can be defined by means of a time evolution operator

φs1,s0 : T ×X → X(s0,x0) 7−→ x1 = φs1,s0 (s0,x0)

(2)

The pair (s0,x0) is the initial condition (initial time, initial state)

Time parameter ”s“ is commonly denoted by symbol t (resp. k or n) for continuous-time (resp. discrete- time) systems for which T ⊆ R (resp. T ⊆ Z).

If φsj ,s0 is time-invariant (∀sj ∈ T ), then φ only depends on the difference τ =sj − s0 and is simply denoted φτ

The time evolution operator is also referred to as a functional, a mapping ormap, a function, depending on the context


Dynamical Systems


The evolution of a system over the time (set T ) can be given by a set (a collec-tion) of time evolution operator(s)

Remark: Definition 2 also includes variable structure systems (aspart of Hybrid dynamical systems), in the sense that φ may changeover the time (along the time space T ), with respect, at least, to thepiecewise continuity.

A time evolution operator (also referred to as an update rule) can be a solutionof a differential equation, a recurrence relation / difference equation, an integralequation, a delayed equation, a partial differential equation, ... or a mixture ofthem.


Dynamical Systems


Basic properties

Definition 3 (Cr−diffeomorphism)

A map φ : U ⊂ Rn → V = φ(U) ⊂ Rn (where U and V are two open subsets / twomanifolds) is a Cr− diffeomorphisma if and only if:

φ is Cr , r ≥ 1 (r times continuously differentiable)

φ a bijective map

φ−1 is Cr

aThat is an isomorphism in the category of smooth manifolds.


Dynamical Systems


Determinism and predictability

DeterminismDeterminisma: Concept that events are linked by causality principleb, in such a waythat any state of a phenomenon/system/process is completely, or at least to somelarge degree, determined by prior states.

aalso commonly referred to as absolute determinism, causal determinism or Laplace determinismbprinciple which induces the reaction to the action.

Definition 4 (Determinism)

A time evolution operator is deterministic if

1. φs2,s1 φs1,s0 = φs2,s0 ∀s0,s1,s2 ∈ T (3)

(where denotes the composition)

2. φs,s = Id (identity) ∀s ∈ T "no time variation, no evolution"

In other words, for any initial condition x0 ∈ X , if x1 = φs1,s0 (x0) and x2 = φs2,s1 (x1)then x2 = φs2,s0 (x0)


Dynamical Systems


RemarkTime evolution operator must exist for all time s ∈ T ,

set of times T must be, at least, a semigroupa (acting on state space X)Usually T is an additive semigroup such that:

1. 0 ∈ T (4)

2. ∀s1,s2 ∈ T , s1 + s2 ∈ T (5)

aan algebraic structure consisting of a set together with an associative binary operation.

In most cases, evolution operator is time-invariant, leading Def. 3 to be rewritten as:

1. φτ2 φτ1 = φ(τ1+τ2) ∀τ1,τ2 ∈ T

with τ2 = s2− s1 and τ1 = s1− s0 s0,s1,s2 ∈ T

2. φ0 = Id (identity) "no time variation, no evolution"


Dynamical Systems


About predictabilityDeterministic systems may not have (long-term) predictable evolutions

Determinism together with causality principle→ same causes induce the same effects

ThusEvolution of a deterministic dynamical system should be possibly predictable(for long time scales)

ButThis supposes that each state can be known to an arbitrarily high precision

ThereforeLong-term predictability then becomes impossible in case of systems withhigh sensitivity to initial conditions or disturbance effects (such as chaoticsystems) as :

small variations of initial conditions can give rise to highly different evo-lutions (long-time behavior of solutions)small numerical errors in a state may lead to an exponential error growthover the time.


Dynamical Systems


RemarkContrarily to some encountered ideas, Heisenberg uncertainty principle (inquantum mechanics) as well as chaos theory (including works of Poincaré, VonNeumann, ...) do not contradict neither the determinism concept nor the causal-ity principle. They simply point out the unpredictable character of some pro-cesses or systems.

About repeatabilityRepeatability of an experimentation implies the initial conditions to be identical. Ifsuch an identity can exist in mathematics or computer science (as constants arewell-defined), causes can at best be similar or “pseudo-identical” when dealingwith real phenomena / systems / processes (with states ∈ Rn).

About invertibility of the time evolution operator (recall)As the set of times T is a topological group, then each φt , t ∈ R is invertible(since it follows from the definition of time-evolution operator that φ−t = φ

−1t )


Dynamical Systems


Dynamical systems (mathematical definition)Definition 5 (Dynamical System)

A dynamical system is a triplet (X ,T ,φ), where X is the state space (or phasespace), T is an additive (semi-)group, and φ is a mapping

φ : T ×X → Xsuch that, for any x ∈ X ,

1. φs2,s1 (x)φs1,s0 (x) = φs2,s1 (φs1,s0 (x)) = φs2,s0 (x), ∀s0,s1,s2 ∈ T

2. φs,s(x) = x , ∀s ∈ T

For a time-invariant evolution operator

1. φτ2 (x)φτ1 (x) = φτ2 (φτ1 (x)) = φτ2+τ1 (x) ∀τ1,τ2 ∈ T

with τ2 = s2− s1 and τ1 = s1− s0 s0,s1,s2 ∈ T

2. φ0(x) = x

If T is a group, then the dynamical system is said to be an invertible dynamicalsystem.


Dynamical Systems


ExamplesConsider the following first-order linear differential equation as modeling the be-havior of a continuous-time system

dx(t)dt

= αx(t), α ∈ R,x ∈ R and initial condition x(0) = x0

Solving the differentialequation leads to theunique, global solution:x(t) = x0eαt

1x

dx = αdt

x0 is the initial condition at t = 0, and xt is the value at time t∫ xt

x0

1x

dx =∫ t

0αdt

[ln x]xtx0 = α t ⇔ ln

xt

x0= αt

xt = x0eαt

Then, according to definition 5 , state space X = R, set of times T = R and evo-lution operator φt (x) = x0eαt (∀t ∈ T ) is a continuous-time dynamical system.


Dynamical Systems


Consider the (well-known) Hénon’s discrete-time mathematical model given bythe following coupled difference equations [H76]

xk+1 = yk + 1−ax2k ,

yk+1 = bxk(6)

where a ∈ R and b ∈ R are two parameters, and xk ∈ R, yk ∈ R are the statevariables (evaluated at discrete time k ≥ 0).Then, we can define the evolution operator φ as a map:

φ : T ×X → X

ζn 7→ φ(ζn) = ζn+1

with ζn = [xn yn]T , n ∈ T , and φ(ζn) =[yn + 1−ax2

n bnxn]T

Noting that, starting from an initial state ζ0 = [x0 y0]T , we have ζn = φn(ζ0)where φn is the n’th iterate of φ (for n ∈ T ). Then, according to definition 5,X = R2, T = N and ζnn∈T (given by φ) is a discrete-time dynamical system.


Dynamical Systems


Terminology: flow/map, orbit and phase portraitConsider the evolution of a dynamical system (X ,T ,φ) as given by a 1-parameterfamily φss∈ T of mapping φs : X → X , where s ∈ T should be thought as theparameter.Definition 6 (flow) Continuous-time Dynamical Systems

A flow is a one-parameter family φtt∈R of deterministic transformations of X

Flow and mapφ is the flow over X when T ⊆ R, and a semi-flow if T is restricted to non-negative reals (R+)

For discrete-time dynamical systems, instead of flow, Φ is usually referred to asa map or a cascade, and the restriction to N is a semi-cascade.

In certain situations one might also consider local flows, which are defined only insome subset

dom(φ) = (t,x) / t ∈]ax ,bx [,ax < 0 < bx , x ∈ X ⊂ R×X

called the flow domain of φ


Dynamical Systems


Informally, an orbit may be regarded as the trajectory of a particle that was initiallypositioned at x . The (forward) orbit or trajectory of a state x is then thetime-ordered collection of states that follow from x using the evolution rule (the flow(continuous-time case) or the map (discrete-time case)).

Definition 7 (Orbit / Trajectory)

Given x in X, the set O+φt

= φ(t,x), t ∈ T is called the orbit of x under the flow Φ.

For a deterministic rule with discrete-time, the forward orbit of x0 is the sequencex0,x1,x2, ...,

When both state space and time are continuous, the forward orbit is a curvex(t), t ≥ 0 (that is an integral curve). If the flow is generated by a vector field,then the orbits are the images of its integral curves.


Dynamical Systems


For some systems, the orbits may depends on initial conditionsDefinition 8 (Phase portrait)

A phase portrait is a geometric representation of the trajectories of a dynamical sys-tem in the phase plane.

Example (Phase portrait of a Van der Pol equation)

Fig. 12: Phase portrait of a Van der Pol equation


Dynamical Systems


Example (with details)Consider The FitzHugh-Nagumo modela of a biological neuron

dudt

= F(u,w) = u + a0u3 + w

dwdt

= G(u,w) = b0 + b1u−w(7)

where a0, b0 and b1 are some system parametersThe temporal evolution of the state variables [u w ]T can be visualized in the so-called phase plane. From a starting point [u(t) w(t)]T the system will move ina time ∆t to a new state (u(t + ∆t),w(t + ∆t))T which has to be determined byintegration of the differential equations (7). For ∆t sufficiently small, the displacement(∆u,∆w)T is in the direction of the flow(

∆u∆w

)=

(uw

)∆t

which can be plotted as a vector field in the phase plane: The phase portrait of thesystem.

asimplified version of the Hodgkin-Huxley modelLaurent LAVAL ISCDE 2011 – Dynamical systems and chaos : Some theoretical preliminaries 34 / 64

Dynamical Systems


NullclinesLet us consider the set of points with u = 0, called the u-nullcline. The directionof flow on the u-nullcline is in direction of (0, w)T , since u = 0. Hence arrows inthe phase portrait are vertical on the u-nullcline. Similarly, the w-nullcline is definedby the condition w = 0 and arrows are horizontal. The fixed points of the dynamics,defined by u = w = 0 are given by the intersection of the u-nullcline with the w-nullcline (see Figure B).

Fig. 13: Phase portrait and nullclines


Part III

Continuous-time Dynamical Systems with differentialequations as basis



Basic results and definitions related to DE

Consider the first-order, nonautonomous differential equation:

dx(t)dt

= f (t,x(t)) x(t) ∈ Rn (8)

where f is a vector field such that f ∈ C0(Ω,Rn), the domain Ω of f is an open set:Ω⊂ (T ×Rn), and T is a nondegenerate interval of R (← continuous-time case).

Definition 9 (Solution of a diff. equation)

A solution of (8) is a pair of Sφ =

Tφ,φ

with function φ : Tφ ⊂R×Rn→Rn, whereTφ is a (nondegenerate) interval of R, and such that:

1 φ is n-times differentiable on Tφ (here, first-order system⇒ n = 1 )2 The grapha G(φ) =

(t,φ(t)) , t ∈ Tφ

⊂ Ω

3 ∀t ∈ Tφ:dφ(t)

dt= f (t,φ(t))

aThe collection of all ordered pairs (t,φ(t)).



Problem statement: An Initial Value Problem (IVP)Regarding to differential equation (8), the problema is to find a (or the !) solutionpassing by a specified point (t0,x0) ∈ Ω⊂ (T ×Rn) called the initial condition

t0 ∈ Tφ (Tφ ⊆ T )φ(t0) = x0

aSometimes referred to as a Cauchy’s problem.

RemarkWith respect to (8), this problem (IVP) is equivalent to find a solution x(t)such that:

dx(t)dt

= f (t,x(t))

x(t0) = x0

(9)



Definition 10 (Local solution)

A local solution to Initial Value Problem (9) is a solution to problem (8) such that Tφ

is a neighborhood of t0, and φ(t0) = x0.

Definition 11 (Extension)

Given two solutions(φ,Tφ

)and

(ψ,Tψ

), solution

(ψ,Tψ

)is called an extension of(

φ,Tφ

)if Tψ ⊃ Tφ , and ∀t ∈ Tφ,ψ(t) = φ(t)

Definition 12 (Maximal solution)

A local solution(φ,Tφ

)is said to be the maximal solution to problem (8) if this

solution can not be extended.

in other words: Tφ is the (maximal interval) such that φ is a local solution to the IVP.

TheoremAny local solution admits one maximal solution.

Definition 13 (Global solution)

A solution (φ,Tφ) is a global solution to IVP (9) if it is a local solution and Tφ = T

Remark : A global solution is also a maximal solution (as having no extension)Laurent LAVAL ISCDE 2011 – Dynamical systems and chaos : Some theoretical preliminaries 39 / 64


Remark : Regarding to IVP (9), a solution x(t) = xt to diff. equation (8) – assumingthis solution exists – is defined by:∫ xt

x0

1f (ζ)

dζ = t− t0, t, t0 ∈ T (10)

where f is assumed to be continuous and invertible on its domain

Definition 14 (Integral curve)

An integral curve is a (specific) solution to an ordinary differential equation or systemof equations .

Remarks and terminologyIf the differential equation is represented as a vector field, then the correspond-ing integral curves are tangent to the field at each point.

Integral curves are known by various other names, depending on the natureand interpretation of the differential equation or vector field. In dynamical sys-tems theory, the integral curves are referred to as trajectories or orbits.



Autonomous and nonautonomous systems

For autonomous differential equations (or systems) with time as independentvariable, the trajectories (orbits) are independent from initial time t0. Contrarily,such a dependence exists for nonautonomous diff. equations.

Example 1Consider the system defined by the following set of autonomous differential equations

dx(t)dt

= y(t),

dy(t)dt

=−x(t),

and initial conditions: x(t0) = 1, y(t0) = 0

The (global) solution to this IVP is given by: x(t) = cos(t− t0), y(t) =−sin(t− t0).The system trajectory (orbit) in the phase plane (y vs x) is then a circle defined byrelation: x2(t) + y2(t) = 1, and is passing through point (1,0) for any initial time t0.



Example 2Let us now consider a nonautonomous system given by

dx(t)dt

= y(t)

dy(t)dt

=−x(t) + t ← explicit dependence on the independent variable t

and initial conditions: x(t0) = 1, y(t0) = 0

In such a case, the system trajectory (orbit) depends on the initial time

If t0 = π/2, the solution is given by:

x(t) = cos(t)+(1− π

2)sin(t)+ t, y(t) =−sin(t)+(1− π

2)cos(t)+1

If t0 = π, the solution is now:

x(t) = (π−1)cos(t)+ sin(t)+ t, y(t) =−(π−1)sin(t)+ cos(t)+1

Finally, if t0 = 2π then:

x(t) = (1−2π)cos(t)− sin(t)+ t, y(t) =−(1−2π)sin(t)− cos(t)+1

All the orbits pass through the initial point but they are all different



Transformations of differential equations

Using very weak assumptions (“separability”, ...), any nonautonomous differen-tial equation can be expressed as an autonomous one.

Example

Consider a nonautonomous differential equation:.x = f (t,x(t)).

By posing,Z =

[xt

]← (state vector transformation)

we obtain a (new) autonomous differential equation:.Z = F(Z ) with F(Z ) =

[f (Z )

1

](Attention !! may be not bounded, excepted for time-periodic function f (t,x))

Also true for a diff. equation depending on a parameter: x = f (x ,λ), λ ∈ Rm



Any one-dimensional equation of order n (x(n) = f (t,x , x , ...,x(n−1)) is equiva-lent to an n−dimensional first-order system (a set of first-order differential equa-tions).

ExampleThe second-order, autonomous, Van der Pol equation

d2y(t)dt2 − ε

(1− y2(t)

) dy(t)dt

+ y(t) = 0

is equivalent to dy1(t)

dt= y2

dy2(t)dt

= −y1(t) + ε(1− y2

1 (t))

y2(t)



Existence and uniqueness of solutions

Recall the IVP problem .x = f (t,x) ∀t ∈ T ⊂ Rx(t0) = x0

(11)

with f : Ω⊂ R×Rn→ Rn, (t0,x0) ∈ Ω

2 questionsIs there, at least, a solution ?Is this solution unique ?

Existence

Theorem(Cauchy-Peano) If f is a continuous function defined on U an open subset of Ω =T ×Rn then there exists a neighborhood J of t0 in T and a function φ ∈ C1(J,Rn)such that: ∀t ∈ J,

.φ(t) = f (t,φ(t)) , φ(t0) = x0



Carathéodory’s existence theoremCarathéodory’s theorem extends Peano’s one by showing existence of solutions (ina more general sense) for some differential equations with, possibly, discontinuousright-hand sideConsider the following IVP problem .

x = f (t,x) ∀t ∈ T ⊂ Rx(0) = 0 ← the initial condition

(12)

where f is defined on a rectangular domain of the form,

Ω = (t,x) ∈ R×Rn : ‖t− t0‖ ≤ a,‖x− x0‖ ≤ b

and f is the Heaviside step function defined by

H(t) =

0, if t ≤ 0,1, if t > 0



Regarding to Peano’s theorem, there is no solution as f is not differentiable at t = 0However, it makes sense to consider the ramp function

x(t) =∫ t

0H(s)ds =

0, if t ≤ 0;t, if t > 0

as a solution of the differential equation.

Carathéodory idea: To allow for solutions that are not everywhere differentiable byconsidering the following definition

Definition 15 (Solution in the extended sense)

A function x is called a solution in the extended sense of the differential equationx = f (t,x) with initial condition x(t0) = x0 if x is absolutely continuousa, x satisfiesthe differential equation almost everywhere and x satisfies the initial condition.

aThe absolute continuity of x implies that its derivative exists almost everywhere.



Theorem(Carathéodory’s existence theorem)Consider the differential equation

x(t) = f (t,x(t)), x(t0) = x0,

with f on the rectangular domain Ω. If f satisfies the three conditions

the function f (•,x) is measurable for every x with ‖x− x0‖ ≤ b

the function f (t,•) is continuous for every t with ‖t− t0‖ ≤ a,

there is a Lebesgue-integrable function m on [t0−a, t0 + a] such that

‖f (t,x)‖ ≤m(t) ∀(t,x) ∈ Ω

then the differential equation has a solution in the extended sense in a neighborhoodof the initial condition.



Uniqueness

Existence of a local solution but what about uniqueness ?

Consider the following example:

dxdt

= 3x2/3, and domain Ω = R×R

According to Peano’s theorem, there exist at least one (local) solution as, for instance,φ : R → R

t 7→ (t− c)3

But, for a given initial condition (t0,x0) there exists many solutions.(FIGURE)



Definition 16 (Lipschitz continuity)

Given two metric spaces (U,dU) and (V ,dV ), where dU (resp. dV ) denotes the metricon set U (resp. on set V ).A function,

f : U→ V

is called Lipschitz continuous if there exists a real constant K ≥ 0 such that, for allx1 and x2 in U,

dV (f (x1), f (x2))≤ K dU (x1,x2)

If 0≤ K < 1 the function is called a contraction.

Definition 17 (Lipschitz continuity – metric space (Rn,‖•‖) )

Consider a function f : Ω⊂ R×Rn→ Rn and a metric space (Rn,‖•‖), f is said tobe k−Lipschitz continuous in Ω for x if:

∀(t,x1),(t,x2) ∈ Ω, ‖f (t,x1)− f (t,x2)‖ ≤ k ‖x1− x2‖ k ∈ R+



Considering the IVP (11)

Theorem(Cauchy-Lipschitz theorem) (Picard-Lindelöf theorem) If f : Ω ⊂ R×R→ Rn iscontinuous (∈ C0(Ω,Rn)), k-lipschitz in x within the cylinder D = Ta×Br with:

Ta = t ∈ R, |t− t0| ≤ a ,Br = x ∈ Rn,‖x− x0‖ ≤ r, and bounded in D, then there exists a unique solution Sφ to the initial value problemwith Tφ = t ∈ R, |t− t0| ≤ α and α = min

(a, r

k

)Theorem(shortened version) Consider the initial value problem

x(t) = f (t,x(t)), x(t0) = x0, t ∈ [t0− ε, t + ε]

Suppose f is Lipschitz continuous in x and continuous in t. Then, for some valueε > 0, there exists a unique solution x(t) to the initial value problem within the range[t0− ε, t0 + ε].

(there exist many interesting corollaries)



Dependence on initial conditions and/or parameters

Problem statementConsider the following IVP

dx(t)dt

= f (t,x(t),λ) ← λ is a system parameter

x(t0) = x0, λ = λ0

(13)

Assume that solution φ(t, t0,x0,λ0) exists and is unique. We wish to study the depen-dence (sensitivity) of that solution on t0, x0 and λ0 → focus on

φ(t, t0 + ∆t0,x0 + ∆x0,λ0 + ∆λ0) (14)

where ∆• represents small variations of quantity • around the nominal value.(See Generalized Gronwall’s inequality (Gronwall’s lemma) for depen-dence on the initial condition.)

Assuming that solution (14) is continuous and differentiable (there exist some theo-rems to prove it), there exist several approaches: variational methods, regular pertur-bation theory, ...

A usual (deterministic) approach To linearizeLaurent LAVAL ISCDE 2011 – Dynamical systems and chaos : Some theoretical preliminaries 52 / 64


φ(t, t0 + ∆t0,x0 + ∆x0,λ0 + ∆λ0) ≈ φ(t, t0,x0,λ0) +∂φ

∂t0(t, t0,x0,λ0)︸︷︷︸

A

∆t0

+Jx φ(t, t0,x0,λ0)︸︷︷︸B

∆x0

+Jλφ(t, t0,x0,λ0)︸︷︷︸C

∆λ0 + · · ·

where J• denotes the Jacobian matrix, and A, B, C are the sensitivity terms.

A: Sensitivity to initial time variations

B: Sensitivity to trajectory variations→ focus on eigenvalues of Jx

C: Sensitivity to parameter variations→ bifurcations→ focus on eigenvalues ofJλ

...

To be continued



A simple exampleConsidering the Van der Pol equation

d2x(t)dt2 − ε

(1− x2(t)

) dx(t)dt

+ x(t) = 0

with initial conditions: x(0) = x0, x(0) = x0 and ε = ε0.A simple transformation leads to

y1(t) = y2

y2(t) = −y1(t)ε(1− y2

1 (t))

y2(t)

By Posing Y =

[y1

y2

], the IVP problem becomes Y = F(Y ) with

F(Y ) =

[0 1−1 ε(1− x2)

]︸︷︷︸

J

Y with initial conditionY (0) = Y0 =

[x0

x0

],ε0

The trajectory depends on eigenvalues of J which itself depends on parameter ε



TrajectoriesConsider a map F as a Cr -diffeomorphism which represents the dynamics of thediscrete-time system xk+1 = F(xk ) = ΦD(k ,x), x ∈ X (the state space)

Consider a function f as a Cr -diffeomorphism which represents the dynamics of thecontinuous-time system x(t) = f (x(t)) = ΦC(t,x(t)), x ∈ X

Definition 18 (Fixed point)

x∗ ∈ X is a fixed point for F (resp. f ) if and only if F(x∗) = x∗ (resp. f (x∗(t)) = x∗(t))

Also called: stationary point, equilibrium point, singular point, ...

Definition 19 (Periodic point))

x∗ ∈ X is a periodic point (in time) for F (resp. f ) if and only if there exists τ ∈ N(resp. τ∈R+) such that F τ(x∗) = x∗ and F(x∗) 6= x∗ (resp. f (x∗(t +τ)) = x∗(t) andf (x∗(t)) 6= x∗(t) . The smallest τ which verifies such an equality is called the primeperiod of x∗.

A periodic point of a map (resp. a function) is a point which the system returns toafter a certain number of map iterations (resp. a certain amount of time).



Definition 20 (Invariant set)

A set Λ ⊂ Rn is said to be invariant for the map ΦD (resp. the flow φC ) if ∀x ∈ Λ,∀k ∈ Z (resp. ∀t ∈ R), ΦD(k ,x)⊂ Λ (resp. ΦC(t,x)⊂ Λ).

This means that the orbits, starting from x ∈ Λ remain in Λ

Definition 21 (Positively invariant set)

A set Λ is said to be positively invariant if:

∀k ∈ Z+,ΦD(k ,Λ)⊂ Λ (discrete-time case)

∀t ∈ R+,ΦC(t,Λ)⊂ Λ (continuous-time case)

That are invariant sets with set of times restricted to positive values.



In the study of dynamical systems, a limit set is the state a dynamical systemreaches after an infinite amount of time has passed, by either going forward orbackwards in time. Limit sets are then important because they can be used tounderstand the long term behavior of a dynamical system.

Definition 22 (ω/α – Limit point)

A point x∗ ∈ X is an α−limit point of the orbit of ΦD (resp. ΦC ) passing through x ifthere exists a sequence (ki )i∈N→−∞ (resp. a time t →−∞) such that:

limi→+∞

ΦD(ki ,x) = x∗ (resp. limt→+∞

ΦC(t,x) = x∗)

(x is generally the initial condition x0)

A point x∗ ∈ X is an ω−limit point of the orbit of ΦD (resp. ΦC ) passing through x ifthere exists a sequence (ki )i∈N→+∞ (resp. a time t →+∞) such that:

limi→+∞

ΦD(ki ,x) = x∗ (resp. limt→+∞

ΦC(t,x) = x∗)

In general limits sets can be very complicated as in the case of chaotic attractors



Let us denote O and orbit passing through point x

Definition 23 (α/ω – limit set)

The set of all α−limit points of x (i.e. for a given orbit O) is the α− limit set ofx (i.e. for O), denoted Lα (x) (or Lα (O))

The set of all ω−limit points of x (i.e. for a given orbit O) is the ω− limit set ofx (i.e. for O), denoted Lω (x) (or Lω (O))

Remark: These sets are invariant sets

Definition 24 (Limit set)

Let us denote,

Lα (E): the set of all α− limit sets of points of a domain E

Lω (E): the set of all ω−limit sets of points of a domain E

The set of all (α or ω) limit points of a domain E , Lα (E) ∪ Lω (E) is called the limitset of E .



Some propertiesA solution (trajectory / orbit) is divergent in the positive direction (i.e. for increas-ing t or k ) if this orbit has no ω− limit point

A trajectory (orbit) is said to be asymptotic in the positive direction if there existssome ω− limit point which are excluded from this solution

A trajectory (orbit) is stable in the positive direction (in the sense of Poisson andLyapunov) if there exit some ω – limit points belonging to this solution

Any fixed point (i.e. stationary point) is itself its α and ω limit point

Any point of a periodic solution is a α and ω limit point; Such a solution is thenstable (in the sense of Poisson and Lyapunov).

Definition 25 (Asymptotic stability of limit sets)

A limit set L is asymptotically stable if there exists an open neighborhood ν of Lsuch that the ω− limit set of all points in ν is L



Limit cyclesFirst, let us recall

The orbit of a periodic solution x (x0, t) is a closed curve within the phase space, andx (ti + τ,x0) = x (ti ,x0) (where τ is the period)

Definition 26 (Limit cycle)

A limit cycle is a closed orbit γ such that

∃x /∈ γ, γ⊂ Lα(x) or γ⊂ Lω(x)

In other words, if the ω− limit set (resp. α− limit set) is disjunct from the orbit γ, thatis Lω(γ)∩ γ = (resp. Lω(γ)∩ γ =), we call Lω(γ) (resp. Lα(γ) is a ω− limitcycle (resp. ω− limit cycle)

Another interesting definitionDefinition 27 (Limit cycle)

A periodic solution is a limit cycle if this solution is the α (or ω) limit set of anothersolution for the system.



Example

Fig. 14: Phase portrait of a Van der Pol equation



Attractors and basins of attraction

An attractor is a set towards which a dynamical system evolves over time.

Geometrically, an attractor can be a point, a curve, a manifold, or even a compli-cated set with a fractal structure known as a chaotic attractor.

A trajectory of the dynamical system in the attractor does not have to satisfy anyspecial constraints except for remaining on the attractor.

Definition 28 (Attractors – Repeller – Basin of attraction)

An attractor is an asymptotically stable ω−− limit set

A repeller (or repellor) is an unstable α−− limit set

A basin of attraction for an attractor A is the set of all points whose the ω−−limit set ⊂ A

In other words, a basin of attraction is the set of initial conditions leading to long-timebehavior that approaches that attractor



Chaotic attractor

DefinitionIf X is a metric space, f : X → X is sensitive to initial conditions if

Discrete-time case: ∃λ > 0 such that ∀ε > 0, ∃n ∈ N,∀x0 ∈ X such thatd(x0,x)≤ ε and d(f n(x0), f n(x))≥ λ with n independent from x0

Continuous-time case: ∃λ > 0 such that ∀ε > 0, ∃t ∈ R+,∀x0 ∈ X such thatd(x0,x)≤ ε and d(ΦC(t,x0),ΦC(t,x))≥ λ with t independent from x0

Tentative definition (Chaotic attractor)

A chaotic attractor is an attractor such that:

There exist some (unstable) periodic trajectories of all periods τi , i ∈ N withcontinuous case: τi=1...∞ = i ∗ τf , where τf is the basic/fundamental perioddiscrete case: τi=1...∞ = i

it is sensitive to initial conditions

the sum of Lyapunov exponents is negative (→ dissipativity) and, at least, oneLyapunov exponent is positive.



J. Bernstein, A theory for everything, Copernicus Series, Copernicus, 1996.

Michel Hénon, A two-dimensional mapping with a strange attractor, Communi-cations in Mathematical Physics 50 (1976), 69–77.

Nick Huggett, Zenos paradoxes, The Stanford Encyclopedia of Philosophy (Ed-ward N. Zalta, ed.), winter 2010 ed., 2010.


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Dynamical systems and chaos : Some theoretical ...Dynamical systems and chaos : Some theoretical preliminaries From trajectories to trajectories Laurent LAVAL [email protected]