Nonlinear Dynamical Systems Lecture - 01

Transcription

1 Nonlinear Dynamical Systems Lecture - 01 Alexandre Nolasco de Carvalho August 08, 2017

2 Presentation Course contents Aims and purpose of the course Bibliography

3 Motivation To explain what is a dynamical system (or semigroup) imagine that we wish to predict the value of the variable x at future times and denote the space of all possible values of this variable by X. The variable x may describe a large number of quantities from many areas of knowledge (physics, biology, economics, etc.). As examples we mention the position of a body in the space (a vector in R 3 ) or a temperature distribution in a body Ω (a function defined in Ω and taking values in R). Consequently, the space X may be finite of infinite dimensional.

4 A dynamical system is a family of operators {S(t, s) : t s}, from X into itself in such a way that, given that the value of the variable at time s is x, S(t, s)x denotes the value of the variable at a later time t. The knowledge of the dynamical system allow us to predict, at any future time, the values of the variable that we know at the present time, for each possible value of the variable x in X. Clearly, this family of operators must obey certain compatibility conditions, which are: i) S(t, t) = I X, where I X is the identity in X, and ii) S(t, s)s(s, τ)x = S(t, τ)x for all t s τ and x X.

5 If t, s are taken in Z, we say that {S(t, s) : t s} is a discrete dynamical system and if t, s are taken in R we say that it is a continuous dynamical system. To fix ideas let us imagine that we are working with a continuous dynamical system. The space X, in general, is a metric space and we will denote its metric by dist(, ) : X X [0, ). In general we also ask that it satisfies a continuity property; that is, iii) {(t, s, x) R R X : t s} (t, s, x) S(t, s)x X is continuous.

6 Definition Let X be a metric space and C(X ) the space of continous maps from X into itself. A dynamical system on X is a family {S(t, s) : t s} C(X ) such that i) S(t, t)x = x for all t R and x X, ii) S(t, σ) S(σ, s) = S(t, s) for all t σ s, iii) {(t, s, x) R R X : t s} (t, s, x) S(t, s)x X is continuous.

7 We will distinguish two main types of dynamical systems. The autonomous dynamical systems are those that satisfy S(t, s) = S(t s, 0), for all t s. The remaining evolution processes will be called non-autonomous. Note that, for an autonomous evolution process, the evolution of the state x, from the initial time s to the state S(t + s, s)x at time t + s, is independent of the initial time s and depends only elapsed time t.

8 For autonomous evolution processes we define T (t) = S(t, 0) C(X ) and the family {T (t): t 0} satisfies (i ) T (0)x = x, for all x X, (ii ) T (t)t (s) = T (t + s) for all t, s 0 and (iii ) the map [0, ) X (t, x) T (t)x X is continuous. A family {T (t): t 0} with the above properties is called a semigroup. Clearly, from a semigroup {T (t): t 0} we can define an autonomous evolution process {S(t, s): t s} making S(t, s) = T (t s), t s. We will use both terms indifferently.

9 Dynamical systems frequently appear associated to ordinary, partial or functional (evolving) differential equations. To fix our ideas, consider the Cauchy problem: dx + Ax = f (t, x), t > s dt, (1) x(s) = x 0 X where X is a Banach space and A is a linear operator (possibly unbounded) and f is a nonlinear forcing term dominated by A. Here are all ODE s, the semilinear parabolic equations, the Navier-Stokes equations, the wave equations, the Cahn-Hilliard equation, the Schrödinger equation, and many others.

10 Assume that for each s R and x 0 X there is a unique continuous function x(, s, x 0 ): [s, ) X which is the weak solution (related to the way it satisfies the equation) of (1). Assume also that the map P X (t, s, u 0 ) x(t, s, x 0 ) X is continuous, where P = {(t, s) R 2 : t s}. In this case if (t, s) P, we define S(t, s) C(X ) by S(t, s)x 0 = x(t, s, x 0 ), and the operator S(t, s) evolves the phase space X from the initial time s to the later time t and the properties of an evolution process are satisfied.

11 If f is independent of t, the evolution depends of a single vector field (whereas if depends on t there are infinitely many of them - one for each instant of time). Hence, the evolution of an initial state x depends only on the elapsed time t s and not particularly on the initial and final times, s and t. Hence S(t, s)x = S(t s, 0)x, (t, s) P and x X, and the equation generates a semigroup.

12 Example Consider the initial value problem ẋ = f (t, x) x(s) = x s R n (2) where f : R R n R n is a continuously differentiable function. Under these conditions, the following holds

13 Theorem For each x s R n and s R, there exists τ > s and continuously differentialbe function [s, τ) t ξ(t) R n such that ξ(s) = x s and ξ(t) = f (t, ξ(t)) for all t (s, τ). This function is called a solution of the initial value problem (2) and has the following properties If there is a σ > s and continuously differentiable function η : [s, σ) R n such that η(s) = x s and η(t) = f (t, η(t)) for all t (s, σ) then ξ(t) = η(t) for all t [0, min{σ, τ}) There is a τ(s, x s ) > s and a solution [s, τ s,xs ) t x(t, s, x s ) R n of (2) such that either τ(s, x s ) = or lim sup t τ(s,xs) x(t, s, x s ) =. If E = {(t, s, x) R R R n : s t τ(s, x)}, the function E (t, s, x s ) x(t, s, x s ) R n is continuous.

14 Denote by x(t, s, x s ), t [s, τ(s, x s )) the unique solution of (2). Assume that threre is a constant M > 0 such that f (t, x) x < 0, whenever x M, and for all t R. (3) Then, if x(t, s, x s ) is a solution of (2), d dt x 2 = 2f (t, x) x. It follows from (3) and of Theorem 2 that τ(s, x s ) = for all s R and x s R n.

15 Under these conditions define S(t, s) : R n R n, t s, by S(t, s)x s = x(t, s, x s ), x s R n. It is clear, in view of Theorem 2, that the only condition we need to verify to conclude that {S(t, s) : t s} is a dynamical system (with X = R n ) is the condition ii) of Definition 1. This condition follows from the uniqueness of solutions of (2) given in Theorem 2 noting that x(t, σ, x(σ, s, x s )) e x(t, s, x s ) are both solutions of ẋ = f (t, x) x(σ) = x(σ, s, x s ) R n.

16 If f (t, x) = g(x) for all x R n ; that is, f is independent of t, then t x(t + τ, τ, x s ) and t x(t + σ, σ, x s ) are both solutions of ẋ = g(x) x(0) = x s R n. In this case {T (t) : t 0} given by T (t)x 0 = x(t, 0, x 0 ) is a semigroup (with X = R n ). Exercise Explain why the above reasoning cannot be extended to the case when f is time dependent?

17 Now that we understand what is a dynamical system we will try to determine what will seek to understand from them. Firstly we will insist that the dynamical systems studied here include models that are more general then those from ordinary differential equations, they may even come from other fields. Hence, assume that X is a metric space or, in situations for which the vector space structure is required, a Banach space (finite or infinite dimensional). Secondly, given the nature of our space of states X, we wish to study those dynamical systems which are dissipative (in the above example the solutions eventually enter the bal or radius M).

18 A set in which all solutions enter (uniformly in bounded subsets of X ) after a given time is called absorbing set. We will consider those dynamical systems which possess an absorbing set. Among those, we consider the ones for which the intersection of all absorbing sets is a compact set (called attractor). The states in the attractor are called asymptotic states and the states outside the attractor are transient states. Concerning attractors, we will study conditions for its existence, to determine what is inside it, to understand how the bounded sets are attracted to it, to determine how complex it is and how it behaves under perturbation.

19 Let X be a metric space and d : X X [0, ) its metric. Denote by C(X ) the set of continous maps from X into itself. We will write T to denote Z or R, T + = {t T : t 0}, T = {t T : t 0}, T t = t + T and T + t = t + T +. Given K X e r > 0, the r neighbourhood of K is the set defined by O r (K) := {x X : d(x, K) < r}.

20 Definition A semigroup is a family {T (t) : t T + } C(X ) such that T (0)x = x for all x X, T (t + s) = T (t) T (s), [0, ) X (t, x) T (t)x X is continuous. When T = Z the third condition is automatically satisfied and, since T (n) = T (1) n, writing T := T (1), the discrete semigroups may be written in the form {T n : n N} C(X ).

21 For a semigroup {T (t) : t T + }, a point x X and a subset B X, define: For t T, the image of B under T (t), T (t)b :={T (t)x :x B}; The positive orbit of B, γ + (B):= t T + T (t)b; The partial orbit between to numbers of T +, t < t, γ + [t,t ] (B) := t s t T (s)b; The orbit of T (t)b, γ t + (B) := T (s + t)b = T (s)b. s T + s T + t The function T + t T (t)x X is a solution through x of the semigroup {T (t) : t T + }. In the case T + = N, the solution through x satisfy, for T := T (1), the discrete initial value problem x n+1 = Tx n, n N, (4) x 0 = x.

22 Definition A semigroup {T (t) : t T + } is said eventually bounded if for each bounded set B X there exists t B T + such that γ + t B (B) is bounded. We say that {T (t) : t T + } is bounded if γ + (B) is bounded whenever B is bounded.

23 Remark It does not follows from the fact that T C(X ) that T takes bounded subsets of X into bounded subsets of X, due to the fact that bounded subsets of X are not necessarily relatively compact. Hence, the semigroup {T (t) : t T + } is not necessarily bounded. To obtain that {T (t) : t T + } is bounded, we need to assume that {T (t) : t T + } is eventually bounded and that γ + [0,T B ](B) is bounded for each T B T + and for all bounded subset B of X. If we assume that T = Z and T = T (1) takes bounded subsets into bounded subsets, considering the semigroup {T n : n N}, we have that γ + [n,n ](B) is bounded for each bounded subset B of X and n n N.