Perturbations on autonomous and non-autonomous systems Francisco - PowerPoint PPT Presentation

Perturbations on autonomous and non-autonomous systems Francisco Balibrea balibrea@um.es Departamento de Matem´ aticas Universidad de Murcia (Spain) ICDEA, Barcelona. July 2012

Two introductory examples x n +1 = ax n where a > 0 x n +1 = a + x n x n − 1 where also a > 0 2 / 39

perturbations If in both equations we perturb the parameters x n +1 = ( a + p n ) x n x n +1 = ( a + p n ) + x n x n − 1 we obtain non-autonomous systems which can be formulated by x n +1 = f n ( x n ) 3 / 39

Non-autonomous discrete systems (n.a.d.s.) That is by ( X , f ∞ ) where f ∞ = ( f n ) ∞ n =0 and f n ∈ C ( X , X ) for all n ( X , f ∞ ) is called a non-autonoumous discrete system ( n.a.d.s. ) 4 / 39

(n.a.d.s.) We use the notation f n i = f i +( n − 1) ◦ f i +( n − 2) ◦ .... f i +2 ◦ f i +1 ◦ f i with i ≥ 0 , n > 0 and f 0 i = Identity on X and Tr f ∞ ( x 0 ) = ( f n 0 ) ∞ n =0 = ( x n ) ∞ n =0 5 / 39

(n.a.d.s.) We are dealing with the stability or instability in the Lyapunov sense of such systems 6 / 39

Lyapunov exponents for autonomous systems They were introduced by Aleksandr Lyapunov in 1892 in his Doctoral Memoir: The general problem of the stability of motion It is a extended practice, especially in experimental and applied dynamics, to associate the idea of orbits having a positive Lyapunov exponent with instability and negative Lyapunov exponent with stability of orbits in dynamical system. Stability and instability of orbits are defined in topological terms while Lyapunov exponents is a numerical characteristic calculated all along the orbit 7 / 39

definition of Lyapunov exponents Definition Let f : R → R be a C 1 -map. For each point x 0 the Lyapunov exponent of x 0 , λ ( x 0 ) is n − 1 1 1 � n log( | ( f n ) ′ ( x 0 ) | ) = lim n →∞ log( | f ′ ( x j ) | ) λ ( x 0 ) = lim n →∞ n j =0 where x j = f j ( x 0 ) (if the limit exists). 8 / 39

stability and instability in Lyapunov sense Definition The forward trajectory Tr f ( x 0 ) is said to be Lyapunov stable if for any ǫ > 0 there is δ > 0 such that whenever | y − x 0 | < δ is | f n ( y ) − f n ( x 0 ) | < ǫ for all n ≥ 0. 9 / 39

stability and instability in Lyapunov sense Lyapunov instability is equivalent to sensitivity dependence on initial conditions (sdic) Definition Tr f ( x 0 ) exhibits (sdic), if there exists ǫ > 0 such that given any δ > 0 there is y holding | y − x 0 | < δ and N > 0 such that | f n ( y ) − f n ( x 0 ) | ≥ ǫ for all n ≥ N 10 / 39

stability and instability in Lyapunov sense In the following examples, we consider the trajectories of 0 of two maps and obtain that we can have instability trajectories with negative Lyapunov exponents and stable trajectories with positive Lyapunov exponents 11 / 39

Map f introduced by Demir and Ko¸ cak 1.0 0.8 0.6 f ( x ) 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 x Figure: Map f 12 / 39

Map g introduced by Demir and Ko¸ cak 1.0 0.8 0.6 f ( x ) 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 x Figure: Map g 13 / 39

The strong Lyapunov exponent is 1 n Σ k + n − 1 log( | f ′ ( x j ) | ) Φ( x ) = lim n →∞ j = k if this limit exists uniformly with respect to k ≥ 0 14 / 39

Results: 1 Let f ∈ C 1 ( I ). If the forward trajectory of x ∈ I has positive strong Lyapunov exponent, then the orbit has (sdic) 2 Let f ∈ C 1 ([0 , 1)). If the forward trajectory of x ∈ [0 , 1) has negative strong Lyapunov exponent, then the orbit is Lyapunov stable 15 / 39

BC used the notion of Lyapunov exponents for non-autonomous systems on R and C 1 − maps for the difference equation x n +1 = a n x n as an immediate extension of the formula to calculate the Lyapunov exponents in the autonomous case (if the limit exists) as n − 1 1 1 � n log | ( f n 0 ) ′ ( x ) | = lim n →∞ log | f ′ λ ( x ) = lim n →∞ j ( x j ) | n j =0 where x j = f j 0 ( x ) 16 / 39

Lyapunov exponents for the non-autonomous case They considered the case when a n = a + p ( n ) where p n = [ a + ǫ ( b n + β c n ) holding a > 1 but closed to 1, 0 < β > 1 and √ b n = 2 sin n √ c n = 2 sn [2 K ( m )( n + Θ) /π ; m ] with ǫ > 0 and m the modulus of the elipticity of senam map 17 / 39

Lyapunov exponents for the non-autonomous case The Lyapunov exponent has the following values: 1 If β = 0, then if log a > 1 2( ǫ a ) 2 then the system has for all initial conditions on (0 , ∞ ) constant positive Lyapunov exponents and has (dsic) 2 If β � = 0, then for fixed modulus m and in some range of Θ, the system has also constant positive Lyaunov exponents. Also it is proved the system has (dsci) 18 / 39

Stability and instability of orbits in periodic non-autonomous systems Take a periodic block composed of the maps f and g , { f 0 , f 1 , ..., f m − 1 } where p < m of them are the map f and the rest g and consider the non-autonomous periodic system of period m = p + q where f i = f for i = 0 , 1 , ..., p − 1 and f j = g for j = p , ..., m − 1 If we compute the Lyapunov exponent 0 of such periodic non-autonomous system we have for the f n 0 map For n = km + 1 is k ( p − q ) + 1 log2 km + 1 n = km + 2 is k ( p − q ) + 2 log2 km + 2 ... n = km + p is k ( p − q ) + p log2 km + p ... n = ( k + 1) m is ( k + 1)( p − q )log2 19 / 39

Stability and instability of orbits in periodic non-autonomous systems When n → ∞ , the Lyapunov exponent of the trajectory of 0 is λ (0) = p − q log2 m 20 / 39

Stability and instability of orbits in non-periodic non-autonomous systems When we choose a non-periodic block of maps f and g the orbit of 0 continues being instable if the map g appears infinite times Theorem (BC) Let f ∞ a non-periodic sequence of maps f and g. If the map g appears infinite times, then the trajectory of 0 is Lyapunov instable. 21 / 39

Let X ⊂ R m and d any metric on it. If ( x n ) ∞ n =0 and ( x ′ n ) ∞ n =0 are two trajectories starting from nearby initial states x 0 and x ′ 0 and write δ x n = x ′ n − x n . If f has continuous partial derivatives in every x i , then, iterating the map, we have the linear approximation ( DF ( x ) denotes the differential of the map F : R n → R n at the point x ). n − 1 � δ x n ≃ Df n ( x 0 ) δ x 0 = ( Df ( x i ) δ x 0 i =0 where the ( i , j ) element of the matrix Df ( x ) is given by ∂ f i ∂ x j and where f i and x j are the components of f and x in local coordinates on X 22 / 39

Given a matrix A , we denote by A t the transpose of A . Let the matrix ( Df n ( x 0 ) t )( Df n ( x 0 )) where Df n ( x 0 ) = Df ( x n − 1 )( Df ( x n − 2 ) ... ( Df ( x 1 ) Df ( x 0 ) have eigenvalues in x 0 given by µ i ( n , x 0 ), for i = 1 , 2 , ..., m such that µ 1 ( n , x 0 ) ≥ µ 2 ( n , x 0 ) ≥ ... ≥ µ m ( n , x 0 ). Then the i th local Lyapunov exponent at x 0 is defined by: 1 λ i ( x 0 ) = lim 2 n log( | µ i ( n , x 0 ) | ) n →∞ if this limit exists. In [ ? ] it is possible to state conditions for the existence of such limit. Now we recall the notions of instability and stability in the Lyapunov sense 23 / 39

Markus-Lyapunov Fractal We consider the logistic equation x n +1 = r n (1 − x n ) and the sequence of blocks BBBBB .... where B = 112112 .... and 112 = r 1 r 1 r 2 24 / 39

Markus-Lyapunov Fractal 2.jpg Figure: Fractal Markus-Lyapunov 25 / 39

Markus-Lyapunov Fractal 3.jpg Figure: Fractal Markus-Lyapunov 26 / 39

We propose two dynamical systems, one defined in [0 , 1] 2 which has a forward trajectory with a positive Lyapunov exponent but not having sensitive dependence on initial conditions and other defined in [0 , 1) 2 which has a forward trajectory with a negative Lyapunov exponent but having sensitive dependence on initial conditions. The examples are two dimensional versions of those mentioned in the introduction. The maps we are using are examples of permutation maps. 27 / 39

Example We are going to obtain a continuous function F = ( f , g ) in [0 , 1] 2 such that the forward trajectory of (0 , 0) has a positive Lyapunov exponent, but has not has no sensitive dependence on initial conditions. 28 / 39

a) The map f : [0 , 1] → [0 , 1] was introduced in [ ? ] 1  2 x − 1 + a n < x ≤ b n , x ∈  2 n +1      f ( x ) = 5 n +2 − 22 2 1 2 · 5 n +2 − 11( x − b n ) + 1 + 10 n +1 − b n < x ≤ a n +1   2 n +1    1 x = 1  with a n = 1 − 2 − n − 10 − n − 1 , b n = 1 − 2 − n + 10 − n − 1 , n = 0 , 1 , 2 , ... . b) Now we define another map g : [0 , 1] → [0 , 1] 3 x + 1 0 ≤ x ≤ 1    2 15       127 x + 7 6 2 15 < x ≤ 1 1 1   10 − 2 −    635 100       3 x + 1 5 2 n +1 (2 n − 1) g ( x ) = 2 − a n < x ≤ b n      5 n +2 − 33  3 1    2 · 5 n +2 − 11( x − b n ) + 1 + 10 n +1 − b n < x ≤ a n +1   2 n +1   29 / 39    