Perturbations on autonomous and non-autonomous Two introductory examples systems
Francisco Balibrea
[email protected] Departamento de Matem´aticas xn+1 = axn Universidad de Murcia (Spain) where a > 0
a + xn xn+1 = xn 1 − where also a > 0
ICDEA, Barcelona. July 2012
2 / 39 perturbations Non-autonomous discrete systems (n.a.d.s.)
If in both equations we perturb the parameters
xn+1 = (a + pn)xn
That is by (X , f ) where f = (fn)n∞=0 and fn C(X , X ) for all n ∞ ∞ ∈ (a + pn) + xn (X , f ) is called a non-autonoumous discrete system (n.a.d.s.) xn+1 = ∞ xn 1 − we obtain non-autonomous systems which can be formulated by
xn+1 = fn(xn)
3 / 39 4 / 39 (n.a.d.s.) (n.a.d.s.)
We use the notation
n We are dealing with the stability or instability in the Lyapunov sense of fi = fi+(n 1) fi+(n 2) ....fi+2 fi+1 fi − ◦ − ◦ ◦ ◦ such systems 0 with i 0 , n > 0 and fi = Identity on X and ≥ n Trf ∞ (x0) = (f0 )n∞=0 = (xn)n∞=0
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Lyapunov exponents for autonomous systems definition of Lyapunov exponents
They were introduced by Aleksandr Lyapunov in 1892 in his Doctoral Definition Memoir: The general problem of the stability of motion 1 Let f : R R be a C -map. For each point x0 the Lyapunov exponent of → It is a extended practice, especially in experimental and applied dynamics, x0, λ(x0) is to associate the idea of orbits having a positive Lyapunov exponent with n 1 instability and negative Lyapunov exponent with stability of orbits in 1 n 1 − λ(x0) = lim n log( (f )0(x0) ) = lim n log( f 0(xj ) ) dynamical system. Stability and instability of orbits are defined in →∞ n | | →∞ n | | topological terms while Lyapunov exponents is a numerical characteristic Xj=0 j calculated all along the orbit where xj = f (x0) (if the limit exists).
7 / 39 8 / 39 stability and instability in Lyapunov sense stability and instability in Lyapunov sense
Lyapunov instability is equivalent to sensitivity dependence on initial conditions (sdic) Definition Definition The forward trajectory Trf (x0) is said to be Lyapunov stable if for any Trf (x0) exhibits (sdic), if there exists > 0 such that given any δ > 0 > 0 there is δ > 0 such that whenever y x0 < δ is | − | there is y holding y x0 < δ and N > 0 such that f n(y) f n(x ) < for all n 0. | − | | − 0 | ≥ f n(y) f n(x ) | − 0 | ≥ for all n N ≥
9 / 39 10 / 39 stability and instability in Lyapunov sense Map f introduced by Demir and Ko¸cak
1.0
0.8
In the following examples, we consider the trajectories of 0 of two maps 0.6 ) x ( and obtain that we can have instability trajectories with negative Lyapunov f exponents and stable trajectories with positive Lyapunov exponents 0.4
0.2
0.0
0.0 0.2 0.4 0.6 0.8 1.0
x
Figure: Map f
11 / 39 12 / 39 Map g introduced by Demir and Ko¸cak
1.0
0.8 The strong Lyapunov exponent is 1 0.6 k+n 1 Φ(x) = limn Σj=k − log( f 0(xj ) ) )
x →∞ n
( | | f 0.4 if this limit exists uniformly with respect to k 0 ≥
0.2
0.0
0.0 0.2 0.4 0.6 0.8 1.0
x
Figure: Map g
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BC used the notion of Lyapunov exponents for non-autonomous systems on R and C 1 maps for the difference equation − Results: xn+1 = anxn 1 Let f C 1(I ). If the forward trajectory of x I has positive strong ∈ ∈ Lyapunov exponent, then the orbit has (sdic) as an immediate extension of the formula to calculate the Lyapunov exponents in the autonomous case (if the limit exists) as 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 n 1 1 n 1 − λ(x) = limn log (f0 )0(x) = limn log fj0(xj ) →∞ n | | →∞ n | | Xj=0 j where xj = f0 (x)
15 / 39 16 / 39 Lyapunov exponents for the non-autonomous case Lyapunov exponents for the non-autonomous case
They considered the case when The Lyapunov exponent has the following values: an = a + p(n) 1 If β = 0, then if where 1 loga > ( )2 pn = [a + (bn + βcn) 2 a then the system has for all initial conditions on (0, ) constant holding a > 1 but closed to 1, 0 < β > 1 and ∞ positive Lyapunov exponents and has (dsic) √ 2 If β = 0, then for fixed modulus m and in some range of Θ, the bn = 2sinn 6 system has also constant positive Lyaunov exponents. Also it is proved the system has (dsci) cn = √2sn[2K(m)(n + Θ)/π; m] with > 0 and m the modulus of the elipticity of senam map
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Stability and instability of orbits in periodic Stability and instability of orbits in periodic non-autonomous systems non-autonomous systems
Take a periodic block composed of the maps f and g, f0, f1, ..., fm 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 fi = f for i = 0, 1, ..., p 1 and f = g for j = p, ..., m 1 − j − If we compute the Lyapunov exponent 0 of such periodic non-autonomous When n , the Lyapunov exponent of the trajectory of 0 is n system we have for the f0 map → ∞ For p q λ(0) = − log2 m k(p q) + 1 n = km + 1 is − log2 km + 1 k(p q) + 2 n = km + 2 is − log2 km + 2 ... k(p q) + p n = km + p is − log2 km + p ... (k + 1)(p q) 19 / 39 20 / 39 n = (k + 1)m is − log2 (k + 1)m Stability and instability of orbits in non-periodic non-autonomous systems m Let X R and d any metric on it. If (xn)∞ and (x0 )∞ are two ⊂ n=0 n n=0 trajectories starting from nearby initial states x0 and x00 and write δx = x x . If f has continuous partial derivatives in every x , then, n n0 − n i iterating the map, we have the linear approximation (DF (x) denotes the When we choose a non-periodic block of maps f and g the orbit of 0 differential of the map F : Rn Rn at the point x). → continues being instable if the map g appears infinite times n 1 n − Theorem δxn Df (x0)δx0 = ( Df (xi )δx0 ' (BC) Let f a non-periodic sequence of maps f and g. If the map g Yi=0 ∞ appears infinite times, then the trajectory of 0 is Lyapunov instable. where the (i, j) element of the matrix Df (x) is given by ∂fi and where f ∂xj i and xj are the components of f and x in local coordinates on X
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Markus-Lyapunov Fractal Given a matrix A, we denote by At the transpose of A. Let the matrix
n t n (Df (x0) )(Df (x0)) where
n Df (x0) = Df (xn 1)(Df (xn 2)...(Df (x1)Df (x0) We consider the logistic equation − − have eigenvalues in x0 given by µi (n, x0), for i = 1, 2, ..., m such that µ (n, x ) µ (n, x ) ... µ (n, x ). Then the ith local Lyapunov xn+1 = rn(1 xn) 1 0 ≥ 2 0 ≥ ≥ m 0 − exponent at x0 is defined by: and the sequence of blocks BBBBB.... where B = 112112.... and 112 = r r r 1 1 1 2 λi (x0) = lim log( µi (n, x0) ) n →∞ 2n | | 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 24 / 39 Markus-Lyapunov Fractal Markus-Lyapunov Fractal
2.jpg 3.jpg Figure: Fractal Markus-Lyapunov Figure: Fractal Markus-Lyapunov
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We propose two dynamical systems, one defined in [0, 1]2 which has a forward trajectory with a positive Lyapunov exponent but not having Example sensitive dependence on initial conditions and other defined in [0, 1)2 We are going to obtain a continuous function F = (f , g) in [0, 1]2 such which has a forward trajectory with a negative Lyapunov exponent but that the forward trajectory of (0, 0) has a positive Lyapunov exponent, but having sensitive dependence on initial conditions. The examples are two has not has no sensitive dependence on initial conditions. dimensional versions of those mentioned in the introduction. The maps we are using are examples of permutation maps.
27 / 39 28 / 39 2 a) The map f : [0, 1] [0, 1] was introduced in [?] The map F (x, y) = (f (y), g(x)) is continuous in [0, 1] , because f y g are → 1 continuous in [0, 1]. 2x 1 + an < x bn, x [0,We1] consider the trajectory of (0, 0): − 2n+1 ≤ ∈ (0, 0), (x1, y1), (x2, y2), f (x) = n+2 5 22 2 1 { ···} − (x bn) + 1 + bn < x an+1 In every points of the trajectory, the map is differentiable (except (0, 0)) 2 5n+2 11 − 10n+1 − 2n+1 ≤ · − 1 x = 1 Since f y g are differentiable maps on right of 0, we define with a = 1 2 n 10 n 1, b = 1 2 n + 10 n 1, n = 0, 1, 2, .... n − − − n − − − + + 0 limy 0+ f (y) 0 2 − − − DF (0 , 0 ) = → = b) Now we define another map g : [0, 1] [0, 1] limx 0+ g(x) 0 3 0 → → 1 1 6 0 3x + 0 x DF (x , y ) = DF 2(0, 0) = 2 ≤ ≤ 15 1 1 0 6 6 7 2 1 1 1 0 12 x + < x DF (x , y ) = DF 3(0, 0) = 127 10 635 15 2 100 2 2 18 0 − ≤ − and g(x) = 1 5 n 3x + (2 1) an < x bn n n 2 − 2n+1 − ≤ 2n 6 0 2n 1 0 2 6 DF (0, 0) = DF − (0, 0) = · 0 6n 3 6n 0 5n+2 33 3 1 · for n = 1, 2, .... n+2− (x bn) + 1 + n+1 n+1 bn < x an+1 2 5 11 − 10 − 2 ≤29 / 39 30 / 39 · − 1 x = 1 n t n Now we compute the eigenvaluesn n 1 (DF ) )(DF n): n 1 with an = 1 2− 10− − , bn = 1 2− + 10− − , n = 1, 2, .... For n >1 we− have − −
2 n 2n 1 t 2n 1 3 6 0 (DF − (0, 0)) DF − (0, 0) = · 0 22 6n · Therefore, and the maximum value of the eigenvalues of such matrix is 1 1 λ1(0, 0) = lim log( µ1(n, (0, 0)) ) = log 6 > 0 1 n 2n 2 µ(2n 1, (0, 0)) = ((n + 1) log 3 + n log 2) →∞ | | − 2n 1 − it is easy to prove that the forward trajectory of (0, 0) has not sensitive For n > 1 we have dependence on initial conditions 6n 0 DF 2n(0, 0) = 0 6n whose eigenvalue is 1 µ(2n, (0, 0)) = (log 3 + log 2) 2
31 / 39 32 / 39 The map is continuous at every (x, y) I 2. ∈ k To see it, let > 0, we can chose k such that 1/2 < . As f 0(y) > 0 and g 0(x) > 0 on the forward orbit of (0, 0), si se considera la distancia del m´aximo: 1 1 F k (0, 0) = (1 , 1 ) m k Example − 2 k0m − 2 then We are going to obtain a continuous function G = (f 2, g) in [0, 1)2 such 1 1 1 k k that the forward trajectory of (0, 0) has a negative Lyapunov exponent, F (x, y) F (0, 0) ( k , k ) = k < | − | ≤ | 2 2 | 2 but it has sensitive dependence on initial conditions. for n k and 0 < x < δ¯ it remains to prove that the last inequality holds ≥ for n < k, but it is made using that F j is continuous and then, given > 0 there exists δ such that if 0 < (x, y) < δ , F j (x, y) F j (0, 0) < for j | | j | − | j = 1, ..., n 1. Then if we take − k k δ = min δ1, ..., δn 1, δ¯ and 0 < x < δ F (x, y) F (0, 0) < for all k > 0 − ⇒ | − |
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2 2 a) f : [0, 1) [0, 1) is defined by The map G(x, y) = (f (y), g(x)), is continuous in [0, 1) since f and g → are continuous in [0, 1). 1 1 Let us consider the trajectory of (0, 0), denoted by x + 0 x < 7/16 or an x < bn 2 2 ≤ ≤ (0, 0), (x , y ), (x , y ), { 1 1 2 2 ···} n+1 n+1 1 n n 1 with f (x) = (2 4 2− )(x + 2− 2 4− − 1) bn x < cn − − − · − ≤ 2n 1 1 2n 1 1 1 G (0, 0) = 1 , 1 , G − (0, 0) = 1 , 1 1 2 n 2 2 4 n 3 − 23n − 23n − 23n 1 − 23n 2 − − − − n n 1 − − − n 1 − · n 2 (x + 2− 2 4− − 1) cn x < an+1 2− − 9 4− − − · − ≤ for n = 1, 2, ... − · where a= 1 2 n 4 n 1, b = 1 2 n + 4 n 1, Similarly to the former example, we have n − − − − − n − − − − c = 1 2 n + 2 4 n 1 for n = 1, 2, .... n − − · − − b) g : [0, 1] [0, 1] is defined by 0 1/4 3/4 0 → DG(0, 0) = DG(x , y ) = DG 2(0, 0) = 3 0 1 1 0 3/4 1 1 2 3x + 0 x 3 0 3/4 2 ≤ ≤ 15 DG(x2, y2) = DG (0, 0) = 2 3 /4 0 6 7 2 1 1 1 and in general we have x + < y 127 10 − 635 15 ≤ 2 − 100 n n 2n (3/4) 0 2n 1 0 3 (1/4) DG (0, 0) = n DG − (0, 0) = n · 1 5 0 (3/4) 3 (1/4)3 0 g(x) = 3x + (2n 1) a < x 35 /b 39 · 36 / 39 2 − 2n+1 − n ≤ n for n = 1, 2, .... 5n+2 33 3 1 − (x b ) + 1 + b < x a n+2 n n+1 n+1 n n+1 2 5 11 − 10 − 2 ≤ · − 1 x = 1 where a= 1 2 n 10 n 1, b = 1 2 n + 10 n 1, para n − − − − − n − − − − n = 1, 2, .... Now we compute the eigenvalues of (DG n)t DG n for n = 1, 2, ...: · 2n 1 Eigenvalues of DG − (0, 0) are Therefore
3n 3n 1 32n 1 1 3 0 − 0 λ (0, 0) = lim log( µ (n, (0, 0)) ) = log < 0 n 1 0 2(n 1) 1 1 4 − 4n 4 n 2n | | 2 4 n 1 n = − 2(n 1) →∞ 3 − 3 3 0 0 − n n 1 0 2n 4 4 − 4 It is left to prove that the forward trajectory of (0, 0) has sensitive dependence on initial conditions, that is, there exists > 0 such that for therefore every δ > 0 there exists d((x, y), (0, 0)) < δ and k > 1 holding k k 1 d(G (x, y) G (0, 0)) > . (n log 3 (n 1) log 4) − 2n 1 − − Now we compute the distance to the maximum. Taking = 3/8, then for − 2n every δ > 0 there exists (x, y) such that d((x, y), (0, 0)) < δ, that is, Eigenvalues of DG (0, 0): x < δ, y < δ and k 2 it is hold that f k (x) < 1/2 or f k (y) < 1/2 and by ≥ The related matrix is diagonal and then the we have the double other hand we have f k (0) > 7/8. Therefore 3 n eigenvalue . Consequently we have 4 d(G k (0, 0) G k (x, y)) > − n (log 3 log 4) 2n −
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The previous construction for the two dimensional case on I 2 or [0, 1) [0, 1) = B can be extended to similar constructions on I n, Bn or × Tn using general versions of the permutation maps considered in a paper from BL.
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