Snap-back repellers in rational difference equations Definition Let - PowerPoint PPT Presentation

Introduction Marotto chaos in RDE Numerical examples References Snap-back repellers in rational difference equations Definition Let f : R n → R n be a (continuous) function. An order one Antonio Cascales, Francisco Balibrea difference equation (DE) is: antoniocascales@yahoo.es balibrea@um.es k ≥ 0 x k +1 = f ( x k ) (1) Departamento de Matem´ aticas Universidad de Murcia (Spain) A solution of the DE is a sequence ( x k ) k ⊆ R n obtained from (1) taking x 0 ∈ R n . The equation is rational (RDE) if f is a rational function. 18th ICDEA Barcelona. July 2012 2 / 35 Introduction Marotto chaos in RDE Numerical examples References Introduction Marotto chaos in RDE Numerical examples References Generalized Li-Yorke chaos or Marotto chaos Let f : R n → R n be a continuous function. x k +1 = f ( x k ) is Li-Yorke chaotic in a generalized sense or Marotto chaotic if: LY1 There exists N ∈ N such that ∀ p ≥ N there are (prime) If the sequence ( x k ) k has a finite number of elements, we say that periodic sequences with period p . x 0 is an element of the forbidden set of (1). LY2 There exists an uncountable set S ⊂ R n holding f ( S ) ⊂ S and • The forbidden set is empty when Dom f = R n . non containing periodic points, such that for every x � = y ∈ S , • If f is rational, the forbidden set includes the poles of f . lim k →∞ || f k ( x ) − f k ( y ) || > 0 and for any period p of f A general open problem in RDE is to determine the forbidden set. lim k →∞ || f k ( x ) − f k ( p ) || > 0 LY3 There is an uncountable subset S 0 ⊂ S , such that for every x 0 � = y 0 ∈ S 0 is lim k →∞ || f k ( x 0 ) − f k ( y 0 ) || = 0 A Li-Yorke pair is ( x , y ) verifying [LY2] and [LY3]. 3 / 35 4 / 35

Introduction Marotto chaos in RDE Numerical examples References Introduction Marotto chaos in RDE Numerical examples References Notion of snap-back repeller (SBR) Theorem (Marotto, 1978 and 2004) Let f ∈ C 1 in a neighborhood of a fixed point z ∗ of f . We say that z ∗ is a SBR if: The existence of a SBR implies Marotto chaos. A similar result, without derivatives and only continuity was proved SBR1 All the eigenvalues of Df ( z ∗ ) have modulus greater than 1 by Peter Kloeden in 1981. In the proof, he used the fact that ( z ∗ is a repeller) continuous one-to-one mappings have continuous inverses on SBR2 There exist a finite sequence x 0 , x 1 , . . . , x M such that compact spaces and the Brouwer fixed point theorem. In this case, x k +1 = f ( x k ), x M = z ∗ and x 0 � = z ∗ belongs to a repelling saddle points can be considered as well as snap-back repellers neighborhood of z ∗ . Moreover, | Df ( x k ) | � = 0 for when Marotto’s result can not be applied. 0 ≤ k ≤ M − 1. It is apparent that the existence of a snap-back repeller for a one-dimensional map f is equivalent to the existence of a point of period 3 for the map f n for some positive integer n . Remark that a SBR is a special kind of homoclinic point in the sense of Louis Block (1978). 5 / 35 6 / 35 Introduction Marotto chaos in RDE Numerical examples References Introduction Marotto chaos in RDE Numerical examples References Open questions Motivation I We consider systems of difference equations x n +1 = P 1 ( x n , y n ) Q 1 ( x n , y n ) 1) Does Marotto’s theorem remain true for a RDE? y n +1 = P 2 ( x n , y n ) 2) How estimate the forbidden set of a RDE? Q 2 ( x n , y n ) where P 1 , Q 1 , P 2 , Q 2 are quadratic polynomials in x n , y n and look for conditions for the existence of chaotic behavior. For some particular values of the parameters, Mazrooei and Sebdani have recently given an example of the former system, where the origin (0 , 0) is a snap-back repeller. So there will be generalized Li-Yorke chaos if Marotto’s rule can be applied to this new frame. 7 / 35 8 / 35

Introduction Marotto chaos in RDE Numerical examples References Introduction Marotto chaos in RDE Numerical examples References Marotto chaos in RDE: LY1 property Motivation II In a RDE, we define SBR in the same way that in the continuous case. Models containing RDE of this kind are founded in population dynamics and also in the application of the Newton’s method to Remark that none of the elements of the finite sequence x 0 , x 1 ,... , polynomial equations. x M = z ∗ belongs to the forbidden set. Easy examples are: 1 • inverse parabolas, x k +1 = x 2 Theorem (LY1) k − r 1 • inverse logistic equation, x k +1 = In a RDE, the existence of a SBR implies the existence of infinite rx k (1 − x k ) periods (LY1). 9 / 35 10 / 35 Introduction Marotto chaos in RDE Numerical examples References Introduction Marotto chaos in RDE Numerical examples References Sketch of proof Sketch of proof II The key idea is fix a special compact neighborhood B ǫ ( z ∗ ) and to These concepts appear in Li-Yorke and Marotto’s works. In the apply Brouwer’s fixed point theorem to each function of the form RDE setting, they remain valid because of the local continuity and f − k ◦ g differentiability of the iteration function. An important remark is that the repelling character of z ∗ assures where that f − 1 ( A ) ⊆ A where A is a convenient neighborhood of z ∗ , and • g is the local inverse of f M in a neighborhood of z ∗ . therefore f − k ( x ) is well defined for all x ∈ A and k > 0. • f − k is the k -times composition of a local inverse of f near z ∗ . 11 / 35 12 / 35

Introduction Marotto chaos in RDE Numerical examples References Introduction Marotto chaos in RDE Numerical examples References Definition (Itinerary) Second part of Marotto’s theorem Let f : R n → R n a function. A sequence of compact subsets of R n , { I k } + ∞ k =0 , is an itinerary of f when f ( I k ) ⊇ I k +1 . We wonder if it is true or not the existence of an uncountable set of Li-Yorke pairs when the RDE has a SBR. Itineraries are the tool that allow us to locate special solutions of a difference equation behaving in a particular way. Indeed, they are We claim that the answer is not, but we have not yet been able to used several times in the proof of the second part of Marotto’s find a counterexample. theorem, where we find the result: Lemma Our claim is based on the necessity of an additional topological If f is continuous, and { I k } + ∞ k =0 is an itinerary, there exist x 0 ∈ R n condition if we want to obtain similar results to those of Marotto. such that f k ( x 0 ) ∈ I k , ∀ k ≥ 0 . We have called it the compact preimage property (CPP) . 13 / 35 14 / 35 Introduction Marotto chaos in RDE Numerical examples References Introduction Marotto chaos in RDE Numerical examples References Definition (CPP property) We say that a function f has the compact pre-image property Itineraries’ lemma in the real line is an application of the (CPP) if for every compact subset K in the range of f , there exist intermediate value property. a compact L such that f ( L ) = K. • If f has CPP, this is also the case for f k . In the n th-dimensional case the lemma was proved by Phil Diamond in 1976. • If f : R → R is continuous, the Intermediate Value Property proves that f is CPP. In the RDE setting the problem is more complicated because we • If f : R n → R n is continuous, the CPP property is also true have in addition the presence of the the forbidden set and the non (Diamond). continuity in the maps. Lemma k =0 is an itinerary, there exist x 0 ∈ R n such If f is CPP, and { I k } + ∞ that f k ( x 0 ) ∈ I k , ∀ k ≥ 0 . 15 / 35 16 / 35

Introduction Marotto chaos in RDE Numerical examples References Introduction Marotto chaos in RDE Numerical examples References Example of non-CPP function: reducible case Another non-CPP function 4 x 5 − 3 x 3 3 · ( x − 1 / 2) 2 f ( x ) = 1 f ( x ) = 4 x 4 − 7 x 2 + 3 x ( x − 1) 2 17 / 35 18 / 35 Introduction Marotto chaos in RDE Numerical examples References Introduction Marotto chaos in RDE Numerical examples References Main result LY2 and LY3: Sketch of proof. The definition of SBR allows to construct two compact sets U , V of R n and N ∈ N such that: • U ∩ V = ∅ Theorem A RDE with the CPP property having a SBR is chaotic in the • V ⊂ f N ( U ) sense of Marotto. • U , V ⊂ f N ( V ) Therefore, any sequence { I k } + ∞ k =0 with I k = U , V and with the restriction that I k = U implies I k +1 = V , is an itinerary of f N . The set of all these itineraries define an uncountable subset of R n which can be refined to obtain the scrambled set of Li-Yorke pairs. 19 / 35 20 / 35