Attractors and Commutation Sets in Hénon-like Diffeomorphisms

In this work we display Hénon-like attractors that emerge and appear in diffeomorphisms generated by embedding of one-dimensional endomorphisms.We show the properties of basin of attraction, and identify various types of attractors and commutation sets which are associated with these diffeomorphisms. Numerically presented scenarii of the creation and destruction of these attractors via bifurcations are illustrated.


Introduction
Hénon map is considered as the simplest diffeomorphism possessing important properties that contributed greatly to our understanding of complex and chaotic dynamics.We study a diffeomorphism in dependence of four parameters, and which can be considered as a purely artificial model coming from the embedding of a one-dimensional noninvertible map into a two-dimensional invertible one with constant Jacobian.
The dynamics involves various transitions by bifurcations.In this respect, it can be compared with classical examples such as the generalized Hénon maps.On the one hand, our study concerns a noninvertible map embedded into the invertible one and local bifurcations.On the other hand, we have to deal with global bifurcations of observable sets, such as crises of attractors or metamorphoses 10 W. Selmani and I. Djellit of basin boundaries, which are the most easily detected and probably the most often described in scientific literature.
There has been growing interest for homoclinic and heteroclinic phenomena, they have been the most studied objects in dynamical systems.From the qualitative point of view homoclinic phenomena are of interest because they represent a possible source for complex dynamics.It has been recognized that connecting orbits and their bifurcations play an important role in the qualitative theory of dynamical systems.Strong effort has gone into describing the different bifurcations that can occur in terms of genericity and into determining the different types of behavior in systems undergoing homoclinic and heteroclinic bifurcations.
A very influential work is used here, concerning the importance of commutation sets done by Mira and Gracio who have developed the fascinating role of these sets to delimit chaotic attractors (see [3]).Here the dynamical features are explored by numerical methods.Also, in several cases we find interesting dynamical objects predicted by the theory and global phenomena in the parameter plane.This kind of scanning has been made for giving a first idea about bifurcation organization.
Let's start with "the generalized logistic map " in dimension 1, the system presents a population evolution model, which generalizes the logistic models that are proportional to the beta densities with the shape parameters p and q, such that p, q > 1, and the growth rate r.
The complex dynamical behavior of these models is studied in the plane (r, p) using explicit methods when the parameter r increases.Anticipating the future evolution of population's dynamics is one of the most important issue in several domains, such as biological, ecological, social or economical sciences.
Rocha and al. in [4] introduced some basic concepts and results on probability density functions.They showed that the sequence is proportional to Beta density Beta(2, 2) for x ∈ [0, 1] and r > 0 .This sequence is a simplified population model.For a small initial condition, that's mean a low population, the growth rate in years n is exponential.For a large initial condition, the population is more important for the same space and the same food, so it will be increase.Also they have studied the complex dynamical behavior of some models of the main following form which are proportional to Beta(p, q) densities, where the variable x ∈ [0, 1] and the parameters p, q > 1.In the particular case of q = 2, these models are typically used to study of whales population and forest fires.The parameter p measures the difficulty of the mating process.
They have considered an extension of the function Beta and the density Beta to approach the dynamical system of Verhulst, which symbolizes the study of the birth and death processes of a population of one species, represented for p ∈ N − {1, 2} by the map: K and r = r * K p−2 , we obtained: In other way, Verhulst in [7] has considered N (t n ) the number of individuals at time t n , N (t n ) ≥ 0 with, N (t n+1 ) = f (N (t n )) where f determined by the birth and death rates in the population.We put ) is large because of natural bounds on the amount of available space and food.We present a simple model r is the growth coefficient and k a positive constant.We introduce some basic rescaling x n = rN (t n )/(k(1 + r)) and a = 1 + r the equation (1.1) becomes ( The equation (1.2) is called the quadratic equation, logistic equation or Verhulst equation.
This paper is devoted to present a numerical investigation of a two-dimensional diffeomorphism on the dynamical properties of its basins of attractions, regular and chaotic attractors, the bifurcation structures and the mechanisms that assure chaotic dynamics by extending works.We focuse on the topological structure of trajectories around eventual cycles and fixed points and to illustrate its phaseparameter portraits.

Endomorphisms depending of two parameters (r, p)
In this section we provide a few overviews of the analysis of the family of endomorphisms f r,p : [0, 1] → [0, 1] depending of two parameters p ∈ N − {1, 2} and r > 0, and defined by: p and r are chosen such that , p ∈]1, p M ] and r ∈]0, r(p M )], with p M and r(p M ) correspond to the maxima values of p and r.
Let c be a critical point of f r,p which satisfies the following conditions: • f ′ r,p (c) = 0 and f ′′ r,p (c) < 0 meaning that f r,p is strictly increasing in [0, c[ and strictly decreasing in ]c, 1]; and f ′ r,p (x) = 0, ∀x = c,; • f 0,p (c) = 0 and f r(pM ),p (c) = 1; with c = p−1 p ; • f r,p (0) = f r,p (1) = 0; Singer in [6] used the concept of negative Schwartzian derivative to discuss the existence of stable periodic orbits.The Schwarz derivative of f r,p (x) is Singer concluded that a function f which is C 1 -unimodal and for which S(f(x)) < 0 for all x ∈]0, 1[−{c} has at most one stable periodic orbit plus possibly a stable fixed point in the interval [0, 1].
is strictly decreasing on [0, c] and strictly increasing on [c, 1]; and f r,p is once continuously differentiable with f ′ r,p (x) = 0 when x = c.
The maximum value of the parameter p is p M = 20, it is the largest value for which we consider that the model can be realistic.The value r(p M ) is the value of the parameter r corresponding to the full shift for p = p M .We consider that 1 < p ≤ p M = 20 and 0 < r ≤ 53.001.For any fixed value p > 1, if r = 0, then there is no curves.
For example, we can verify that unimodal maps f r,p have the fixed point x * = 0 for r > 0 and p > 1.However, for p = 1.1 and p = 1.5, we can verify that these maps have another positive fixed point besides 0. For p ∈ [2, p M ] and r > 6.721, there are the fixed point zero and two other fixed points (cf. Figure 1).Figure 1: The fixed points for different values of (r, p) according to [1].
Attractors and Commutation Sets 13 Rocha and al. in [1,5] have shown that there exists a relation between the parameters p, q and c such that for all r ∈]r 1 , r 2 [, we can assure the existence of a unique attractor for x ∈ [0; 1] with then x f is the only one positive fixed point, c = p−1 p is the critical point of order 0.

Embedding of one-dimensional noninvertible map into a two-dimensional invertible map
Let T be a two-dimensional Hénon-like map: The endomorphism f r,p = rx p−1 (1 − x) is embedding into the diffeomorphism T with b = 0, and jacobian is J = −b.For a family of recurrence (2.2), the continuously passing of properties, for b = 0 to b = 0 and b sufficiently small, is obtained in the sense that we find identical cycles associated with the structures of bifurcation as in [3,4].Figure 2   At once b is not small enough, an attractor type fixed point appears and coexists with a cycle of order two.
2.2.bidimensional map depending of three parameters (r, p, q) In this part, we consider three parameters (r, p, q).The system T b will be: Taking into account [3], Mira describes some properties of the two-dimensional invertible systems T b that can be expressed as associated with noninvertible one-dimensional maps T 0 such that when the parameter b is equal to the critical value, i.e.: b = 0 then T 0 : x ′ = f r,p,q (x).These properties are also about the stable and the instable fixed points and the concept of commutation set in invertible maps which are useful for interpreting such problems and fundamental in the definition of bifurcations leading to important modification of attractors and their basins of attraction.
Recall that a closed and invariant set A, is called an attracting set if some neighborhood U of A exists such that T (U ) ⊂ U , and is the total basin (or simply: basin of attraction, or influence domain) of the attracting set A. In general, several types of attractors, e.g.fixed points, invariant closed curves, chaotic attractors, may coexist in the same mapping.This non-uniqueness also indicates that the routes to chaos depend on initial conditions and are therefore non-unique and depend on the values of the parameters.The basins of attraction D, defining the initial conditions leading to a certain attractor, may be a complex set.
The critical set of rank-one, said CS, is the geometrical locus of points x having at least two coincident preimages.The critical set CS i of rank-(i+1), i > 0, is the rank-i image of the set CS 0 ≡ CS.
Definition 2.2.Let S be a saddle fixed point and U a neighborhood of S. The local unstable set W u loc (S) of S ∈ U is given by: and the global unstable set W u (S) of S is given by: The local stable set W s loc (S) of S ∈ U is given by: and the global stable set W s (S) of S is given by:W s (S) = ∪ k≥0 T k [W s loc (S)].
Definition 2.3.The point Q is said to be homoclinic to the nonattracting fixed point S (or homoclinic point of S) if Q ∈ W u (S) ∩ W s (S).
Let M be another nonattracting fixed point.A point when k increases, and Q belongs to the local unstable set W u loc (S).
Definition 2.4.[3] We define the commutation sets E i as .  , 4 represent two attraction basins associated with two attractors for q = 2 and p = 3 in the phase space.In Figure 3, The red basin is associated with the fixed point and the green one is associated with the 2 -cycle represented by C 1 and C 2 points, q 1 and q 2 are two saddle points such that q 2 is on the boundary of the attraction basin of the attractive fixed point and q 1 is inside of the 2-cycle attraction basin.

Bifurcations and complexity
First, we consider the one-dimensional maps T 0 : x ′ = f r,p,q (x), and we study their dynamics.The complexity described by the family of f r,p,q (x) is analyzed in terms of the parameter r > 0, and the shape parameters p > 1 and q > 1, related with growth-retardation phenomena.
This family is subject to spontaneous existence which can be guaranteed for the value of the growth parameter where x f is the only positive fixed point (for more details see [5]).With r 2 , we have a set of globally stable fixed points where the population growth remains stable, r 2 is given by r 2 = p 2 q − p 2 − 2pq + 2p + q − 1 (p + q − 2) 3 c p (1 − c) q , with p, q > 1 where c = p−1 p+q−2 is the critical point.Globally, the iterates of the map f r,p,q (c) are always attracted to the positive fixed point sufficiently near of the attractive point, designated by stability region 1, where r satisfies r 1 < r < r 2 .
Otherwise for the values of the parameter r such that r 2 < r < r 3 , the stability of the region 2 is characterized by the beginning of period-doubling.The value r 3 satisfies this condition For r = r 4 we can observe complex dynamical behavior.It is defined by For other values of the parameters p and q, due to certain factors, the population can take the risk of extinction.
In two-dimensional diffeomorphisms T b , we consider other tools for analyzing the complexity and chaotic patterns of behavior.The notion of commutation sets instead of critical sets helps us to acquire such information about extension and period-doubling.We fix p = q = 2, the system (6) is considered in [3], by combining of properties of T b and T 0 As mentioned in proposition 3.1, commutation sets are of primordial importance for the study of critical singularities.This method consists to approach attractors due to principal and secondary fold points (see Figure 6).
The inverse map T −1 b is defined by the relations: Let E ′ 0 be the line y = b/2, which is given by J = 0 and by the same method applied to T b .The commutation curves are defined by E ′ n = T −n b (E ′ 0 ).

Invariant manifolds of fixed points
The stable invariant manifold W s (S) constitutes in general the frontier of the basin of attraction of an associated attractor and the unstable one converges to this attractor.Contact bifurcations between the attractor and its attraction basin may correspond to homoclinic or heteroclinic bifurcations.Homoclinic orbits and heteroclinic orbits are important concepts in the study of bifurcation of structures and chaos.Many chaotic behaviors of a complex system are related to these kinds of trajectories in the system.A homoclinic orbit is an orbit that is doubly asymptotic to a fixed point, or is a closed trajectory asymptotic to itself.A heteroclinic orbit is a trajectory that connects two fixed points or cycles of saddle type.In this part, we are interested in the existence or nonexistence of homoclinic orbits and heteroclinic orbits of this considered system.
For fixed parameter values, attractors's basins are illustrated in the phase space.When there exist several attractors, it is possible to define a global basin, that means the set of initial conditions giving rise to bounded iterated sequences, independently of the fact that they converge to one attractor or another.
We fix b = 0.2, we draw the invariant manifolds of a saddle point which is inside the basin for r = 2.When we investigate the heteroclinic tangency between unstable and instable manifolds by numerical way, we use saddle points.and the unstable manifold of q 2 intersect and then heteroclinic points appear.In Figure 9, only the invariant manifolds of saddle points are shown.For b = 0.21, we find Hénon attractor that is growing with r , has a contact with the frontier of its basin and disappears by contact bifurcation, p = 3.5 and q = 2.5 (see Figure 10).

Conclusion
After embedding a one-dimensional noninvertible map into a larger system which becomes invertible, we reviewed and used some analysis techniques as commutation sets and heteroclinic solutions together to detect chaotic behaviors.
presents information on stability region for the fixed point (blue domain), and the existence region for attracting cycles of order k exists (k ≤ 14) .The black regions (k = 15) correspond to chaotic behavior.

Figure 7
Figure 7 for r = 2 and Figure8for r = 5 show how of the stable manifold of q 1 and the unstable manifold of q 2 intersect and then heteroclinic points appear.In Figure9, only the invariant manifolds of saddle points are shown.