Dynamical properties of a 2-D non-invertible system

The non-invertible systems are very useful in practical applications. The study of the non-invertible systems has important value since a large number of genetics studies in biology, physics, engineering, and economic systems have been widely carried out found to exhibit a class of non-invertible systems. This short paper proposes a new simple four-term 2-D polynomial chaotic system with only one quadratic nonlinearity and describes its some interesting dynamical properties. Moreover, the stability of the fixed point and chaotic motions are investigated using analytical and numerical methods. Our 2-D polynomial system displays new chaotic attractors via the quasi-periodic route to chaos for certain values of its parameter of bifurcation.


Introduction
*Many papers have described 2-D chaotic invertible system with a quadratic inverse and constant Jacobian (Aziz-Alaoui et al., 2001;Miller and Grassi, 2001), one of the most famous is the smooth two-dimensional Hénon system (Hénon, 1976) and studied in detail by others (Hénon, 1969;Benedicks and Carleson, 1991;Cao and Liu, 1998;Marotto, 1979). In this context, the study of the noninvertible systems has important value since, a large number of genetics researches in biology (Bi and Ruan, 2013;Gałach, 2003), physics (Benerjee and Verghese, 2001), engineering (Tse, 2003), economics (Bischi and Tramontana, 2010), and applied mathematics (Mammeri, 2018) systems have been widely carried out found to exhibit a class of noninvertible quadratic systems. This short paper proposes a new simple 2-D non-invertible discrete chaotic system (2) with one bifurcation parameter, and that has only one nonlinear term (Mammeri, 2017). In section 1, a rigorous proof of the existence of some interesting properties of the system (2) on open, a connected subset is given, in section 2, a detailed dynamical behavior of this system (2) is further investigated numerically in term of a single bifurcation parameter. The final section concludes the letter.
It is well known that the general form twodimensional quadratic systems were made by Zeraoulia and Sprott (2010), where the 2-D quadratic systems are classified according to their number of nonlinearities. Also, many examples are given. And the first case of one nonlinearity is defined by: (1) In this paper, the new simplest two-dimensional quadratic system with only one cross-product nonlinear term is presented as follows: where ( , ) ∈ ℝ² and ∈ ℝ + * is the bifurcation parameter. For = 0 the system (2) reduces to a two-dimensional linear system. On the other hand, the system (2) permits the construction of a new family of attractors dependent on the bifurcation parameter and initial conditions.

Qualitative properties of the system
In the following section, we will prove some propositions in order to rigorously demonstrate the existence of some interesting properties of the system (2) on the largest open, connected subset. Let us define the following subset: = {( , ) ∈ ℝ ²: 1 − − > 0}.
Proposition 1: The system (2) is invertible if Proof: The determinant of the Jacobi matrix of the system (2) evaluated at a point ( , ) is ( , ) = (− + 1 − ) and we consider the finite number of points ( ₀, ₀), ( ₁, ₁), . . . , ( , ), . . . , ( , ) of an orbit of the system (2) and let us define the following matrix, then one has, we use the results available on linear algebra, then one has, where, then we have,

Proposition 2: The open subset
is the largest open connected and includes (0, 0).

Proposition 3:
The system (2) is of class ∞ on the subset Ω.
Proof: Because the coordinates of the system (2) is polynomial.
Proposition 5: The system (2) is one to one on .

Proposition 8:
In the positive quadrant, all orbits of the system (2) are bounded.
Proof: It follows from the system (2) that +1 ≤ and +1 ≤ for all positive integer since > 0, > 0 and > 0, thus we have For all positive integer . Then the sequences are monotone decreasing and so are bounded from above by ₀. It follows that the orbit of the system (2) is bounded.

Bifurcation properties of the system
The following section further investigates the dynamical behaviors of the chaotic system (2), including the stability of fixed point and bifurcations, Lyapunov exponents, bifurcation diagram, and Phases portraits.

Local stability conditions
The only fixed point of the system (2) is (0, 0). The Jacobi matrix of the system (2) evaluated at a point ( , ) is given by: and ( , ) = (1 − − ), at the fixed point (0,0), the Jacobi matrix is given by: The characteristic polynomial of the Jacobi matrix of the system (2) calculated at the fixed point , which takes the form: ( ) = ² − + , according to the criterion available in Elaydi (1996), we conclude that the fixed point A of the system (2) is asymptotically stable if and only if the following conditions hold: 1 + 1 + > 0,1 − 1 + > 0,1 − > 0 or, equivalently,

Numerical results
In this subsection, we will illustrate some observed chaotic attractors, the dynamical behaviors of the system (2) are investigated numerically. Fig. 1 shows the bifurcation diagram and the diagram of the variation of Lyapunov exponent of the system (2) by varying the parameter . For the range 0.5 ≤ ≤ 1.42. It can be observed from Fig. 1 that system (2) undergoes the following dynamical behaviors as increases:  For 0.5 ≤ < 1, system (2) is a fixed point.  For = 1, the fixed point loses stability at = 1, and we have the following two eigenvalues ₁ = 1+ √3 2 and ₂ = 1− √3 2 , thus |λ i(1≤i≤2) |=1. At this value, a Hopf bifurcation occurs, and via the quasiperiodic route to chaos, chaotic behavior can be observed. Fig. 1a shows a diagram description of this scenario of chaos.  For 0.5 < ≤ 1.42, system (2) is chaotic via the quasi-periodic route to chaos, and there are several quasi-periodic windows. If we fix the parameter to the value = 1.40 at the point, the dynamical behavior of the system (2) is chaotic, which is verified by the corresponding largest Lyapunov exponent is positive, as shown in Fig. 1b. The corresponding chaotic attractor is shown in Fig. 2d.
a: Bifurcation diagram of the system (2)

Conclusion
This paper is devoted to the rigorous proof of the existence of some interesting dynamical properties of the system using the standard methods available in most kinds of literature on analysis mathematics. Also, the dynamics of the system are described numerically in some detail.