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In this paper, we investigate Mira 2 map in parameter-space (A-B) and obtain some interesting dynamical behaviors. According to the parameter space of Mira 2 map, we take A and B as some groups of values and display complex dynamical behaviors, including period-1, 2, 3, 4, 5, ???, 38, ??? orbits, Arnold tongues observed in the circle map [7], crisis, some chaotic attractors, period-doubling bifurcation to chaos, quasi-period behaviors to chaos, chaos to quasi-period behaviors, bubble and onset of chaos.

Mira first introduced Mira 1 and 2 maps in [

Mira 2 map [

{ x n + 1 = A x n + y n , y n + 1 x n 2 + B . (1)

where A and B are real.

Though more dynamical behaviors of Mira 2 map (1) had gotten someone’s less attention, we studied Mira 2 map and got many interesting dynamical behaviors, such as the conditions of the existence for fold bifurcation, flip bifurcation, Naimark-Sacker bifurcation and chaos in the sense of Marroto of this map in [

The paper is organized as follows. In Section 2, we give the parameter space of dynamical behaviors of Mira 2 map (1) in ( A − B ) plane. And in Section 3, the numerical simulations bifurcations in ( A − x ) and ( B − x ) planes for different values, the computation of maximum Lyapunov exponent corresponding to bifurcation diagram and the phase portraits at neighborhood of critical values are given.

In this section, we give the parameter space of dynamical behaviors of Mira 2 map (1) in ( A − B ) plane.

In order to show more dynamics of Mira 2 map (1), we take A and B as the parameters and observe the motions of Mira 2 map (1) according to the initial condition ( x 0 , y 0 ) = ( 0.001,0.05 ) of Mira 2 map (1). After computing some groups of the value scopes and the length of the grid of A and B , we find that there exist almost all dynamical motion of Mira 2 map (1) for the parameter interval A × B = [ − 2,2 ] × [ − 4,0.5 ] and it takes relatively less time. The parameter-space of Mira 2 map (1) is shown in

( A , B ) in

In

on the boundary-the line B = A 2 − 3 4 -of period-1 (cyan) region, as a result of

Naimark-Sacker bifurcations of period 1 (we give the condition of the existence of Naimark-Sacker bifurcation in [

Now we present some numerical simulation results to show other interesting dynamical behaviors of Mira 2 map (1). According to the parameter space of Mira 2 map (1) in

Case (1). Fixing A = 0 , and − 2 ≤ B ≤ − 0.5 ;

Case (2). Fixing A = 0.1 , and − 1.705 ≤ B ≤ 0.2 ;

Case (3). Fixing A = 0.5 , and − 1.173 ≤ B ≤ − 0.4 ;

Case (4). Fixing A = 0.85 , and − 0.785 ≤ B ≤ − 0.3 ;

Case (5). Fixing B = − 2.2 , and − 1.682 ≤ A ≤ − 1.57 ;

For case (1) The bifurcation diagram of Mira 2 map (1) for A = 0 in ( B , x ) plane and the corresponding maximal Lyapunov exponents are given in

chaos occur with B decreasing and chaos region abruptly disappears as B = − 1.4746, − 1.6243, − 1.749 , respectively. And when B decrease to − 2 , the chaos region turns to an attractor in infinity (unbounded attractor).

For case (2) The bifurcation diagram of Mira 2 map (1) for A = 0.1 in ( B , x ) plane and the corresponding maximal Lyapunov exponents are given in

For case (3) The bifurcation diagram of Mira 2 map (1) for A = 0.5 in ( B , x ) plane and the corresponding maximal Lyapunov exponents are given in

B = A 2 − 3 4 = − 0.5 . At B = − 0.8025 , quasi-period region disappears to period-5

windows, and at B = − 0.8915 , period-5 window becomes 15 period-doubling to chaos. Figures 5(c)-(f) are shown chaotic attractors at B = − 0.913, ( M L E = 0.0106, F D = 1.0723 ) , B = − 0.94, ( M L E = 0.0269, F D = 1.2055 ) , B = − 1, ( M L E = 0.045, F D = 1.5670 ) and B = − 1.167 ( M L E = 0.0845,1.5278 ) , respectively.

For case (4) The bifurcation diagram of Mira 2 map (1) for A = 0.85 in ( B , x ) plane and the corresponding maximal Lyapunov exponents are given in

B = − 0.724 ( M L E = 0.007, F D = 1.0815 ) and B = − 0.77 ( M L E = 0.0423, F D = 1.4178 ) , quasi-period orbit at B = − 0.649 , and period-21 orbit at B = − 0.72 , respectively.

For case (5) The bifurcation diagram of Mira 2 map (1) for B = − 2.2 in ( A , x ) plane and the corresponding maximal Lyapunov exponents are given in

attractor in infinity suddenly converges to quasi-period orbit. And as A increasing, quasi-period behaviors, period-orbits which include period-3, 8, 11, 17, 19, 20, 21, 25, etc., and chaotic behaviors alternatively appear. When A increase from A = − 1.5798 to A = − 1.5797 , chaos disappears and period-3 orbit appear. We observe that 3 pieces of Naimark-Sacker bifurcation occur at A = − 1.5707 . As A increasing to A = − 1.5707, quasi-period behaviors suddenly disappear and the unbounded attractor appears. The phase portraits of quasi-period orbit, chaotic attractor, period-orbit of Mira 2 map (1) are shown in Figures 7(c)-(g) for A = − 1.6878 , A = − 1.665 ( M L E = 0.0057, F D = 1.1717 ) , A = − 1.617 , A = − 1.5798 ( M L E = 0.016, F D = 1.1055 ) and A = − 1.5797 , respectively.

In this paper, we study Mira 2 map in parameter-space (A-B) and obtain some interesting dynamical behaviors. According to the parameter space of Mira 2 map, we take A and B as some groups of values and display complex dynamical behaviors.

This work was supported by the National Science Foundations of China (10671063 and 61571052).

Jiang, T. and Yang, Z.Y. (2017) Bifurcation of Parameter-Space and Chaos in Mira 2 Map. Journal of Applied Mathematics and Physics, 5, 1899-1907. https://doi.org/10.4236/jamp.2017.59160