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 JAMP  Vol.5 No.9 , September 2017
Bifurcation of Parameter-Space and Chaos in Mira 2 Map
Abstract: 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.

1. Introduction

Mira first introduced Mira 1 and 2 maps in [1], 1996. And in [2], Styness et al. attained a deeper understanding of the phenomenon―a transition from one chaos regime to another chaos regime via crisis―for B falling in the interval B c [ 2.0501226960083, 2.05012267960082 ] (where B c denotes the critical value of the parameter B) and other parameter A = 1.5 .

Mira 2 map [1] has the functional form

{ 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 [3]. In this paper using numerical simulations [4], we obtained the distribution of dynamics in the parameter plane, the maximum Lyapunov exponent [5], fractal dimension [6] and more 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 bifurcations to chaos, quasi-period behaviors to chaos, chaos to quasi-period behaviors, bubble, on set of chaos.

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.

2. Bifurcation in the Parameter-Space

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 Figure 1. It is an isoperiodic diagram obtained by discretizing the parameter interval A × B = [ 2,2 ] × [ 4,0.5 ] in a grid of 800 × 900 points equally spaced. This corresponds in Figure 1 to a same resolution in both A and B axes, that is Δ A = Δ B = 0.005 . For each point

Figure 1. The parameter-space of Mira 2 map (1).

( A , B ) in Figure 1, an orbit of initial condition ( x 0 , y 0 ) = ( 0.001,0.05 ) converges to a chaotic attractor indicated by c, or to a quasi-periodic orbit indicated by q p , or to a n-period motion indicated by n, or to an attractor in infinity (unbounded attractor) indicated by , after a transient of 5000 iterates.

In Figure 1, we can see quasiperiodic motion (purple region) is born exactly

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 [3]). There is a collection of periodic regions embedded in the quasiperiodic (purple) region not all of these observed clearly with the scale utilized in Figure 1. In two plots of Figure 2 one sees magnifications of the two regions inside of the boxes I and II of Figure 1, the first located in the range 1.8 a 0.6, 3 b 0.8, and the second in the range 0.5 a 1, 2 b 0.5. In Figure 2(a), period-1 (cyan) region and period-3 (green) region have well defined boundary. For parameter values taken along the boundary line, pitchfork bifurcation occurs, and parameter b decreasing and passing through out period-3 (green) region Naimark-Sacker bifurcation occurs. In Figure 2(b), one sees periodic similar to the Arnold tongues observed immersed in purple region in the circle map [7].

3. Bifurcation and Chaos in Numerical Simulations

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 Figure 1, we take A and B as follows:

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 Figure 3(a) and Figure 3(b), respectively. From Figure 3(a), we see period-doubling to

(a) (b)

Figure 2. Magnification of the boxes (a) I, and (b) II in Figure 1.

(a) (b)

Figure 3. Bifurcation diagram and Lypunov exponents of Mira 2 map (1). Here A = 0 .

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 Figure 4(a) and Figure 4(b), respectively. In Figure 4(a), Mira 2 map (1) undergoes a Naimark-Sacker bifurcation from period-1 orbit at B = 0.7 . At B decreasing to B = 1.0022 , quasi-period region suddenly disappears and six pieces of period-doubling to chaos occur. In the interval B ( 1.705, 1.22 ) , period-doubling, Naimark-Sacker bifurcation and quasi-period behaviors are immersed in chaos region. The phase portraits of Mira 2 map (1) are shown in Figures 4(c)-(g), respectively. In Figures 4(c)-(e), the size of chaotic attractors at B = 1.29 ( MaxLyapunovExponent( M L E ) = 0.0394, FractalDimension ( F D ) = 1.4692 ) , B = 1.34 ( M L E = 0.0559, F D = 2.2151 ) , and B = 1.6 ( M L E = 0.0836 ) , increases with B decreasing. And the quasi-period orbits and its amplification are shown in Figure 4(f) and Figure 4(g), respectively.

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 Figure 5(a) and Figure 5(b), respectively. As B decreasing, Mira 2 map (1) undergoes a Naimark-Sacker bifurcation from period-1 window at

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 Figure 6(a) and Figure 6(b), respectively. And the amplifications of (a) at 0.665 < B < 0.61 and 0.785 < B < 0.68 are shown in Figure 6(c) and

(a) (b) (c) (d)
(e) (f) (g)

Figure 4. Bifurcation diagram and Lypunov exponents of Mira 2 map (1). Here A = 0.1 . (c)-(f) Phase portraits of Mira 2 map (1) at B = 1.29 , B = 1.34 , B = 1.6 and B = 1.535 . (g) The amplification of (f).

(a) (b) (c) (d)
(e) (f)

Figure 5. Bifurcation diagram and Lypunov exponents of Mira 2 map (1). Here A = 0.5 . (c)--(f) Phase portraits of Mira 2 map (1) at B = 0.913 , B = 0.94 , B = 1 and B = 1.167 .

Figure 6(d), respectively. In Figure 6(a), Mira 2 map (1) undergoes a Naimark-Sacker bifurcation from period-1 window at B = A / 2 3 / 4 = 0.325 . As B decreasing to B = 0.5535 , quasi-period behaviors suddenly disappear and period-6 window appears. In Figure 6(c), we observe that quasi-period behaviors and period windows alternatively appear, including period-18, 20, 21, 27, 28, 31, 33, 43, 53, etc. And, as B decreasing to B = 0.6632 , 7 pieces of inverse period-doubling to chaos appear, and in Figure 6(d), chaos region and period-doubling alternatively appear. The phase portraits of Mira 2 map (1) in Figures 6(e)-(i) are chaotic attractors at B = 0.6448 ( M L E = 0.0047, F D = 1.1148 ) ,

(a) (b) (c) (d)
(e) (f) (g) (h)
(i)

Figure 6. Bifurcation diagram and Lypunov exponents of Mira 2 map (1). Here A = 0.85 . (c) and (d) The amplification of (a). (e)-(i) Phase portraits of Mira 2 map (1) at B = 0.6448 , B = 0.649 , B = 0.72 , B = 0.724 and B = 0.77 .

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 Figure 7(a) and Figure 7(c), respectively. As A increasing to A = 1.682 , the

(a) (b) (c) (d)
(e) (f) (g)

Figure 7. Bifurcation diagram and Lypunov exponents of Mira 2 map (1). Here B = 2.2 . (c)-(g) Phase portraits of Mira 2 map (1) at A = 1.6878 , A = 1.665 , A = 1.617 , A = 1.5798 and A = 1.5797.

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.

4. Conclusion

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.

Acknowledgements

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

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

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