† Corresponding author. E-mail:
Project supported by the National Natural Science Foundation of China (Grant Nos. 11672257, 11632008, 11772306, and 11972173), the Natural Science Foundation of Jiangsu Province of China (Grant No. BK20161314), the 5th 333 High-level Personnel Training Project of Jiangsu Province of China (Grant No. BRA2018324), and the Excellent Scientific and Technological Innovation Team of Jiangsu University.
We study a novel class of two-dimensional maps with infinitely many coexisting attractors. Firstly, the mathematical model of these maps is formulated by introducing a sinusoidal function. The existence and the stability of the fixed points in the model are studied indicating that they are infinitely many and all unstable. In particular, a computer searching program is employed to explore the chaotic attractors in these maps, and a simple map is exemplified to show their complex dynamics. Interestingly, this map contains infinitely many coexisting attractors which has been rarely reported in the literature. Further studies on these coexisting attractors are carried out by investigating their time histories, phase trajectories, basins of attraction, Lyapunov exponents spectrum, and Lyapunov (Kaplan–Yorke) dimension. Bifurcation analysis reveals that the map has periodic and chaotic solutions, and more importantly, exhibits extreme multi-stability.
The phenomenon that a dynamical system displays more than one attractor under different initial conditions is called multi-stability. Multi-stability has been extensively investigated in the literature since it was found in many areas, such as physics, chemistry, biology, and economics, see e.g. Refs. [1–5]. Some multi-stability is undesirable since the system can be perturbed to an undesired attractor in the presence of noise. So control of multi-stability has received considerable attention.[6,7] For this purpose, Liu and Páez Chávez[6] studied the control of coexisting attractors in an impacting system based on linear augmentation. On the other hand, multi-stability can be used to obtain a desired performance of a system without changing its parameters.[4,8] For example, Liu and Páez Chávez[8] investigated the control of multistability in a vibro-impact capsule system driven by a harmonic excitation using numerical continuation. The motion of the capsule system can be controlled by switching between its two stable coexisting attractors via a proper selection of the initial conditions. Very recently, hidden coexisting attractors, whose’s basins of attraction are not connected with unstable equilibria, have been explored extensively, see e.g. Refs. [9,10].
There is a special type of multi-stability, namely, extreme multi-stability which means that the number of coexisting attractors tends to infinite. In Refs. [11–14], infinitely many coexisting attractors that belong to extreme multi-stability have been explored extensively by using different types of couplings. In Refs. [15,16], the authors presented a method for designing an appropriate coupling scheme for two dynamical systems in order to realize extreme multi-stability. Then extreme multi-stability was first observed in experiment.[17] Apart from the coupling method, there are several other methods to generate extreme multi-stability. Some researchers introduced one,[18–24] two,[25–28] and multiple[29] ideal memristors into nonlinear systems to show infinitely many coexisting attractors. In Ref. [27], a novel five-dimensional two-memristor-based dynamical system was constructed by introducing two memristors with cosine memductance and extreme multi-stability was obtained. Other researchers constructed new chaotic systems with extreme multi-stability using proper nonlinear functions[30–36] or state feedback controller.[37,38] For example, in Refs. [30,31], the offset boosting method was developed to construct a self-reproducing system from a unique class of variable-boostable systems, and then extreme multi-stability was obtained. In Ref. [32], infinitely many strange attractors on a three-dimensional spatial lattice were obtained in a new dynamical system based on Thomas’ system by using the disturbed offset boosting of sinusoidal functions with different spatial periods. Jafari et al.[35] constructed a new five-dimensional chaotic system displaying hidden attractors and extreme multi-stability. In Ref. [37], a novel four-dimensional chaotic system presenting extreme multi-stability was derived by using a simple state feedback controller in a three-dimensional chaotic system.
Since nonlinear maps have broad applications in different disciplines including economics, biology, and engineering, smooth and non-smooth maps have been studied extensively in the literature.[5,39–53] Jiang et al.[40,41] explored coexisting hidden chaotic attractors in a class of two-dimensional and three-dimensional maps. Then Jiang et al.[42] studied the multi-stability in a class of two-dimensional chaotic maps with closed curve fixed points by showing three cases of coexisting attractors. In Refs. [44–46], the authors investigated complex chaotic dynamics of several fractional-order chaotic maps, and in Ref. [47] fractional-order chaotic maps were used for image encryption. In Refs. [48,49], a sine chaotification model (SCM) was introduced to enhance the complexity of one-dimensional and two-dimensional chaotic maps, respectively. However, it is rare to observe extreme multi-stability in nonlinear maps. In Ref. [50], infinite coexistence of a piecewise-linear continuous map was proved directly by explicitly computing periodic solutions in the infinite sequence. In Ref. [51], the equivalence between infinitely many asymptotically stable periodic solutions of n-dimensional piecewise-linear continuous maps and their subsumed homoclinic connections was established.
This paper aims to explore some simple chaotic maps with infinitely many coexisting attractors, whose fixed points are infinite and unstable, by performing an exhaustive computer search.[52,53] The main work of this paper is summarized as follows. (1) A novel class of two-dimensional chaotic maps with infinitely many coexisting attractors is developed. (2) The chaotic attractors are numerically analyzed by using basin of attractions, the Lyapunov exponent spectrum (Les), and the Kaplan–Yorke dimension. (3) Bifurcation analysis of the map is conducted to show different types of infinitely many coexisting attractors, including infinitely many coexisting periodic solutions and infinitely many coexisting chaotic solutions. The rest of this paper is organized as follows. In Section
Lai et al.[36] obtained an extremely simple chaotic system with infinitely many coexisting attractors by introducing a sinusoidal function into the Sprott B system. Following the idea of Lai et al.,[36] we introduce a sinusoidal function into a class of two-dimensional maps to generate infinitely many coexisting attractors. By using a computer searching program,[52,53] we study a novel class of two-dimensional maps written as
The fixed points (x*,y*) of the map (
The Jacobian matrix of the map (
The map (
![](cpb_29_6_060501/cpb_29_6_060501_ieqn1.gif)
![](cpb_29_6_060501/cpb_29_6_060501_ieqn2.gif)
![](cpb_29_6_060501/cpb_29_6_060501_ieqn3.gif)
![](cpb_29_6_060501/cpb_29_6_060501_ieqn4.gif)
![](cpb_29_6_060501/cpb_29_6_060501_ieqn5.gif)
![](cpb_29_6_060501/cpb_29_6_060501_ieqn6.gif)
![](cpb_29_6_060501/cpb_29_6_060501_ieqn7.gif)
![](cpb_29_6_060501/cpb_29_6_060501_ieqn8.gif)
When the initial value is chosen at (1, –3), the map (25) displays a chaotic attractor as shown in Fig.
Figure
By the translational symmetry, the map (
In order to show the complex dynamics of the chaotic map (
As can be seen from Fig.
Figure
In order to show the complex dynamics of the map (
Chaotic dynamics of a class of two-dimensional maps with homogenous extreme multi-stability was studied in this paper. A computer searching program was used to explore some simple chaotic maps by using phase portraits. Numerical methods, including computations of basins of attraction and the Lyapunov exponent spectrum, and bifurcation analysis, were used to demonstrate the complex dynamical behaviors of these maps. The maps also have infinitely many coexisting attractors in different types, such as coexisting periodic attractors and coexisting chaotic attractors. The coexisting attractors have the same shape but at different positions. The proposed chaotic maps can be used to generate chaotic signals for applications of chaos-based information engineering, such as data and image encryption.[47] Future works will focus on investigation of the high dimensional maps with infinitely many coexisting attractors and the construction of maps showing heterogeneous extreme multi-stability.
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