Approximating hidden chaotic attractors via parameter switching.

In this paper, the problem of approximating hidden chaotic attractors of a general class of nonlinear systems is investigated. The parameter switching (PS) algorithm is utilized, which switches the control parameter within a given set of values with the initial value problem numerically solved. The PS-generated attractor approximates the attractor obtained by averaging the control parameter with the switched values, which represents the hidden chaotic attractor. The hidden chaotic attractors of a generalized Lorenz system and the Rabinovich-Fabrikant system are simulated for illustration.

Lyapunov Exponents of a Discontinuous 4D Hyperchaotic System of Integer or Fractional Order.

In this paper, the dynamics of local finite-time Lyapunov exponents of a 4D hyperchaotic system of integer or fractional order with a discontinuous right-hand side and as an initial value problem, are investigated graphically. It is shown that a discontinuous system of integer or fractional order cannot be numerically integrated using methods for continuous differential equations. A possible approach for discontinuous systems is presented. To integrate the initial value problem of fractional order or integer order, the discontinuous system is continuously approximated via Filippov's regularization and Cellina's Theorem. The Lyapunov exponents of the approximated system of integer or fractional order are represented as a function of two variables: as a function of two parameters, or as a function of the fractional order and one parameter, respectively. The obtained three-dimensional representation leads to comprehensive conclusions regarding the nature, differences and sign of the Lyapunov exponents in both integer order and fractional order cases.

Research Frontier in Chaos Theory and Complex Networks.

In recent years, as natural and social sciences are rapidly evolving, classical chaos theoryand modern complex networks studies are gradually interacting each other with a great joineddevelopment [...].

Chaos control of Hastings-Powell model by combining chaotic motions.

In this paper, we propose a Parameter Switching (PS) algorithm as a new chaos control method for the Hastings-Powell (HP) system. The PS algorithm is a convergent scheme that switches the control parameter within a set of values while the controlled system is numerically integrated. The attractor obtained with the PS algorithm matches the attractor obtained by integrating the system with the parameter replaced by the averaged value of the switched parameter values. The switching rule can be applied periodically or randomly over a set of given values. In this way, every stable cycle of the HP system can be approximated if its underlying parameter value equalizes the average value of the switching values. Moreover, the PS algorithm can be viewed as a generalization of Parrondo's game, which is applied for the first time to the HP system, by showing that losing strategy can win: "losing + losing = winning." If "loosing" is replaced with "chaos" and, "winning" with "order" (as the opposite to "chaos"), then by switching the parameter value in the HP system within two values, which generate chaotic motions, the PS algorithm can approximate a stable cycle so that symbolically one can write "chaos + chaos = regular." Also, by considering a different parameter control, new complex dynamics of the HP model are revealed.

Noise induced complexity: patterns and collective phenomena in a small-world neuronal network.

The effects of noise on patterns and collective phenomena are studied in a small-world neuronal network with the dynamics of each neuron being described by a two-dimensional Rulkov map neuron. It is shown that for intermediate noise levels, noise-induced ordered patterns emerge spatially, which supports the spatiotemporal coherence resonance. However, the inherent long range couplings of small-world networks can effectively disrupt the internal spatial scale of the media at small fraction of long-range couplings. The temporal order, characterized by the autocorrelation of a firing rate function, can be greatly enhanced by the introduction of small-world connectivity. There exists an optimal fraction of randomly rewired links, where the temporal order and synchronization can be optimized.

Random parameter-switching synthesis of a class of hyperbolic attractors.

The parameter perturbation methods (the most known being the OGY method) apply small wisely chosen swift kicks to the system once per cycle, to maintain it near the desired unstable periodic orbit. Thus, one can consider that a new attractor is finally generated. Another class of methods which allow the attractors born, imply small perturbations of the state variable [see, e.g., J. Güémez and M. A. Matías, Phys. Lett. A 181, 29 (1993)]. Whatever technique is utilized, generating any targeted attractor starting from a set of two or more of any kind of attractors (stable or not) of a considered dissipative continuous-time system cannot be achieved with these techniques. This kind of attractor synthesis [introduced in M.-F. Danca, W. K. S. Tang, and G. Chen, Appl. Math. Comput. 201, 650 (2008) and proved analytically in Y. Mao, W. K. S. Tang, and M.-F. Danca, Appl. Math. Comput. (submitted)] which starts from a set of given attractors, allows us, via periodic parameter-switching, to generate any of the set of all possible attractors of a class of continuous-time dissipative dynamical systems, depending linearly on the control parameter. In this paper we extend this technique proving empirically that even random manners for switching can be utilized for this purpose. These parameter-switches schemes are very easy to implement and require only the mathematical model of the underlying dynamical system, a convergent numerical method to integrate the system, and the bifurcation diagram to choose specific attractors. Relatively large parameter switches are admitted. As a main result, these switching algorithms (deterministic or random) offer a new perspective on the set of all attractors of a class of dissipative continuous-time dynamical systems.