Hidden Attractors and Dynamical Behaviors in an Extended Rikitake System

Abstract

In this paper, an extended Rikitake system is studied. Several issues, such as Hopf bifurcation, coexistence of stable equilibria and hidden attractor, and dynamics analysis at infinity are investigated either analytically or numerically. Especially, by a simple linear transformation, the wide range of hidden attractors is noticed, and the Lyapunov exponents diagram is given. The obtained results show that the unstable periodic solution generated by Hopf bifurcation leads to the hidden attractor. The existence of hidden attractors that may render the system's behavior unpredictable not only depends on the value of system parameters but also on the value of initial conditions. The phenomena are important and potentially problematic in engineering applications.

A fractional-order form of a system with stable equilibria and its synchronization

“…Interestingly, system (1) has two stable equilibrium points, and attractors in this system are 'hidden attractors' because the basin of attraction for a hidden attractor is not connected with any unstable fixed point [27][28][29]. Recently, a hidden attractor has been discovered in different systems such as Chua's system [27], model of drilling system [32], extended Rikitake system [36], and DC/DC converter [38]. The identification of hidden attractors in practical applications is important to avoid the sudden change to undesired behavior [29].…”

“…It is worth noting that systems with stable equilibria are systems with 'hidden attractors' [26][27][28][29]. Hidden attractors have received considerable attention recently because of their roles in theoretical and practical problems [30][31][32][33][34][35][36][37][38]. Different definitions and main properties of fractional calculus have been reported in the literature [47][48][49][50].…”

Perpetual Points: New Tool for Localization of Coexisting Attractors in Dynamical Systems

“…From the very beginning, it has been stated that perpetual points may be useful in localization of co-existing attractors. Such property of considered objects may be particularly relevant in the systems with hidden oscillations [Dudkowski et al, 2016;Chaudhuri & Prasad, 2014;Leonov et al, 2011;Leonov & Kuznetsov, 2013;Wei & Zhang, 2014;Wei et al, 2015;Wei et al, 2015b;Wei et al, 2016;Wei et al, 2016a]. Multistability of states have been recently widely observed in many dynamical models and thoroughly studied by the researchers [Pisarchik & Feudel, 2014].…”

Periodic parametric perturbation control for a 3D autonomous chaotic system and its dynamics at infinity

“…Furthermore, researchers frequently encounter chaos control and chaos synchronization problems in many physical chaotic systems due to the sensitivity of chaos, and they have developed many methods and techniques over the last few decades, such as feedback control, pulse control, adaptive control, passivity control [10,24,25,26] and different kinds of synchronization like complete synchronization (CS, i. e. Identical synchronization (IS)), phase synchronization (PS), lag synchronization (LS), anticipatory synchronization (AS), GS, and multiplexing synchronization (MS) [1, 9]. As we know, the main method used in previous studies is numerical computation, and the qualitative analysis of orbit structure and periodic orbits have not been well studied [6,18,22,29].…”