Hidden Attractors and Dynamics of a General Autonomous van der Pol–Duffing Oscillator

Abstract

In this paper, a general autonomous van der Pol–Duffing oscillator is studied. Several issues, such as periodic bifurcations and the dynamical structures of the system are investigated either analytically or numerically. Especially, a phenomenon of hidden attractors is noticed and an algorithm for the location of hidden attractors is given. The obtained results show that hidden attractors exist around chaotic attractors.

On differences and similarities in the analysis of Lorenz, Chen, and Lu systems

“…For example, hidden attractors are attractors in the systems with no equilibria or with only one stable equilibrium (a special case of multistable systems and coexistence of attractors). 8 Recent examples of hidden attractors can be found in [10,28,34,52,57,58,65,66,79,80,83,84]. Multistability is often an undesired situation in many applications, however coexisting self-excited attractors can be found by the standard computational procedure.…”

Perpetual points and hidden attractors in dynamical systems

“…The possible final states of the system -known as its attractors -are the key concept of this theory. Recently, new types of attractors known as the hidden [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] and rare [16][17][18][19][20][21] are thoroughly studied.…”

“…In [12] the authors study a time-delayed system, and in [13] hidden oscillations of autonomous van der PolDuffing oscillator are described. In the control theory [4,14] this type of states is also analyzed.…”

Self-Excited and Hidden Attractors Found Simultaneously in a Modified Chua's Circuit

“…an attractor with a basin of attraction is associated with an unstable equilibrium. However, the attractors of some particular nonlinear dynamical systems are hidden because the generations have nothing to do with any unstable equilibrium [Leonov et al, 2011[Leonov et al, , 2012Zhao et al, 2014]. Due to existing hidden attractors, some particular dynamical systems related to line equilibrium, or no equilibrium, or stable equilibrium have received much attention in recent years [Jafari & Sprott, 2013;Li & Sprott, 2014;Pham et al, 2014;Wei et al, 2014].…”