Is that Really Hidden? The Presence of Complex Fixed-Points in Chaotic Flows with No Equilibria

Abstract

In this letter we investigate the role of complex fixed-points in finding hidden attractors in chaotic flows with no equilibria. If these attractors could be found by starting the trajectory in the neighborhood of complex fixed-points, maybe it would be better not to call them hidden.

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.…”

“…The hidden attractors -the ones whose basins of attraction do not intersect with the neighborhood of any equilibrium, have been found in Chua's circuit [1][2][3][4], Lorenz type systems [5][6][7], chaotic flows [8][9][10][11] and others. In [12] the authors study a time-delayed system, and in [13] hidden oscillations of autonomous van der PolDuffing oscillator are described.…”

Multistability: Uncovering hidden attractors

“…For example, the hidden attractor is the periodic or chaotic attractor in the system without equilibria or with the only stable equilibrium (a special case of multistability and coexistence of attractors). Various examples of hidden attractors are presented in [18][19][20][21][22][23][24][25].Contrary to the self-exited attractors for numerical localization of hidden attractors it is necessary to develop special analytical-numerical procedures, since there are no similar transient processes leading to such attractors from the neighborhoods of the unstable fixed points [26,27]. …”

“…For example, the hidden attractor is the periodic or chaotic attractor in the system without equilibria or with the only stable equilibrium (a special case of multistability and coexistence of attractors). Various examples of hidden attractors are presented in [18][19][20][21][22][23][24][25].…”