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

Abstract

Perpetual points (PPs) are special critical points for which the magnitude of acceleration describing dynamics drops to zero, while the motion is still possible (stationary points are excluded), e.g. considering the motion of the particle in the potential field, at perpetual point it has zero acceleration and non-zero velocity. We show that using PPs we can trace all the stable fixed points in the system, and that the structure of trajectories leading from former points to stable equilibria may be similar to orbits obtained from unstable stationary points. Moreover, we argue that the concept of perpetual points may be useful in tracing unexpected attractors (hidden or rare attractors with small basins of attraction). We show potential applicability of this approach by analysing several representative systems of physical significance, including the damped oscillator, pendula and the Henon map. We suggest that perpetual points may be a useful tool for localization of co-existing attractors in dynamical systems.

Memristor‐Based Canonical Chua’s Circuit: Extreme Multistability in Voltage‐Current Domain and Its Controllability in Flux‐Charge Domain

“…For the circuit parameters given in Table 1, the normalized parameters , , , and in (14) are the same as those given in (5). With these determined parameters, the initial conditions-dependent extreme multistability in the memristor-based canonical Chua's circuit can be effectively controlled by adjusting the system parameters , 1 , 2 , and 3 .…”

“…Consequently, three eigenvalues of the model (14) at are yielded by solving the following characteristic polynomial:…”

“…When the normalized system parameters a, b, c, and for the model (14) are fixed as given in (5) and the relationship of −2 √ 10 ≤ ≤ 2 √ 10 is satisfied, three equilibrium points consisting of one zero equilibrium point and two nonzero equilibrium points are solved from (16) as…”

“…Extreme multistability is a fantastic kind of multistability, which makes a nonlinear dynamical circuit or system supply great flexibility for its potential uses in chaos-based engineering applications [10][11][12], but also raises new challenges for its control of the existing multiple stable states [11][12][13][14]. Generally, multistability is confirmed in hardware experiments by randomly switching on and off experimental circuit supplies [9,[15][16][17][18][19][20][21] or by MATLAB numerical or PSPICE/PSIM circuit simulations [4][5][6][7][8][22][23][24][25][26][27][28].…”

Experimental investigation of perpetual points in mechanical systems

“…The lack of unstable fixed points in the neighborhood of hidden attractors led to the hypothesis that PPs could serve as a tool to locate hidden attractors [3,4]. This concept has been widely investigated recently, and despite some limitations [12], PPs prove to be a useful tool to localize hidden and rare attractors in multiple types of systems [3,[13][14][15]. Especially in electrical systems there is a need to search for efficient tool to localize non-trivial solutions.…”