Statistical Measures and Diffusion Dynamics in a Modified Chua's Circuit Equation With Multiscroll Attractors

Abstract

In this paper the focus is set on a modified Chua's circuit model equation with saw-tooth function in place of piece-wise linear function of Chua's circuit displaying multi-scroll chaotic attractors.We study the characteristic properties of first passage times (t FPT s) to nth scroll chaotic attractor, residence times (t RT s) on a scroll attractor and returned times (t RET s) to the middle-scroll attractor.t FPT s exhibit a series of Gaussian-like distribution followed by a long tail continuous distribution. t RT s and t RET s show completely discrete distribution. Power-law variation of mean values of t FPT s, t RT s and t RET s with a control parameter is found. On the other hand, mean values of t FPT s and t RET s have linear dependence with the number of the scroll attractors for fixed values of the control parameter. For the system with infinite scroll chaotic attractors normal diffusive motion occurs.In the normal diffusion process the mean square displacement grows linearly with time.

Widening of the basins of attraction of a multistable switching dynamical system with the location of symmetric equilibria

“…Consider a larger value of δ, for example δ = 55. With this value the symmetric equilibria located at X * p,n = (±36.6666, 0, 0) T are very far from each other comparing to the size of the attractor S p , as it can be appreciated in Figure 4 a), where the projection onto the (x 1 , x 2 ) plane of a trajectory of the attractor of the system (3) with (7) and δ = 55 is depicted. Notice the equilibria on the positive side marked with red asterisk, which due to the scale of the graphic these equilibria seems to be located almost in the same place.…”

“…where N A corresponds to the number of initial conditions that trajectories asymptotically go to the attractor A regarding the full grid of points in N , both considered from the numerical simulation. The results of this can be appreciated in the Table 1 where the values are evaluated through the integration method Thus, the relation of the system regarding the commutation law (7) and its parameter δ prompts to the variation of the distance between the two double scroll attractor. This will be addressed next.…”

“…Therefore, resulting in an increasing unstable oscillation that cannon be bounded to the system's attractor. On the other hand, if the system had presented more equilibria at this side of the commutation law, the trajectory could have been pulled due to the strength of each M s (regarding the position in which the trajectory is located) and bound the trajectory towards the location of the attractor, which in the system with eqs (7). for δ = 1/2 results in the four scrolls attractor A M .…”

“…Section of the basins of attraction in the plane x 3 = 0 generated by eq. (3) and(7) with a) δ = 1/2; b) δ = 1; c) δ = 3; d) δ = 5 and e) δ = 15 onto the (x 1 , x 2 ) plane. (36.6666, 0, 0) T .…”

Vibrational and Ghost-Vibrational Resonances in a Modified Chua's Circuit Model Equation

“…In recent years, the study of the n-scroll Chua's circuit has received a great deal of interest. For example, adaptive control of chaotic dynamics [Zou et al, 2006], existence of n-scroll chaotic attractors, global stability of equilibrium points and feedback control laws for synchronization , stabilization and synchronization of n-scroll chaotic orbits ], controlling of unstable equilibrium points and periodic orbits [Boukabou et al, 2009], diffusion dynamics and characteristic features of first passage times to nth scroll attractor and residence times on a scroll attractor [Sakthivel et al, 2012] were analysed. Furthermore, the influence of the number of equilibrium points on the characteristics of stochastic and coherence resonances [Arathi et al, 2013] have been investigated.…”