Coexistence of Chaos with Hyperchaos, Period-3 Doubling Bifurcation, and Transient Chaos in the Hyperchaotic Oscillator with Gyrators

Abstract

In this paper, the dynamics of the paradigmatic hyperchaotic oscillator with gyrators introduced by Tamasevicius and co-workers (referred to as the TCMNL oscillator hereafter) is considered. This well known hyperchaotic oscillator with active RC realization of inductors is suitable for integrated circuit implementation. Unlike previous literature based on piecewise-linear approximation methods, I derive a new (smooth) mathematical model based on the Shockley diode equation to explore the dynamics of the oscillator. Various tools for detecting chaos including bifurcation diagrams, Lyapunov exponents, frequency spectra, phase portraits and Poincaré sections are exploited to establish the connection between the system parameters and various complex dynamic regimes (e.g. hyperchaos, period-3 doubling bifurcation, coexistence of attractors, transient chaos) of the hyperchaotic oscillator. One of the most interesting and striking features of this oscillator discovered/revealed in this work is the coexistence of a hyperchaotic attractor with a chaotic one over a broad range of system parameters. This phenomenon was not reported previously and therefore represents a meaningful contribution to the understanding of the behavior of nonlinear dynamical systems in general. A close agreement is observed between theoretical and experimental analyses.

An Improved Memristive Diode Bridge‐Based Band Pass Filter Chaotic Circuit

“…It is worth noting that the stability depends on the memristor initial condition in a memristive dynamical circuit, leading to the occurrence of coexisting multiple attractors [9,13]. The coexistence of different kinds of attractors, called multistability, reveals a rich diversity of stable states in nonlinear dynamical systems [12,[24][25][26][27][28][29][30][31][32] and makes the system offer great flexibility, which can be used for image processing or taken as an additional source of randomness used for many information engineering applications [32][33][34][35][36][37]. Therefore, it is very attractive to seek for a simple memristive chaotic circuit that has the striking dynamical behavior of coexisting multiple attractors.…”

Asymmetric Double Strange Attractors in a Simple Autonomous Jerk Circuit

“…More interestingly, our model (4) represents one of the simplest autonomous 3D systems reported to date, capable of exhibiting asymmetric double strange attractors (see Sections 4 and 5) depending uniquely on the choice of initial conditions [3,4,17].…”

“…Such systems exhibit pairs of mutually symmetric attractors that merge to form a single symmetric one via the well-known attractor merging crisis as a parameter is varied. However, asymmetric multistability (i.e., coexistence of nonsymmetric attractors) is also reported in systems without any symmetry such as Colpitts oscillator [3], Newton-Leipnik system [17], and hyperchaotic oscillator with gyrators [4]. In the present contribution, we consider the dynamics of an extremely simple chaotic jerk circuit recently introduced by Sprott [18] with particular attention on the chaos mechanism as well as the possibility of multiple coexisting attractors.…”

“…They have done many works in various fields of science and engineering such as electrical circuits [1][2][3][4][5], laser systems [6,7], biological systems [8], and chemical reactions [9]. Systems with only one attractor are called monostable systems.…”

A Novel Image Encryption Scheme Based on Self-Synchronous Chaotic Stream Cipher and Wavelet Transform

“…However, because of the high correlation and redundancy of adjacent pixels of the digital image, some international standard encryption algorithms are not suitable for image encryption, including 3DES (Triple Data Encryption Algorithm), IDEA (International Data Encryption Algorithm), and AES (Advanced Encryption Standard), etc. On the other hand, the chaotic nonlinear dynamic system has some good characteristics, such as positive Lyapunov exponents, ergodicity, sensitivity to initial conditions, topological transitivity, and unpredictability [2][3][4][5], and was widely applied in the field of cryptography and secret communication.…”