New Chaotic Dynamical System with a Conic-Shaped Equilibrium Located on the Plane Structure

Abstract

Featured Application: Theoretical foundations provided in this contribution demonstrate that the existence of chaos is not bound to mathematical models and electronic circuits with singular equilibrium points. The practical part of this work proves that a structurally stable hidden chaotic attractor can be both excited and experimentally observed in a lumped electronic circuit; that is, it can be measured by a digital oscilloscope.Abstract: This paper presents a new autonomous deterministic dynamical system with equilibrium degenerated into a plane-oriented hyperbolic geometrical structure. It is demonstrated via numerical analysis and laboratory experiments that the discovered system has both a structurally stable strange attractor and experimentally measurable chaotic behavior. It is shown that the evolution of complex dynamics can be associated with a single parameter of a mathematical model and, due to one-to-one correspondence, to a single circuit parameter. Two-dimensional high resolution plots of the largest Lyapunov exponent and basins of attraction expressed in terms of final state energy are calculated and put into the context of the discovered third-order mathematical model and real chaotic oscillator. Both voltage-and current-mode analog chaotic oscillators are presented and verified by visualization of the typical chaotic attractor in a different fashion.

Current-Mode Network Structures Dedicated for Simulation of Dynamical Systems with Plane Continuum of Equilibrium

“…47 and a comprehensive study of another di®erent \hyperbolic" case can be found in Ref. 48. The¯rst one can be expressed as (2) with the auxiliary functions…”

A Novel Mega-stable Chaotic Circuit

“…Recently there has been growing attention in finding chaotic systems with special qualities. Systems with no equilibrium [3], [4], with stable equilibria [5], [6], with curves of equilibria [7][8][9], with surface of equilibria [10][11][12], with multi-scroll attractors [13], with hidden attractors [14], [15], with amplitude control [16], [17], with simplest form , having hyperchaos [18][19][20], having fractional order form [21][22][23], with topological horseshoes [24], [25], and with extreme multistability [26][27][28][29], are examples of them. Another major category of chaotic systems includes periodically-forced nonlinear oscillators [30].…”

A New 4-D Chaotic System with Hidden Attractor and its Circuit Implementation

“…On the other hand, a hidden attractor has a basin of attraction which does not contain neighbourhoods of equilibrium points. Classical examples of self-excited attractors are Lorenz system [32] [42]), chaotic systems with infinite number of equilibrium points ( [43]- [44]), chaotic systems with stable equilibrium points ( [45]- [46]) and chaotic systems with line equilibrium ( [47]- [48]). A special case of the hidden attractors is a multi-stability and coexistence of attractors can be searched on ([49]- [52]).…”