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.
Highlights
It was a breakthrough discovery that the irregular motion, with a broadband frequency spectrum with Gaussian-like distribution and extreme sensitivity to changes of the initial conditions, can be observed in the class of autonomous dynamical systems with three degrees of freedom
The proposed mathematical model represents a new contribution to the class of dynamical The proposed mathematical model represents a new contribution to the class of dynamical systems with hyperbolic equilibrium listed in [26], that is, the mathematical with an infinite
The proposed mathematical model represents a new contribution to the class models of dynamical systems systems with hyperbolic equilibrium listed in [26], that is, the mathematical models with an infinite number of equilibrium points that are arranged into a curved object
Summary
It was a breakthrough discovery that the irregular motion, with a broadband frequency spectrum with Gaussian-like distribution and extreme sensitivity to changes of the initial conditions, can be observed in the class of autonomous dynamical systems with three degrees of freedom Such behavior, known as chaotic solution, has often been misinterpreted as a mixture of harmonic signals and noise. Typical examples of very simple chaotic flows can be expressed by a single third-order differential equation as previously addressed [15] Such systems have a direct relation with Newtonian motion dynamics, since individual state variables can be interpreted as position, velocity and acceleration. This process and its superior procedure known as state attractor reconstruction is less precise, returns incorrect results in the case of wrong choices of method input parameters and should be chosen if the mathematical model is unknown, that is, the dynamical system is in the form of a black box
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