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Art of electronics

Hidden attractors in dynamical systems

Using a systematic computer search, nine simple chaotic flows with quadratic nonlinearities were found that have the unusual feature of having a line equilibrium. Such systems belong to a newly introduced category of chaotic systems with hidden attractors that are important and potentially problematic in engineering applications.

Simple chaotic flows with a line equilibrium

In this paper, we discuss self-excited and hidden attractors for systems of differential equations. We considered the example of a Lorenz-like system derived from the well-known Glukhovsky-Dolghansky and Rabinovich systems, to demonstrate the analysis of self-excited and hidden attractors and their characteristics. We applied the fishing principle to demonstrate the existence of a homoclinic orbit, proved the dissipativity and completeness of the system, and found absorbing and positively invariant sets. We have shown that this system has a self-excited attractor and a hidden attractor for certain parameters. The upper estimates of the Lyapunov dimension of self-excited and hidden attractors were obtained analytically.

/pdf/homoclinic-orbits-and-self-excited-and-hidden-attractors-in-5gtgeo8vbh.pdf

Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion

Using the Routh–Hurwitz stability criterion and a systematic computer search, 23 simple chaotic flows with quadratic nonlinearities were found that have the unusual feature of having a coexisting stable equilibrium point. Such systems belong to a newly introduced category of chaotic systems with hidden attractors that are important and potentially problematic in engineering applications.

Simple chaotic flows with one stable equilibrium

The classical attractors of Lorenz, Rossler, Chua, Chen, and other widely-known attractors are those excited from unstable equilibria. From computational point of view this allows one to use standard numerical method, in which after transient process a trajectory, started from a point of unstable manifold in the neighborhood of equilibrium, reaches an attractor and identifies it. However there are attractors of another type: hidden attractors, a basin of attraction of which does not contain neighborhoods of equilibria. Study of hidden oscillations and attractors requires the development of new analytical and numerical methods which will be considered in this paper.

http://www.wseas.us/e-library/conferences/2011/Corfu/SYSTEMS/SYSTEMS-47.pdf

Hidden oscillations in dynamical systems

Hidden oscillations in nonlinear control systems

In this report the existence of hidden attractors in Rabinovich system, phase-locked loop and coupled Chua circuits is considered. It is shown that the existence of hidden attractors may complicate the analysis of the systems and significantly affect the synchronization.

Hidden attractor in the Rabinovich system, Chua circuits and PLL

For continuous and discrete floating phase-locked loops, conditions of local and global stability are obtained. For discrete systems, period-doubling bifurcations are described. The results obtained are applied to the solution of the problem of eliminating the clock skew in digital signal processors.

S. M. Seledzhi

Papers

Hidden oscillations in dynamical systems

Hidden oscillations in nonlinear control systems

Hidden attractor in the Rabinovich system, Chua circuits and PLL

Stability and bifurcations of phase-locked loops for digital signal processors

Hidden oscillations in SPICE simulation of two-phase Costas loop with non-linear VCO