Search results for "chaotic attractor"
showing 7 items of 7 documents
Graphical Structure of Attraction Basins of Hidden Chaotic Attractors : The Rabinovich-Fabrikant System
2019
The attraction basin of hidden attractors does not intersect with small neighborhoods of any equilibrium point. To the best of our knowledge this property has not been explored using realtime interactive three-dimensions graphics. Aided by advanced computer graphic analysis, in this paper, we explore this characteristic of a particular nonlinear system with very rich and unusual dynamics, the Rabinovich–Fabrikant system. It is shown that there exists a neighborhood of one of the unstable equilibria within which the initial conditions do not lead to the considered hidden chaotic attractor, but to one of the stable equilibria or are divergent. The trajectories starting from any neighborhood o…
Approximate renormalization-group transformation for Hamiltonian systems with three degrees of freedom
1999
We construct an approximate renormalization transformation that combines Kolmogorov-Arnold-Moser (KAM)and renormalization-group techniques, to analyze instabilities in Hamiltonian systems with three degrees of freedom. This scheme is implemented both for isoenergetically nondegenerate and for degenerate Hamiltonians. For the spiral mean frequency vector, we find numerically that the iterations of the transformation on nondegenerate Hamiltonians tend to degenerate ones on the critical surface. As a consequence, isoenergetically degenerate and nondegenerate Hamiltonians belong to the same universality class, and thus the corresponding critical invariant tori have the same type of scaling prop…
The chaotic Dadras–Momeni system: control and hyperchaotification
2015
In this paper a novel three-dimensional autonomous chaotic system, the so called Dadras-Momeni system, is considered and two different control techniques are employed to realize chaos control and chaos synchronization. Firstly, the optimal control of the chaotic system is discussed and an open loop feedback controller is proposed to stabilize the system states to one of the system equilibria, minimizing the cost function by virtue of the Pontryagin’s minimum principle. Then, an adaptive control law and an update rule for uncertain parameters, based on Lyapunov stability theory, are designed both to drive the system trajectories to an equilibrium or to realize a complete synchronization of t…
Hidden Strange Nonchaotic Attractors
2021
In this paper, it is found numerically that the previously found hidden chaotic attractors of the Rabinovich–Fabrikant system actually present the characteristics of strange nonchaotic attractors. For a range of the bifurcation parameter, the hidden attractor is manifestly fractal with aperiodic dynamics, and even the finite-time largest Lyapunov exponent, a measure of trajectory separation with nearby initial conditions, is negative. To verify these characteristics numerically, the finite-time Lyapunov exponents, ‘0-1’ test, power spectra density, and recurrence plot are used. Beside the considered hidden strange nonchaotic attractor, a self-excited chaotic attractor and a quasiperiodic at…
Rich dynamics and anticontrol of extinction in a prey-predator system
2019
This paper reveals some new and rich dynamics of a two-dimensional prey-predator system and to anticontrol the extinction of one of the species. For a particular value of the bifurcation parameter, one of the system variable dynamics is going to extinct, while another remains chaotic. To prevent the extinction, a simple anticontrol algorithm is applied so that the system orbits can escape from the vanishing trap. As the bifurcation parameter increases, the system presents quasiperiodic, stable, chaotic and also hyperchaotic orbits. Some of the chaotic attractors are Kaplan-Yorke type, in the sense that the sum of its Lyapunov exponents is positive. Also, atypically for undriven discrete sys…
Complex dynamics, hidden attractors and continuous approximation of a fractional-order hyperchaotic PWC system
2018
In this paper, a continuous approximation to studying a class of PWC systems of fractionalorder is presented. Some known results of set-valued analysis and differential inclusions are utilized. The example of a hyperchaotic PWC system of fractional order is analyzed. It is found that without equilibria, the system has hidden attractors.