Switching between Dissipative and Conservative Behaviors in a Modified Hyperchaotic System with the Variation of Its Parameter

The paper reports a modified 4D autonomous hyperchaotic system with an unusual characteristic. The modified system exhibits dissipative behavior for some ranges of a parameter and conservative behavior for the other ranges of the same parameter. Thus, there is a switching between dissipative and conservative behaviors of the proposed system. In the conservative range, the system exhibits chaotic orbit. Again in the dissipative range, the system, with its considered sets of parameters, exhibits strange attractors. Thus, both the dissipative and conservative behaviors exist in the same system with the switching of its parameter. Such behavior of a system is rarely reported in the literature. Further, the equilibria of the system are located on the surface-shape. The proposed system is implemented and simulated using Field Programmable Gate Array (FPGA) and Multisim simulation softwares.

Simple Chaotic Systems with Specific Analytical Solutions

In this paper, a new structure of chaotic systems is proposed. There are many examples of differential equations with analytic solutions. Chaotic systems cannot be studied with the classical methods. However, in this paper we show that a system that has a simple analytical solution can also have a strange attractor. The main goal of this paper is to show examples of chaotic systems with a simple analytical solution that is unstable so that the chaotic orbit does not track it. We believe the structures presented here are new. Two categories of chaotic systems are described, and their dynamical properties are investigated. The proposed systems have analytic solutions that exist far from the equilibrium. Of course, all strange attractors are dense in unstable periodic orbits, but mostly the equations that describe these orbits are unknown and difficult to calculate. The analytical solutions provide examples where the orbits can be calculated despite their instability.

Multiple Attractor Bifurcation in Three-Dimensional Piecewise Linear Maps

Multiple attractor bifurcations lead to simultaneous creation of multiple stable orbits. This may be damaging for practical systems as there is a fundamental uncertainty regarding which orbit the system will follow after a bifurcation. Such bifurcations are known to occur in piecewise smooth maps, which model many practical and engineering systems. So far the occurrence of such bifurcations have been investigated in the context of 2D piecewise linear maps. In this paper, we investigate multiple attractor bifurcations in a three-dimensional piecewise linear normal form map. We show the occurrence of different types of multiple attractor bifurcations in the system, like the simultaneous creation of a period-2 orbit, a period-3 orbit and an unstable chaotic orbit; a mode-locked torus, an ergodic torus and periodic orbits; a one-loop torus and a two-loop torus; a one-loop mode-locked torus and a two-loop mode-locked torus; a one-piece chaotic orbit and a 3-piece chaotic orbit, etc. As orbits lie on unstable manifolds of fixed points, the structure of unstable manifold plays an important role in understanding the coexistence of attractors. In this work, we show that interplay between 1D and 2D stable and unstable manifolds plays an important role in global bifurcations that can give rise to multiple coexisting attractors.

On a Comprehensive Topological Analysis of Moore Spiegel Attractor

A topological analysis of the attractor associated with the Moore–Spiegel nonlinear system is performed, following the basic idea laid down by Gilmore and Lefranc (2002, The Topology of Chaos, Wiley, Hoboken, NJ). Starting with the usual fixed point analysis and their stability, we proceed to study in detail the process of chaotic orbit extraction with the help of close return map. This is then used to construct the symbolic dynamics associated with it, which is helpful in understanding the sequential change taking place inside the attractor. In the next part, we show how to characterize the evolution of the attractor from its birth to the crisis by finding out the homoclinic orbit and the corresponding unstable manifold. In the concluding part of the paper, we show how all the pertinent information of the attractor can be encoded in the template, leading to the explicit realization of linking numbers and the relative rotation rates. In the concluding section, we have touched upon a new approach to chaotic dynamics, using the flow curvature manifold to display the relative positioning of the attractor in relation to the fixed points and the null lines.

Interaction Between Coexisting Orbits in Impact Oscillators

Impact oscillators exhibit an abrupt onset of chaos close to grazing due to the square-root singularity in their discrete time maps. In practical applications, this large-amplitude chaotic vibration needs to be avoided. It has been shown that this can be achieved if the ratio of the natural frequency of the oscillator ω0 and the forcing frequency is an even integer. But, in practice, it is difficult to set a parameter at such a precise value. We show that in systems with square-root singularity (prestressed impacting surface), there exists a range of ω0 around the theoretical value over which the chaotic orbit does not occur, and that this is due to an interplay between the main attractor and coexisting orbits. We show that this range of forcing frequency has exponential dependence on the amount of prestress as well as on the stiffness ratio of the springs.

Exploring Chaos and Entanglement in the Hénon–Heiles System Using Squeezed Coherent States

Quantum entanglement in the Hénon–Heiles system is analyzed using the squeezed coherent state. Enhancement of quantum entanglement via squeezing is explored in connection with chaotic and regular dynamics of the system. It is found that the entanglement enhancement via squeezing is implicitly linked to the local structure of the classical phase-space and it shows a clear quantum-classical correspondence. In particular, the entanglement enhancement via squeezing is found to be negligible for a highly chaotic orbit compared to the regular and weakly chaotic orbits, and shows a clear correspondence to the degree of chaos present in the classical initial condition. We believe that these results might be useful to develop efficient strategies to enhance entanglement in quantum systems.