Bifurcation and chaos in simple jerk dynamical systems

In recent years, it is observed that the third-order explicit autonomous differential equation, named as jerk equation, represents an interesting sub-class of dynamical systems that can exhibit many major features of the regular and chaotic motion. In this paper, we investigate the global dynamics of a special family of jerk systems {ie075-01}, whereG(x) is a non-linear function, which are known to exhibit chaotic behaviour at some parameter values. We particularly identify the regions of parameter space with different asymptotic dynamics using some analytical methods as well as extensive Lyapunov spectra calculation in complete parameter space. We also investigate the effect of weakening as well as strengthening of the non-linearity in theG(x) function on the global dynamics of these jerk dynamical systems. As a result, we reach to an important conclusion for these jerk dynamical systems that a certain amount of non-linearity is sufficient for exhibiting chaotic behaviour but increasing the non-linearity does not lead to larger regions of parameter space exhibiting chaos.     

Perspectives on relativistic quantum chaos

Quantum Chaos has been investigated for about a half century.It is an old yet vigorous interdisciplinary field with new concepts and interesting topics emerging constantly.Recent years have witnessed a growing interest in quantum chaos in relativistic quantum systems,leading to the still developing field of relativistic quantum chaos.The purpose of this paper is not to provide a thorough review of this area,but rather to outline the basics and introduce the key concepts and methods in a concise way.A few representative topics are discussed,which may help the readers to quickly grasp the essentials of relativistic quantum chaos.A brief overview of the general topics in quantum chaos has also been provided with rich references.     

Controlling chaos with simple limiters

New experimental results demonstrate that chaos control can be accomplished using controllers that are very simple relative to the system being controlled. Chaotic dynamics in a driven pendulum and a double scroll circuit are controlled using an adjustable, passive limiter-a weight for the pendulum and a diode for the circuit. For both experiments, multiple unstable periodic orbits are selectively controlled using minimal perturbations. These physical examples suggest that chaos control can be practically applied to a much wider array of important problems than initially thought possible.     

Strong-coupling diffusion in relativistic systems

Different from the early universe, heavy-ion collisions at very high energies do not reach statistical equilibrium, although thermal models explain many of their features. To account for nonequilibrium strong-coupling effects, a Fokker-Planck equation with time-dependent diffusion coefficient is proposed. A schematic model for rapidity distributions of participant baryons is set up and solved analytically. The evolution from SIS via AGS and SPS to RHIC energies is discussed. Strong-coupling diffusion produces double-peaked spectra in central collisions at the higher SPS momentum of 158 A.GeV/c and beyond.     

Random-phase-induced chaos in power systems

This paper studies how random phase (namely, noise-perturbed phase) effects the dynamical behaviours of a simple model of power system which operates in a stable regime far away from chaotic behaviour in the absence of noise. It finds that when the phase perturbation is weak, chaos is absent in power systems. With the increase of disturbed intensity $\sigma$, power systems become unstable and fall into chaos as $\sigma$ further increases. These phenomena imply that random phase can induce and enhance chaos in power systems. Furthermore, the possible mechanism behind the action of random phase is addressed.     

Effective masses in a simple relativistic energy-band scheme

A relativistic generalization of the effective mass concept is developed within the framework of simplest one-dimensional models by using Dirac equation and the transfer-matrix approach. The numerical analysis performed makes possible a comparison with the conventional quantum mechanical theory.     

Shilnikov sense chaos in a simple three-dimensional system

The Shilnikov sense Smale horseshoe chaos in a simple 3D nonlinear system is studied. The proportional integral derivative (PID) controller is improved by introducing the quadratic and cubic nonlinearities into the governing equations. For the discussion of chaos, the bifurcate parameter value is selected in a reasonable regime at the requirement of the Shilnikov theorem. The analytic expression of the Shilnikov type homoclinic orbit is accomplished. It depends on the series form of the manifolds surrounding the saddle-focus equilibrium. Then the methodology is extended to research the dynamical behaviours of the simplified solar-wind-driven-magnetosphere-ionosphere system. As is illustrated, the Lyapunov characteristic exponent spectra of the two systems indicate the existence of chaotic attractor under some specific parameter conditions.     

Hamiltonian systems of relativistic particles

Hamiltonian relativistic dynamics is formulated. Models can be constructed by writing down Hamiltonians. These Hamiltonians are generating functions leading to the equations of motion. These equations admit a Poisson bracket form, analogous to Heisenberg evolution equations. The Hamiltonians are not the energy. They are rather related to the masses of the interacting particles. In contrast, the energy is the time component of a conserved vector.Among tractable examples we find a relativistic generalization of the harmonic oscillator. An attempt at quantization by means of coupled wave equations is made.     

Multi-strange systems in relativistic mean fields

Characteristics of multi- hypernuclei are investigated within the relativistic mean-field theory. Both linear and nonlinear models and a variety of couplings fitted to ordinary hypernuclei have been investigated. All the parametrizations used in the present work predict qualitatively similar dependence of the studied quantities (rms radii, binding energies, densities) on a number of hyperons.Deceased on May 5, 1991     

Suppression of dynamic chaos in Hamiltonian systems

The paper describes the results of a recent numerical study on the canonical mapping with a sawtooth force. The dynamic effects of the formation of invariant resonance structures of various orders, whose presence prevents the development of global chaos and restricts momentum diffusion in the phase space, are discussed. The dynamic situation near an integer resonance separatrix in the neighborhood of the critical state is studied, and the conditions responsible for the stability of this separatrix in the critical state are determined. Along with the mapping, the related continuous Hamiltonian system is considered. For this system, the separatrix mapping and the Mel’nikov-Arnold integral are introduced, whose analysis facilitates understanding the reasons responsible for the unusual dynamics. This dynamics is shown to be preserved under substantial saw shape changes. Relevant new problems and open questions are formulated.     

Description of physical systems

A general “logical” scheme, containing both classical and quantum mechanics, is developed on the basis of plausible axioms. We introduce the division of states and yes-no measurements into sharp and diffuse ones, and prove that sharp states possess their carriers. Owing to this result, the existence of lattice joins and meets is proved for a wide class of elements of the logic. This “semi-lattice” structure gives the familiar lattice picture for special cases of classical and quantum mechanics. The notion of quantum superposition is introduced in this general scheme. It is proved that if in a theory appear nontrivial quantum superpositions, then this theory is “undeterministic” and vise versa. Further analysis of the pure state space leads to the construction of the canonical embedding of the general logic into an orthomodular complete ortho-lattice. After defining the probability of transition between pure states, the pure state space appears to be a generalization of Mielnik's “probability space” of quantum mechanics.     

Collapse of systems of relativistic particles

We prove in a very simple way that if a system of N non-relativistic particles interacting by Coulomb or gravitational forces has a (negative) binding energy increasing faster than N, the corresponding system necessarily collapses for N big enough if particles are given relativistic kinetic energy. At the same time our method allows to improve considerably a recently obtained sufficient condition on coupling constants for the collapse of ordinary matter.     

From low-dimensional synchronous chaos to high-dimensional desynchronous spatiotemporal chaos in coupled systems

The dynamic behavior of coupled chaotic oscillators is investigated. For small coupling, chaotic state undergoes a transition from a spatially disordered phase to an ordered phase with an orientation symmetry breaking. For large coupling, a transition from full synchronization to partial synchronization with translation symmetry breaking is observed. Two bifurcation branches, one in-phase branch starting from synchronous chaos and the other antiphase branch bifurcated from spatially random chaos, are identified by varying coupling strength epsilon. Hysteresis, bistability, and first-order transitions between these two branches are observed.     

一个简单硬球碰撞问题中的混沌

对于一个在铅垂线上的两个硬球及刚性地面组成的硬球碰撞体系,计算了它的Lyapunov指数．因为该Lyapunov指数的正定性,所以该体系是一个混沌体系．     

Quantization of relativistic systems with constraints

The general dynamical system with constraints is quantized, and the S-matrix is constructed in the most general class of gauges including relativistic ones. In the case when constraints do not form a group a new type of additional diagrams arises securing unitarity of the theory: the four-fermion interaction of ghost fields.     

Supersymmetry and duality in relativistic oscillator systems

A simple mapping is defined which takes the relativistic oscillator on a (1+1)-dimensional spacetime into a dual system consisting of a nonrelativistic oscillator confined to a circular space. The radius of the circle in the dual system measures the size of relativistic effects in the original system. It is shown that both of the dual oscillator systems have supersymmetric partners, and the exact wavefunctions and energy levels of all four of the intertwined systems are found.