Bounds for compact invariant sets of the system describing dynamics of the nuclear spin generator

This paper is concerned with the localization problem of compact invariant sets of the system describing dynamics of the nuclear spin generator. We establish that all compact invariant sets of this system are located in the intersection of a ball with two frusta and compute its parameters. In addition, localization by using the two-parameter set of parabolic cylinders is described. Our results are obtained with help of the iteration theorem concerning a localization of compact invariant sets. One numerical example illustrating a localization of a chaotic attractor is presented as well.     

Bounding a domain containing all compact invariant sets of the permanent-magnet motor system

In this paper we characterize a locus of compact invariant sets of the system describing dynamics of the permanent-magnet synchronous motor (PMSM). We establish that all compact invariant sets of this system are contained in the intersection of one-parameter set of ellipsoids and compute its parameters. In addition, localizations by using a parabolic cylinder, an elliptic paraboloid and a hyperbolic cylinder are obtained. Simple polytopic bounds are derived with help of these localizations. Most of localizations mentioned above remain valid for more specific motor systems; namely, for the interior magnet PMSM and for the surface magnet PMSM. Yet another localization set for the interior magnet PMSM is described. Examples of localization of chaotic attractors existing for some values of parameters of PMSMs are presented as well.     

Localization of invariant compacta of autonomous systems

A class of problems that may be characterized as localization problems are becoming increasingly popular in qualitative theory of differential equations [1–15]. The specific formulations differ, but geometrically all search for phase space subsets with desired properties, e.g., contain certain solutions of the system of differential equations. Such problems include construction of positive invariant sets that contain certain separatrices of the Lorenz system [1], analysis of asymptotic behavior of solutions of the Lorenz system and determination of sets that contain the Lorenz attractor [2–5, 14], as well as determination of sets containing all periodic trajectories [6–13], separatrices, and other trajectories [10, 11]. Such sets may be naturally called localizing sets and it is obviously interesting to study methods and results that produce exact or nearly exact localizing sets for each phase space structure. In this article we focus on localization of the invariant compact sets in the phase space of a differential equation system, specifically the problem of finding phase space subsets that contain all the invariant compacta of the system. Invariant compact sets are equilibria, periodic trajectories, separatrices, limit cycles, invariant tori, and other sets and their finite unions. These sets and their properties largely determine the phase space structure and the qualitative behavior of solutions of the differential equation system.     

Pullback exponential attractors for nonautonomous equations Part II: Applications to reaction-diffusion systems

The existence of a pullback exponential attractor being a family of compact and positively invariant sets with a uniform bound on their fractal dimension which at a uniform exponential rate pullback attract bounded subsets of the phase space under the evolution process is proved for the nonautonomous logistic equation and a system of reaction-diffusion equations with time-dependent external forces including the case of the FitzHugh-Nagumo system.     

Global asymptotic stability analysis by the localization method of invariant compact sets

We study the asymptotic stability and the global asymptotic stability of equilibria of autonomous systems of differential equations. We prove necessary and sufficient conditions for the global asymptotic stability of an equilibrium in terms of invariant compact sets and positively invariant sets. To verify these conditions, we use some results of the localization method for invariant compact sets of autonomous systems. These results are related to finding sets that contain all invariant compact sets of the system (localizing sets) and to the behavior of trajectories of the system with respect to localizing sets. We consider an example of a system whose equilibrium belongs to the critical case.     

Hyperchaotic set in continuous chaos–hyperchaos transition

Topological horseshoes with two-directional expansion imply invariant sets with two positive Lyapunov exponents (LE), which are recognized as a signature of hyperchaos. However, we find such horseshoes in two piecewise linear systems and one smooth system, which all exhibit chaotic attractors with one positive LE. The three concrete systems are the simple circuit by Tamaševičius et al., the Matsumoto–Chua–Kobayashi (MCK) circuit and the linearly controlled Lorenz system, respectively. Substantial numerical evidence from these systems suggests that a hyperchaotic set can be embedded in a chaotic attractor with one positive LE, and keeps existing while the attractor becomes hyperchaotic from chaotic. This paper presents such a new scenario of the continuous chaos–hyperchaos transition.     

Localizing sets for invariant compact sets of continuous dynamical systems with a perturbation

A functional method of localization of invariant compact sets, which was earlier developed for autonomous continuous and discrete systems, is generalized to continuous dynamical systems with perturbations. We describe properties of the corresponding localizing sets. By using that method, we construct localizing sets for positively invariant compact sets of the Lorenz system with a perturbation.     

Bifurcation structure of chaotic attractor in switched dynamical systems with spike noise

High-frequency ripple (spike noise) effects in the qualitative properties of DC/DC converter circuits. This study investigates the bifurcation structure of a chaotic attractor in a switched dynamical system with spike noise. First, we introduce the system dynamics and derive the associated Poincaré map. Next, we show the bifurcation structure of the chaotic attractor in a system with spike noise. Finally, we investigate the dynamical effect of spike noise in the existence region of the chaotic attractor compare with that of a chaotic attractor in a system with ideal switching. The results suggest that spike noise enlarges an invariant set and generates a new bifurcation structure of the chaotic attractor.     

Localization of periodic orbits of the Rössler system under variation of its parameters

The localization problem of compact invariant sets of the Rössler system is considered in this paper. The main interest is attracted to a localization of periodic orbits. We establish a number of algebraic conditions imposed on parameters under which the Rössler system has no compact invariant sets contained in half-spaces z > 0; z < 0 and in some others. We prove that if parameters (a, b, c) of the Rössler system are such that this system has no equilibrium points then it has no periodic orbits as well. In addition, we give localization conditions of compact invariant sets by using linear functions and one quadratic function.     

Localizing sets for invariant compact sets of discrete dynamical systems with perturbation and control

We consider the localization problem for the invariant compact sets of a discrete dynamical system with perturbation and control, that is, the problem of constructing domains in the system state space that contain all invariant compact sets of the system. The problem is solved on the basis of a functional method used earlier in localization problems for time-invariant continuous and discrete systems and also for control systems. The properties of the corresponding localizing sets are described.     

Estimating the ultimate bound and positively invariant set for a synchronous motor and its application in chaos synchronization

To estimate the ultimate bound and positively invariant set for a dynamic system is an important but quite challenging task in general. This paper has investigated the ultimate bound and positively invariant set of a permanent magnet synchronous motor system. We combine the Lyapunov stability theory with the comparison principle method. For this system, we derive a three-dimensional ellipsoidal ultimate bound and positively invariant set for all the positive values of its parameters σ, γ. In addition, the two-dimensional bound with respect to x ? y are established. Then, it is the two-dimensional estimation about x ? z. Finally, the result is applied to the study of completely chaos synchronization. Numerical simulations are presented to show the effectiveness of the proposed chaos synchronization scheme. At the same time, one numerical example illustrating a localization of a chaotic attractor is presented as well. Numerical simulation is consistent with the results of theoretical calculation.     

Bounds for a domain containing all compact invariant sets of the system describing the laser–plasma interaction

In this paper we consider the localization problem of compact invariant sets of the system describing the laser–plasma interaction. We establish that this system has an ellipsoidal localization for simple restrictions imposed on its parameters. Then we improve this localization by applying other localizing functions. In addition, we give sufficient conditions under which the origin is the unique compact invariant set.     

Localization of invariant compact sets in uncertain discrete systems

A functional method for the localization of invariant compact sets in discrete autonomous systems is generalized to discrete systems with uncertainty. We describe the properties of the corresponding localizing sets. By using this method, we construct localizing sets for positively invariant compact sets of the discrete Henon system with uncertainty.     

Functional method for the localization of invariant compact sets in discrete systems

We suggest a method for the localization of invariant compact sets in discrete autonomous systems. We describe the properties of the corresponding localizing sets. By using this method, we construct localizing sets for invariant compact sets of the discrete Henon system.     

Bifurcations and chaos of a discrete mathematical model for respiratory process in bacterial culture

The discrete mathematical model for the respiratory process in bacterial culture obtained by Euler method is investigated. The conditions of existence for flip bifurcation and Hopf bifurcation are derived by using center manifold theorem and bifurcation theory, condition of existence of chaos in the sense of Marotto＇s definition of chaos is proved. The bifurcation diagrams, Lyapunov exponents and phase portraits are given for different parameters of the model, and the fractal dimension of chaotic attractor was also calculated. The numerical simulation results confirm the theoretical analysis and also display the new and complex dynamical behaviors compared with the continuous model. In particular~ we found that the new chaotic attractor, and new types of two or four coexisting chaotic attractors, and two coexisting invariant torus.     

Global random attractors are uniquely determined by attracting deterministic compact sets

It is shown that for continuous dynamical systems an analogue of the Poincaré recurrence theorem holds for Ω-limit sets. A similar result is proved for Ω-limit sets of random dynamical systems (RDS) on Polish spaces. This is used to derive that a random set which attracts every (deterministic) compact set has full measure with respect to every invariant probability measure for theRDS. Then we show that a random attractor coincides with the Ω-limit set of a (nonrandom) compact set with probability arbitrarily close to one, and even almost surely in case the base flow is ergodic. This is used to derive uniqueness of attractors, even in case the base flow is not ergodic. Entrata in Redazione il 10 marzo 1997.     

Localization of invariant compact sets of families of discrete-time systems

The functional method of localization of invariant compact sets developed for continuous- and discrete-time dynamical systems is extended to families of discrete-time systems. Positively invariant compact sets are considered. As an example, the method is applied to the Hénon system with uncertain parameters.     

Some properties for the attractors

For the continuous flows defined on a topological space, we have discussed some properties for the invariant sets and their domains of influence. We have proved the following open problem posed by C. Conley: an attractor in a locally connected compact Hausdorff invariant set has finitely many components. In the meantime, two necessary and sufficient conditions for a set to be an attractor have been given.     

Bifurcations and chaos control in a discrete-time predator-prey system of Leslie type

We investigate the dynamics of a discrete-time predator-prey system of Leslie type. We show algebraically that the system passes through a flip bifurcation and a Neimark-Sacker bifurcation in the interior of $\R^{2}_+$ using center manifold theorem and bifurcation theory. Numerical simulations are implimented not only to validate theoretical analysis but also exhibits chaotic behaviors, including phase portraits, period-11 orbits, invariant closed circle, and attracting chaotic sets. Furthermore, we compute Lyapunov exponents and fractal dimension numerically to justify the chaotic behaviors of the system. Finally, a state feedback control method is applied to stabilize the chaotic orbits at an unstable fixed point.