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1.
A. N. Kanatnikov 《Differential Equations》2010,46(11):1601-1611
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. 相似文献
2.
A. N. Kanatnikov 《Differential Equations》2012,48(11):1461-1469
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. 相似文献
3.
A. N. Kanatnikov 《Differential Equations》2013,49(12):1645-1649
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. 相似文献
4.
A. P. Krishchenko 《Differential Equations》2016,52(11):1403-1410
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. 相似文献
5.
A. P. Krishchenko 《Differential Equations》2017,53(11):1413-1418
We suggest a new method for constructing Lyapunov functions for autonomous systems of differential equations. The method is based on the construction of a family of sets whose boundaries have the properties typical of the level surfaces of Lyapunov functions. These sets are found by the method of localization of invariant compact sets. For the resulting Lyapunov function and its derivative, we find analytical expressions via the localizing functions occurring in the specification of the above-mentioned sets. An example of a system with a degenerate equilibrium is considered. 相似文献
6.
7.
A. N. Kanatnikov 《Differential Equations》2018,54(11):1414-1418
The stability (or asymptotic stability) of equilibria of an time-invariant discrete-time system can be verified with the use of stability and asymptotic stability criteria stated in terms of invariant sets. An earlier proposed method reduces the verification of these criteria to some set operations. However, the method is analytical and hard to implement. We propose another approach to the verification of these criteria based on the functional method for localizing invariant compact sets. 相似文献
8.
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. 相似文献
9.
We generalize the localization method for invariant compact sets of an autonomous dynamical system to the case of a nonautonomous system of differential equations. By using this method, we solve the localization problem for the Vallis third-order dynamical system governing some processes in atmosphere dynamics over the Pacific Ocean. For this system, we construct a one-parameter family of localizing sets bounded by second-order surfaces and find the intersection of all sets of the family. 相似文献
10.
A. P. Krishchenko 《Differential Equations》2018,54(11):1419-1424
Finitely many embedded localizing sets are constructed for invariant compact sets of a time-invariant differential system. These localizing sets are used to divide the state space into three subsets, the least localizing set and two sets called sets of the first kind and the second kind. We prove that the trajectory passing through a point of the set of the first kind remains in this set and tends to infinity. For a trajectory passing through a point of the set of the second kind, there are three possible types of behavior: it either goes to infinity or, at some finite time, enters the least localizing set, or has a nonempty ω-limit set contained in the intersection of the boundary of one of the constructed localizing sets with the universal section of the corresponding localizing function. 相似文献
11.
Konstantin E. Starkov 《Chaos, solitons, and fractals》2009,39(4):1671-1676
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. 相似文献
12.
A. P. Krishchenko 《Doklady Mathematics》2016,94(1):365-368
The asymptotic stability and global asymptotic stability of equilibria in autonomous systems of differential equations are analyzed. Conditions for asymptotic stability and global asymptotic stability in terms of compact invariant sets and positively invariant sets are proved. The functional method of localization of compact invariant sets is proposed for verifying the fulfillment of these conditions. Illustrative examples are given. 相似文献
13.
A. P. Krishchenko 《Computational Mathematics and Modeling》2011,22(4):361-373
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. 相似文献
14.
Strongly negatively invariant compact sets of set-valued autonomous and nonautonomous dynamical systems on a complete metric
space, the latter formulated in terms of processes, are shown to contain a weakly positively invariant family and hence entire
solutions. For completeness the strongly positively invariant case is also considered, where the obtained invariant family
is strongly invariant. Both discrete and continuous time systems are treated. In the nonautonomous case, the various types
of invariant families are in fact composed of subsets of the state space that are mapped onto each other by the set-valued
process. A simple example shows the usefulness of the result for showing the occurrence of a bifurcation in a set-valued dynamical
system. 相似文献
16.
F. Marchetti P. Negrini L. Salvadori M. Scalia 《Annali di Matematica Pura ed Applicata》1976,108(1):211-226
Summary A concept of total stability for continuous or discrete dynamical systems and a generalized definition of bifurcation are
given: it is possible to show the link between an abrupt change of the asymptotic behaviour of a family of flows and the arising
of new invariant sets, with determined asymptotic properties. The theoretical results are a contribution to the study of the
behaviour of flows near an invariant compact set. They are obtained by means of an extension of Liapunov's direct method.
A Dario Graffi nel suo 70° compleanno
Entrata in Redazione il 9 febbraio 1975.
Work performed under the auspices of the Italian Council of Research (C.N.R.). 相似文献
17.
《中国科学 数学(英文版)》2020,(9)
We study various weaker forms of the inverse shadowing property for discrete dynamical systems on a smooth compact manifold. First, we introduce the so-called ergodic inverse shadowing property(Birkhoff averages of continuous functions along an exact trajectory and the approximating one are close). We demonstrate that this property implies the continuity of the set of invariant measures in the Hausdorff metric. We show that the class of systems with ergodic inverse shadowing is quite broad; it includes all diffeomorphisms with hyperbolic nonwandering sets. Second, we study the so-called individual inverse shadowing(any exact trajectory can be traced by approximate ones, but this shadowing is not uniform with respect to the initial point of the trajectory).We demonstrate that this property is closely related to structural stability and ?-stability of diffeomorphisms. 相似文献
18.
Beifang Chen 《Discrete and Computational Geometry》1993,10(1):79-93
The Euler characteristic plays an important role in many subjects of discrete and continuous mathematics. For noncompact spaces,
its homological definition, being a homotopy invariant, seems not as important as its role for compact spaces. However, its
combinatorial definition, as a finitely additive measure, proves to be more applicable in the study of singular spaces such
as semialgebraic sets, finitely subanalytic sets, etc. We introduce an interesting integral by means of which the combinatorial
Euler characteristic can be defined without the necessity of decomposition and extension as in the traditional treatment for
polyhedra and finite unions of compact convex sets. Since finite unions of closed convex sets cannot be obtained by cutting
convex sets as in the polyhedral case, a separate treatment of the Euler characteristic for functions generated by indicator
functions of closed convex sets and relatively open convex sets is necessary, and this forms the content of this paper. 相似文献
19.
《Communications in Nonlinear Science & Numerical Simulation》2010,15(5):1159-1165
In our paper we study the localization problem of compact invariant sets of nonlinear systems. Methods of a solution of this problem are discussed and a new method is proposed which is based on using symmetrical prolongations and the first-order extremum condition. Our approach is applied to the system modeling the Rayleigh–Bénard convection for which the symmetrical prolongation with the Lorenz system has been constructed. 相似文献
20.
Luis N. Coria Konstantin E. Starkov 《Communications in Nonlinear Science & Numerical Simulation》2009,14(11):3879-3888
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. 相似文献