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1.
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
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.
5.
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
6.
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
7.
We consider a system of ordinary differential equations used to describe the dynamics of two coupled single-mode semiconductor
lasers. In particular, we study solutions corresponding to the amplitude synchronization. It is shown that the set of these
solutions forms a three-dimensional invariant manifold in the phase space. We study the stability of trajectories on this
manifold both in the tangential direction and in the transverse direction. We establish conditions for the existence of globally
asymptotically stable solutions of equations on the manifold synchronized with respect to the amplitude.
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 3, pp. 426–435, March, 2008. 相似文献
8.
A. M. Samoilenko 《Ukrainian Mathematical Journal》2011,63(2):278-314
We consider problems of the linear theory of systems of ordinary differential equations related to the investigation of invariant
hyperplanes of these systems, the notion of equivalence for these systems, and the Floquet–Lyapunov theory for periodic systems
of linear equations. In particular, we introduce the notion of equivalence of systems of linear differential equations of
different orders, propose a new formula of the Floquet form for periodic systems, and present the application of this formula
to the introduction of amplitude–phase coordinates in a neighborhood of a periodic trajectory of a dynamical system. 相似文献
9.
Thomas Lorenz 《Journal of Mathematical Analysis and Applications》2018,457(2):1502-1567
Applications in robust control problems and shape evolution motivate the mathematical interest in control problems whose states are compact (possibly non-convex) sets rather than vectors. This leads to evolutions in a basic set which can be supplied with a metric (like the well-established Pompeiu–Hausdorff distance), but it does not have an obvious linear structure. This article extends differential inclusions with state constraints to compact-valued states in a separable Hilbert space H. The focus is on sufficient conditions such that a given constraint set (of compact subsets) is viable a.k.a. weakly invariant. Our main result extends the tangential criterion in the well-known viability theorem (usually for differential inclusions in a vector space) to the metric space of non-empty compact subsets of H. 相似文献
10.
Tony Shardlow 《Numerical Functional Analysis & Optimization》2013,34(9-10):963-989
Consider the unstable manifold of a hyperbolic periodic orbit of an ordinary differential equation under C1 perturbations of the vector field and under approximation by a one-step numerical method, which is at least first order. Trajectories bounded backwards in time near the periodic orbit perturb Hausdorff continuously. This result as applied to numerical perturbations improves on Alouges-Debussche [1], who give only continuity of the unstable maniford, and on Beyn [3], who gives continuity of trajectories only when the periodic orbit is unstable. As a corollary, we find that attractors perturb Hausdorff continuously when the attractor equals a union of locally continuous unstable manifolds of invariant sets 相似文献
11.
首先给出广义Kuramoto_Sivashinsky(GKS)方程周期初边值问题在H2空间惯性集的构造,进而给出并证明GKS方程吸引子的分形结构,同时发现吸引子的一个分形局部化指数型逼近序列·上述结果精细和推进了[1,3,5,7]关于惯性集和吸引子的结论,刻划了吸引子的一种几何结构 相似文献
12.
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
13.
We study periodic solutions and the behavior of phase trajectories of the differential equation of a nonlinear oscillator
with pulse influence at unfixed moments of time.
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 6, pp. 827–834, June, 1999. 相似文献
14.
A. N. Kanatnikov 《Differential Equations》2011,47(7):997-1003
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. 相似文献
16.
Richard Moeckel 《Transactions of the American Mathematical Society》2004,356(11):4395-4425
The collinear relative equilibrium solutions are among the few explicitly known periodic solutions of the Newtonian three-body problem. When the energy and angular momentum constants are varied slightly, these unstable periodic orbits become normally hyperbolic invariant spheres whose stable and unstable manifolds form separatrices in the integral manifolds. The goal of this paper is to construct simple isolating blocks for these invariant spheres analogous to those introduced by Conley in the restricted three-body problem. This allows continuation of the invariant set and the separatrices to energies and angular momenta far from those of the relative equilibrium.
17.
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
18.
Radoslaw Czaja Messoud Efendiev 《Journal of Mathematical Analysis and Applications》2011,381(2):766-765
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. 相似文献
19.
Summary We consider a dynamical system described by an autonomous ODE with an asymptotically stable attractor, a compact set of orbitrary shape, for which the stability can be characterized by a Lyapunov function. Using recent results of Eirola and Nevanlinna [1], we establish a uniform estimate for the change in value of this Lyapunov function on discrete trajectories of a consistent, strictly stable multistep method approximating the dynamical system. This estimate can then be used to determine nearby attracting sets and attractors for the discretized system as done in Kloeden and Lorenz [3, 4] for 1-step methods.This work was supported by the U.S. Department of Energy Contract DE-A503-76 ER72012 相似文献
20.
The present paper is devoted to generalizations of the Dieudonné theorem claiming that the convergence of sequences of regular
Borelian measures is preserved under the passage from a system of open subsets of a compact metric space to the class of all
Borelian subsets of this space. The Dieudonné theorem is proved in the case for which the set functions are weakly regular,
nonadditive, defined on an algebra of sets that contains the class of open subsets of an arbitrary σ-topological space, and
take values in a uniform space.
Translated fromMatematicheskie Zametki, Vol. 62, No. 1, pp. 103–110, July, 1997.
Translated by O. V. Sipacheva 相似文献