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1.
In this paper, we first introduce the measure-theoretic entropy for arbitrary Borel probability measure in nonautonomous case. Then we show that there is certain variational relation for nonautonomous dynamical systems.  相似文献   

2.
I.IntroductionInthelasttwodecades,thetheoriesofan'qnomousinfinitedimensionaldynamicalsystemshavebeenthoroughlystudiedandsystematicallyimprovedl'--'l.Comparatively,thestudiesofnonautonomousonesincreaseslowly.Themaindifficultyliesinthatthesemiflowsgeneratedbythesolutionstoautonomouscasesatisfythesemigroupproperty.whilethoseofnonautonomousonesdonot.Sothemethodsusedtostudytheautonomouscasecan'tbeappropriatefornonautonomouscase.Anditrequiresustoestablisll11on'theoriesandmethods.[5--91havediscussed…  相似文献   

3.
Negatively invariant compact sets of autonomous and nonautonomous dynamical systems on a metric space, the latter formulated in terms of processes, are shown to contain a strictly invariant set and hence entire solutions. For completeness the positively invariant case is also considered. Both discrete and continuous time systems are considered. In the nonautonomous case, the various types of invariant sets are in fact families of subsets of the state space that are mapped onto each other by the process. A simple example shows the usefulness of the result for showing the occurrence of a bifurcation in a nonautonomous system.  相似文献   

4.
Integral manifolds generalize invariant manifolds to nonautonomous ordinary differential equations. In this paper, we develop a method to calculate their Taylor approximation with respect to the state space variables. This is of decisive importance, e.g., in nonautonomous bifurcation theory or for an application of the reduction principle in a time-dependent setting.  相似文献   

5.
In this paper the stability of nonlinear nonautonomous systems under the frequently-acting perturbation is studied. This study is a forward development of the study of the stability in the Liapunov sense; furthermore, it is of significance in practice since perturbations are often not single in the time domain. Malkin proved a general theorem about thesubject. To apply the theorem, however, the user has to construct a Liapunov function which satisfies specified conditions and it is difficult to find such a function for nonlinear nonautonomous systems. In the light of the principle of Liapunov’s indirect method, which is an effective method to decide the stability of nonlinear systems in the Liapunov sense, the authors have achieved several important conclusions expressed in the form of theorems to determine the stability of nonlinear nonautonomous systems under the frequently-acting perturbation.  相似文献   

6.
In this paper the stability of nonlinear nonautonomous systems under the frequently-acting perturbation is studied.This study is a forward development of the study of thestability in the Liapunov sense;furthermore,it is of significance in practice sinceperturbations are often not single in the time domain.Malk in proved a general theoremabout thesubject.To apply the theorem,however,the user has to construct a Liapunovfunction which satisfies specified conditions and it is difficult to find such a function fornonlinear nonautonomous systems.In the light of the principle of Liapunov’s indirectmethod,which is an effective method to decide the stability of nonlinear systems in theLiapunov sense,the authors have achieved several important conclusions expressed in theform of theorems to determine the stability of nonlinear nonautonomous systems under thefrequently-acting perturbation.  相似文献   

7.
In this paper we present an abstract approach to inertial manifolds for nonautonomous dynamical systems. Our result on the existence of inertial manifolds requires only two geometrical assumptions, called cone invariance and squeezing property, and two additional technical assumptions, called boundedness and coercivity property. Moreover we give conditions which ensure that the global pullback attractor is contained in the inertial manifold. In the second part of the paper we consider special nonautonomous dynamical systems, namely processes (or two-parameter semi-flows). As a first application of our abstract approach and for reason of comparison with known results we verify the assumptions for semilinear nonautonomous evolution equations whose linear part satisfies an exponential dichotomy condition and whose nonlinear part is globally bounded and globally Lipschitz.  相似文献   

8.
In this paper we study the continuity of invariant sets for nonautonomous infinite-dimensional dynamical systems under singular perturbations. We extend the existing results on lower-semicontinuity of attractors of autonomous and nonautonomous dynamical systems. This is accomplished through a detailed analysis of the structure of the invariant sets and its behavior under perturbation. We prove that a bounded hyperbolic global solutions persists under singular perturbations and that their nonlinear unstable manifold behave continuously. To accomplish this, we need to establish results on roughness of exponential dichotomies under these singular perturbations. Our results imply that, if the limiting pullback attractor of a nonautonomous dynamical system is the closure of a countable union of unstable manifolds of global bounded hyperbolic solutions, then it behaves continuously (upper and lower) under singular perturbations.  相似文献   

9.
The electromechanical gyrostat is a fourth-order nonautonomous system that exhibits very rich behavior such as chaos. In recent years, synchronization of nonautonomous chaotic systems has found many useful applications in nonlinear science and engineering fields. On the other hand, it is well known that the finite-time control techniques demonstrate good robustness and disturbance rejection properties. This paper studies the potential application of the finite-time control techniques for synchronization of nonautonomous chaotic electromechanical gyrostat systems in finite time. It is assumed that all the parameters of both drive and response systems are unknown parameters in advance. Moreover, the effects of dead-zone nonlinearities in the control inputs are also taken into account. Some adaptive controllers are introduced to synchronize two gyrostat systems in different scenarios within a given finite-time. Two illustrative examples are presented to demonstrate the efficiency and robustness of the proposed finite-time synchronization strategy.  相似文献   

10.
The necessary and sufficient condition of the stability of linear nonautonomous system under the frequently-acting perturbation has been given and proved on the basis of[1]and[2],and the theorem of the equivalence on the uniform and asymptotical stability in the sense of Liapunov and the stability under the frequently-acting perturbation of linear nonautonomous system has been given in this paper.Besides,the analysis of the dynamic stability of robot has been presented by applying the theorem in this paper,which is closer to reality.  相似文献   

11.
This paper is concerned with robust practical synchronization for general second-order nonautonomous systems with parameter mismatch. Some simple yet general algebraic criteria are derived based on practical stability theory of nonautonomous dynamical systems. A?distinctive feature of this work is that the parameter mismatch cannot only be existed in system parameters, but also in external excitation ones. Furthermore, the obtained results are applied to a typical horizontal platform system and the representative forced Van der Pol oscillator. Subsequently, numerical simulations demonstrate the effectiveness of the criteria and the robustness of the control technique.  相似文献   

12.
In this paper, a novel fractional-order terminal sliding mode control approach is introduced to control/synchronize chaos of fractional-order nonautonomous chaotic/hyperchaotic systems in a given finite time. The effects of model uncertainties and external disturbances are fully taken into account. First, a novel fractional nonsingular terminal sliding surface is proposed and its finite-time convergence to zero is analytically proved. Then an appropriate robust fractional sliding mode control law is proposed to ensure the occurrence of the sliding motion in a given finite time. The fractional version of the Lyapunov stability is used to prove the finite-time existence of the sliding motion. The proposed control scheme is applied to control/synchronize chaos of autonomous/nonautonomous fractional-order chaotic/hyperchaotic systems in the presence of both model uncertainties and external disturbances. Two illustrative examples are presented to show the efficiency and applicability of the proposed finite-time control strategy. It is worth to notice that the proposed fractional nonsingular terminal sliding mode control approach can be applied to control a broad range of nonlinear autonomous/nonautonomous fractional-order dynamical systems in finite time.  相似文献   

13.
I.IntroductionSinceEinsteinestablishedgeneralrelativityatthebiginningofthiscentury,differentialgeometry,especiallythemodernditTerentialgeometry,hasbeenextellsivelyappliedtomanyfieldsofphysics.Thestudyofregularholonomicmechanicalsystemsinthemodernsettingofdifferentialgeometryhasahistoryofmorethanthirtyyears.Andtheresearchtendstoperfectgraduallyt'~'l.Sinceearlyin1980'sthegeometrizationaboutconstrainedmechanicalsystemsandsingularmechanicalsystemshasbeenattachedimportanceextensivelyandsomeresult…  相似文献   

14.
The nonautonomous version of the Yakubovich Frequency Theorem characterizes the solvability of an infinite horizon optimization problem in terms of the validity of the Frequency and Nonoscillation Conditions for a linear Hamiltonian system, which is defined from the coefficients of the quadratic functional to be minimized. This paper describes those nonautonomous linear Hamiltonian systems satisfying the required properties. Two groups appear, depending on whether they are uniformly weakly disconjugate or not. It also contains a previous analysis of the long-term behavior of the Grassmannian and Lagrangian flows under the presence of exponential dichotomy, which is required for the classification and has interest by itself.  相似文献   

15.
A normal form for open loop control systems is provided, based on their interpretation as skew product flows and on normal forms for nonautonomous differential equations.  相似文献   

16.
Chen  Yi-Xiang  Ou-Yang  Fang-Yan 《Nonlinear dynamics》2020,100(2):1543-1550
Nonlinear Dynamics - A nonautonomous Gross–Pitaevskii equation with a partially nonlocal nonlinearity and a linear and parabolic potential is discussed, and a projecting expression between...  相似文献   

17.
The long-time behaviour of a two-dimensional nonautonomous nonlinear SchrOdinger equation is considered. The existence of uniform attractor is proved and the up per bound of the uniform attractor' s Hausdorff dimension is given.  相似文献   

18.
Innocenti  Giacomo  Di Marco  Mauro  Forti  Mauro  Tesi  Alberto 《Nonlinear dynamics》2019,96(2):1169-1190
Nonlinear Dynamics - The paper studies bifurcations and complex dynamics in a class of nonautonomous oscillatory circuits with a flux-controlled memristor and harmonic forcing term. It is first...  相似文献   

19.
I.IntroduCtionAtpresent.autollomousinfillitedimensionaldynamicalsystemshavebeenthoroughlystLldicdilltllcory.andwidelyappliedinpracticel'-'1.Forthe11onautonomouscase,[5--91havesttldicd1ilocxistenceanddimensionestimateofattractorsofnonautonomouscase;[12].hasconsideredtileexislellceofinertialmanifolds.Theoretically,inertialmanifoldisaveryusefulInethodtodiscussthelongtimebehaviorofthesolutionstononautonomousinfinitedimcnsiollaldynamicalsystems.Butitcannotbeexpressedexplicitly.Soitisnotconvenient…  相似文献   

20.
IntroductionAttractorsofautonomousevolutionequationshavebeenintensivelystudiedinmathematicalliterature (see,forexample ,books [1 ] ,[2 ] ,[3 ] ,[4 ]andtheliteraturecitedthere) .Thenonautonomousinfinite_dimensionaldynamicalsystemswerelesswellunderstood .In 1 994 ,Chepyzh…  相似文献   

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