Approximation of attractors of semidynamical systems

Kornev, A. N.

doi:10.1070/sm2001v192n10abeh000600

Cited by 3 publications

(7 citation statements)

References 17 publications

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“…Then it is easy to find for the set ½ Ë ´Ì µ [15] the proper estimates. This result is theoretical to a greater extent because it involves finding an image for each element of at the instant Ì ½.…”

Section: àmentioning

confidence: 99%

“…It is possible to explicitly solve it [15,16] for the SDS with the Lyapunov proper function and the finite number of hyperbolic stationary (periodic) points. It is possible to explicitly construct a function of local attraction to the attractor for the appropriate mapping classes [15] in a sufficiently small neighbourhood of these points. The global function can be constructed if in addition it is possible to estimate the residence time of an arbitrary trajectory outside these neighbourhoods.…”

Section: Numerical Investigation Of Global Stabilitymentioning

confidence: 99%

“…The global function can be constructed if in addition it is possible to estimate the residence time of an arbitrary trajectory outside these neighbourhoods. Extending the method [19] of constructing stable and unstable manifolds, in some cases [15] it is also possible to investigate problems with essentially nonhyperbolic points. However, one fails to realize for We denote by Å ·´Ë Çµ the unstable manifold of the subset Ç in the space À, the so-called 'Hadamard outgoing nib':…”

Section: Numerical Investigation Of Global Stabilitymentioning

confidence: 99%

“…Note that in the general case these problems are not equivalent. However, their solution is, in a sense [15], equivalent to finding the rate function ©´Øµ of attraction to the attractor. The existence of ©´Øµ follows from the definition of the global attractor [3].…”

mentioning

confidence: 99%

“…Particular attention was given to this problem in [6]. A priori estimates for ©´Øµ can be derived [15,16] by the general theory of unstable manifolds Å · only for problems with the Lyapunov proper function and the finite number of, possibly, nonhyperbolic points.…”

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confidence: 99%

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On globally stable dynamic processes

Kornev

2002

Russian Journal of Numerical Analysis and Mathematical Modelling

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The problem of describing the whole dynamics on the globall attractor of a semidynamical system is studied. The solution is reduced to finding the function of rate of attraction to an attractor. The method of determing numerically an estimate for this function is presented.The main problem of mathematical modelling of nonstationary processes is to describe the dynamics of the system studied. In the general case this reduces to the construction of a trajectory ´Øµ Ë Ø ¡ by the initial data and the resolving operator Ë´Ø ¡µ of the problem. When solving practical problems the original operator Ë Ø ¡ ¡ and the initial data are replaced by approximate Ë Ø ¡ ¡ and . Thus, the modelled and real trajectories begin to differ with time. The standard estimate of the local rate of discrepancy has the exponential formThe coefficient tends to zero as the accuracy of approximation of the operator and the initial data increases. Thus, the accuracy of modelling can be ensured on a finite interval ¼ Ì´ µ℄. In real processes the errors of modelling, the approximation of the initial data, and the forcing are essentially unremovable. Thus, the value Ì´ µ turns out to be much less than the value of interest. This is the case, for example, when modelling unstable fluid and gas flows in the general climate theory. However, of most interest in these problems is some typical state, which can be observed in the system, rather than the behaviour of a concrete trajectory. The strict definition of typicalness depends on a problem. In this connection there arises a problem of describing all possible limiting states Å, which are realized in the system over long periods of time. Note that this statement is meaningful only for systems with compact set Å. For problems of mathematical physics which deal with noncompact spaces the existence of Å and moreover its compactness are not obvious.The studied approach for partial differential equations was first presented by O. A. Ladyzhenskaya in 1972 in [18]. For the Navier-Stokes two-dimensional problem the compact set Å was constructed which uniformly attracts the arbitrary bounded subset of the initial data in the state space À. The minimality of this set among all sets with this property and the exact invariance with respect to the resolving operator of the problem were proved. It was shown that among all exactly invariant

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Section: àmentioning

confidence: 99%

Section: Numerical Investigation Of Global Stabilitymentioning

confidence: 99%

Section: Numerical Investigation Of Global Stabilitymentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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On globally stable dynamic processes

Kornev

2002

Russian Journal of Numerical Analysis and Mathematical Modelling

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A General Approximation Scheme for Attractors of Abstract Parabolic Problems

Carvalho

Piskarev

2006

Numerical Functional Analysis and Optimization

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We consider semilinear problems of the form u = Au + f (u), where A generates an exponentially decaying compact analytic C 0 -semigroup in a Banach space E , f : E → E is differentiable globally Lipschitz and bounded (E = D((−A) ) with the graph norm). Under a very general approximation scheme, we prove that attractors for such problems behave upper semicontinuously. If all equilibrium points are hyperbolic, then there is an odd number of them. If, in addition, all global solutions converge as t → ±∞, then the attractors behave lower semicontinuously. This general approximation scheme includes finite element method, projection and finite difference methods. The main assumption on the approximation is the compact convergence of resolvents, which may be applied to many other problems not related to discretization.Keywords Abstract parabolic equations; Compact convergence of resolvents; General approximation scheme; Upper and lower semicontinuity of attractors.

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Efficient algorithms for stable manifolds

Ozeritsky

2005

Russian Journal of Numerical Analysis and Mathematical Modelling

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Approximation of attractors of semidynamical systems

Cited by 3 publications

References 17 publications

On globally stable dynamic processes

On globally stable dynamic processes

A General Approximation Scheme for Attractors of Abstract Parabolic Problems

Efficient algorithms for stable manifolds

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