Extended gradient systems: Dimension one

Slijepčević, Siniša

doi:10.3934/dcds.2000.6.503

Cited by 12 publications

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“…The theory of φ-invariant or ergodic measures will not be needed in the following; we only note that all φ-invariant measures are supported on S, which was in a more general framework of "extended gradient systems" proved in [Sli99b]. Now we list further notation:…”

Section: The Induced Semiflow On the Space Of Measuresmentioning

confidence: 99%

“…One of main difficulties when applying the semiflow φ to construction of stationary configurations with given properties, is that it is only formally gradient (the natural "Lyapunov" function A(u) = h(u i , u i+1 ) is divergent; for discussion of various properties of such systems we again refer the reader to [Sli99b]). We can, however, overcome this difficulty by working in the space A * S .…”

Section: Definition 31 a Semiflow ψ On A Set A Is Strictly Gradientmentioning

confidence: 99%

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Construction of invariant measures of Lagrangian maps: minimisation and relaxation

Slijepčević

2001

Math Z

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If F is an exact symplectic map on the d-dimensional cylinder T d ×R d , with a generating function h having superlinear growth and uniform bounds on the second derivative, we construct a strictly gradient semiflow φ * on the space of shift-invariant probability measures on the space of configurations (R d ) Z . Stationary points of φ * are invariant measures of F , and the rotation vector and all spectral invariants are invariants of φ * . Using φ * and the minimisation technique, we construct minimising measures with an arbitrary rotation vector ρ ∈ R d , and with an additional assumption that F is strongly monotone, we show that the support of every minimising measure is a graph of a Lipschitz function. Using φ * and the relaxation technique, assuming a weak condition on φ * (satisfied e.g. in Hedlund's counter-example, and in the anti-integrable limit) we show existence of double-recurrent orbits of F (and F -ergodic measures) with an arbitrary rotation vector ρ ∈ R d , and action arbitrarily close to the minimal action A(ρ).

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Section: The Induced Semiflow On the Space Of Measuresmentioning

confidence: 99%

Section: Definition 31 a Semiflow ψ On A Set A Is Strictly Gradientmentioning

confidence: 99%

Construction of invariant measures of Lagrangian maps: minimisation and relaxation

Slijepčević

2001

Math Z

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“…The existence of a global Lyapunov function for the semigroup generated by a formally gradient system on the set of spatially invariant measures was proved in [37] for small dimensions, n = 1 and n = 2. However, in the construction of this Lyapunov function certain specific facts were used, which have no analogues for n > 2.…”

Section: Formally Gradient Systems Of Reaction-diffusion Equations Anmentioning

confidence: 99%

“…Nevertheless, in spite of the lack of a global Lyapunov function, the existence of the gradient structure (11.1) does lead to very substantial simplification of the space-time dynamics generated by equation (0.1). (See, for example, [22,27,37] regarding a more detailed study of formally gradient systems in small dimensions n = 1 and n = 2. )…”

Section: Formally Gradient Systems Of Reaction-diffusion Equations Anmentioning

confidence: 99%

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Multiparameter semigroups and attractors of reaction-diffusion equations in ${\mathbb R}^n$

Zelik

2004

Trans. Moscow Math. Soc.

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Abstract. The space-time dynamics generated by a system of reaction-diffusion equations in R n on its global attractor are studied in this paper. To describe these dynamics the extended (n + 1)-parameter semigroup generated by the solution operator of the system and the n-parameter group of spatial translations is introduced and their dynamic properties are studied. In particular, several new dynamic characteristics of the action of this semigroup on the attractor are constructed, generalizing the notions of fractal dimension and topological entropy, and relations between them are studied. Moreover, under certain natural conditions a description of the dynamics is obtained in terms of homeomorphic embeddings of multidimensional Bernoulli schemes with infinitely many symbols.

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The Transport Speed and Optimal Work in Pulsating Frenkel–Kontorova Models

Rabar

Slijepčević

2019

Commun. Math. Phys.

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We consider a generalized one-dimensional chain in a periodic potential (the Frenkel-Kontorova model), with dissipative, pulsating (or ratchet) dynamics as a model of transport when the average force on the system is zero. We find lower bounds on the transport speed under mild assumptions on the asymmetry and steepness of the site potential. Physically relevant applications include explicit estimates of the pulse frequencies and mean spacings for which the transport is non-zero, and more specifically the pulse frequencies which maximize work. The bounds explicitly depend on the pulse period and subtle number-theoretical properties of the mean spacing. The main tool is the study of time evolution of spatially invariant measures in the space of measures equipped with the L 1 -Wasserstein metric.

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Extended gradient systems: Dimension one

Cited by 12 publications

References 0 publications

Construction of invariant measures of Lagrangian maps: minimisation and relaxation

Construction of invariant measures of Lagrangian maps: minimisation and relaxation

Multiparameter semigroups and attractors of reaction-diffusion equations in ${\mathbb R}^n$

The Transport Speed and Optimal Work in Pulsating Frenkel–Kontorova Models

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