1 Strongly Order-Preserving Semiflows 6 1.1 Definitions and Basic Results . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2 Nonordering of Omega Limit Sets . . . . . . . . . . . . . . . . . . . . . . 12 1.3 Local Semiflows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.4 The Limit Set Dichotomy . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.5 Q is plentiful . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.6 Stability in Normally Ordered Spaces . . . . . . . . . . . . . . . . . . . . 21 1.7 Stable Equilibria in Strongly Ordered Banach Spaces . . . . . . . . . . . 24 1.8 The Search for Stable Equilibria . . . . . . . . . . . . . . . . . . . . . . . 27

https://link-springer-com-443.webvpn.jmu.edu.cn/content/pdf/10.1007%2F978-1-4419-9863-7_525.pdf

Monotone Dynamical Systems

We study enhancement of diffusive mixing on a compact Riemannian manifold by a fast incompressible flow. Our main result is a sharp description of the class of flows that make the deviation of the solution from its average arbitrarily small in an arbitrarily short time, provided that the flow amplitude is large enough. The necessary and suficient condition on such flows is expressed naturally in terms of the spectral properties of the dynamical system associated with the flow. In particular, we find that weakly mixing flows always enhance dissipation in this sense. The proofs are based on a general criterion for the decay of the semigroup generated by an operator of the form G + iAL with a negative unbounded self-adjoint operator G, a self-adjoint operator L, and parameter A » 1. In particular, they employ the RAGE theorem describing evolution of a quantum state belonging to the continuous spectral subspace of the hamiltonian (related to a classical theorem of Wiener on Fourier transforms of measures). Applications to quenching in reaction-diffusion equations are also considered.

/pdf/diffusion-and-mixing-in-fluid-flow-1qyhrirnpk.pdf

Diffusion and mixing in fluid flow

A reaction-diffusion model for the evolution of dispersal rates is considered in which there is both spatial heterogeneity and temporal periodicity. The model is restricted to two phenotypes because of technical difficulties, but a wide range of mathematical techniques and computational effort are needed to obtain useful answers. We find that the question of selection is a great deal richer than in the autonomous case, where the phenotype with the lowest diffusion is selected for. In the current model either the lower or higher diffuser rate may be selected, or there may be coexistence of phenotypes. The paper raises several open questions and suggests in particular that a mutation-selection multi-phenotypic model would repay study.

The evolution of dispersal rates in a heterogeneous time-periodic environment

Publisher Summary This chapter discusses a restricted but useful class of dynamical systems—,namely, those enjoying a comparison principle with respect to a closed order relation on the state space. Such systems, variously called monotone, order-preserving or increasing, occur in many biological, chemical, physical, and economic models. The chapter introduces the basic definitions and develops the main tools of monotone dynamics. Several results on density of quasiconvergent points are proved and used to establish existence of stable equilibria.

/pdf/chapter-4-monotone-dynamical-systems-1m0rmqsitr.pdf

Chapter 4 Monotone Dynamical Systems

These notes combine an analysis of what the author considers (admittedly subjectively) as the most important trends and developments related to the notion of entropy, with information of more “historical” nature including allusions to certain episodes and discussion of attitudes and contributions of various participants. I directly participated in many of those developments for the last forty three or forty four years of the fifty-year period under discussion and on numerous occasions was fairly close to the center of action. Thus, there is also an element of personal recollections with all attendant peculiarities of this genre.
 
For the full preface, please click the "Full Text" link above.

http://www.personal.psu.edu/axk29/pub/E50-aspublished.pdf

Fifty years of entropy in dynamics: 1958-2007

On dynamical systems with an integral invariant on the torus

Stable Cycles for Attractors of Strongly Monotone Discrete-Time Dynamical Systems

Orbits Connecting Critical Points of Differential Equations Depending on a Parameter

A criterion on compactness of the set of bounded orbits

On a lemma of planar systems

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