This book treats nonlinear dynamics in both Hamiltonian and dissipative systems. The emphasis is on the mechanics for generating chaotic motion, methods of calculating the transitions from regular to chaotic motion, and the dynamical and statistical properties of the dynamics when it is chaotic. The book is intended as a self consistent treatment of the subject at the graduate level and as a reference for scientists already working in the field. It emphasizes both methods of calculation and results. It is accessible to physicists and engineers without training in modern mathematics. The new edition brings the subject matter in a rapidly expanding field up to date, and has greatly expanded the treatment of dissipative dynamics to include most important subjects. It can be used as a graduate text for a two semester course covering both Hamiltonian and dissipative dynamics.

Regular and Chaotic Dynamics

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Philosophical Transactions of the Royal Society of London. for the Year Mdcccxxvii. Part I. Volume Cxvii:355–388, 1827

介绍Plos Computational Biology主编Philip E．Bourne关于论文发表的10务简单法则，包括：读很多论文，客观看待自身工作，选择好的编辑，提高英语水平，忍受退稿，提高论文质量等。以期对医学生和研究人员发表文章提供有益帮助。

Plos Computational Biology主编关于论文获得发表的10条简单法则的评析

Applications of Percolation Theory

Der vorliegende Band enthált sechs Beitrage, in denen punktuell und exemplarisch der Stand der Forschung beleuchtet wird. Ein Beitrag von Codling und Frasinski beschreibt in erster Linie Experimente zum noch wenig verstandenen Phânomen der dissoziativen Ionisation von Molekülen, bei dem diese in einem intensiven Laserfeld mehrere Elektronén verlieren und in Ionen mit hoher kinetischer Energie zerfallen. Ein wesentlicher Teil dieses Beitrags widmet sich einer von den Autoren entwickelten Auswertungstechnik, welche die Korrelationen zwischen den Fiugzeiten emittierter Ionen und Elektronén sichtbar macht. Es folgen drei Beitrage zur Théorie. Schmelcher und Cederbaum beschreiben Effekte starker statischer Felder, die nur dann zu verstehen sind, wenn die Nicht-Separation der Schwerpunktsbewegung im auBeren Magnetfeld korrekt beriicksichtigt wird. Dies fiihrt

Extended Irreversible Thermodynamics

Pattern formation from homogeneity is well-studied, but less is known concerning symmetry-breaking instabilities in heterogeneous media. It is nontrivial to separate observed spatial patterning due to inherent spatial heterogeneity from emergent patterning due to nonlinear instability. We employ WKBJ asymptotics to investigate Turing instabilities for a spatially heterogeneous reaction-diffusion system, and derive conditions for instability which are local versions of the classical Turing conditions We find that the structure of unstable modes differs substantially from the typical trigonometric functions seen in the spatially homogeneous setting. Modes of different growth rates are localized to different spatial regions. This localization helps explain common amplitude modulations observed in simulations of Turing systems in heterogeneous settings. We numerically demonstrate this theory, giving an illustrative example of the emergent instabilities and the striking complexity arising from spatially heterogeneous reaction-diffusion systems. Our results give insight both into systems driven by exogenous heterogeneity, as well as successive pattern forming processes, noting that most scenarios in biology do not involve symmetry breaking from homogeneity, but instead consist of sequential evolutions of heterogeneous states. The instability mechanism reported here precisely captures such evolution, and extends Turing’s original thesis to a far wider and more realistic class of systems.

https://royalsocietypublishing.org/doi/pdf/10.1098/rsif.2019.0621

From one pattern into another: analysis of Turing patterns in heterogeneous domains via WKBJ.

Pullback attractors of non-autonomous stochastic degenerate parabolic equations on unbounded domains

Influence of Curvature, Growth, and Anisotropy on the Evolution of Turing Patterns on Growing Manifolds

Turing-Hopf patterns on growing domains: The torus and the sphere.

The study of pattern-forming instabilities in reaction–diffusion systems on growing or otherwise time-dependent domains arises in a variety of settings, including applications in developmental biology, spatial ecology, and experimental chemistry. Analyzing such instabilities is complicated, as there is a strong dependence of any spatially homogeneous base states on time, and the resulting structure of the linearized perturbations used to determine the onset of instability is inherently non-autonomous. We obtain general conditions for the onset and structure of diffusion driven instabilities in reaction–diffusion systems on domains which evolve in time, in terms of the time-evolution of the Laplace–Beltrami spectrum for the domain and functions which specify the domain evolution. Our results give sufficient conditions for diffusive instabilities phrased in terms of differential inequalities which are both versatile and straightforward to implement, despite the generality of the studied problem. These conditions generalize a large number of results known in the literature, such as the algebraic inequalities commonly used as a sufficient criterion for the Turing instability on static domains, and approximate asymptotic results valid for specific types of growth, or specific domains. We demonstrate our general Turing conditions on a variety of domains with different evolution laws, and in particular show how insight can be gained even when the domain changes rapidly in time, or when the homogeneous state is oscillatory, such as in the case of Turing–Hopf instabilities. Extensions to higher-order spatial systems are also included as a way of demonstrating the generality of the approach.

Andrew L. Krause

Papers

From one pattern into another: analysis of Turing patterns in heterogeneous domains via WKBJ.

Pullback attractors of non-autonomous stochastic degenerate parabolic equations on unbounded domains

Influence of Curvature, Growth, and Anisotropy on the Evolution of Turing Patterns on Growing Manifolds

Turing-Hopf patterns on growing domains: The torus and the sphere.

Turing conditions for pattern forming systems on evolving manifolds