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This article is concerned with the dynamics of glacial cycles observed in the geological record of the Pleistocene Epoch. It focuses on a conceptual model proposed by Maasch and Saltzman [J. Geophys. Res., 95, D2 (1990), pp. 1955-1963, which is based on physical arguments and emphasizes the role of atmospheric CO 2 in the generation and persistence of periodic orbits (limit cycles). The model consists of three ordinary differential equations with four parameters for the anomalies of the total global ice mass, the atmospheric CO 2 concentration, and the volume of the North Atlantic Deep Water. In this article, it is shown that a simplified two-dimensional symmetric version displays many of the essential features of the full model, including equilibrium states, limit cycles, their basic bifurcations, and a Bogdanov-Takens point that serves as an organizing center for the local and global dynamics. Also, symmetry breaking splits the Bogdanov-Takens point into two, with different local dynamics in their neighborhoods.
This article is concerned with the dynamics of glacial cycles observed in the geological record of the Pleistocene Epoch. It focuses on a conceptual model proposed by Maasch and Saltzman [J. Geophys. Res., 95, D2 (1990), pp. 1955-1963, which is based on physical arguments and emphasizes the role of atmospheric CO 2 in the generation and persistence of periodic orbits (limit cycles). The model consists of three ordinary differential equations with four parameters for the anomalies of the total global ice mass, the atmospheric CO 2 concentration, and the volume of the North Atlantic Deep Water. In this article, it is shown that a simplified two-dimensional symmetric version displays many of the essential features of the full model, including equilibrium states, limit cycles, their basic bifurcations, and a Bogdanov-Takens point that serves as an organizing center for the local and global dynamics. Also, symmetry breaking splits the Bogdanov-Takens point into two, with different local dynamics in their neighborhoods.
The phenomenon of resonance will be dealt with from the viewpoint of dynamical systems depending on parameters and their bifurcations. Resonance phenomena are associated to open subsets in the parameter space, while their complement corresponds to quasi-periodicity and chaos. The latter phenomena occur for parameter values in fractal sets of positive measure. We describe a universal phenomenon that plays an important role in modelling. This paper gives a summary of the background theory, veined by examples.
In this survey we look at parabolic partial differential equations from a dynamical systems point of view. With origins deeply rooted in celestial mechanics, and many modern aspects traceable to the monumental influence of Poincaré, dynamical systems theory is mainly concerned with the global time evolution T (t)u 0 of points u 0 -and of sets of such points -in a more or less abstract phase space X. The success of dynamical concepts such as gradient flows, invariant manifolds, ergodicity, shift dynamics, etc. during the past century has been enormous -both as measured by achievement, and by vitality in terms of newly emerging questions and long-standing open problems.In parallel to this development, the applied horizon now reaches far beyond the classical sources of celestial and Hamiltonian mechanics. Applications areas today include physics, many branches of engineering, economy models, and mathematical biology, to name just a few. This influence can certainly be felt in several articles of this volume and cannot possibly be adequately summarized in our survey.Some resources on recent activities in the area of dynamics are the book series Dynamical Systems I -X of the Encyclopedia of Mathematical Sciences [AnAr88, Si89, Ar&al88, ArNo90, Ar94a, Ar93, ArNo94, Ar94b, An91, Ko02], the Handbook of Dynamical System [BrTa02, Fi02, KaHa02], the Proceedings [Fi&al00], and the fundamental books [ChHa82,GuHo83,Mo73,KaHa95].In the context of partial differential equations, the phase space X of solutions u = u(t, x) becomes infinite-dimensional: typically a Sobolev space of spatial profiles u(t) = u(t, ·). More specifically, the evolution of u(t) = T (t)u 0 ∈ X with time t is complemented by the behavior of the x-profiles x → u(t, x) of solutions u. Such spatial profiles could be monotone or oscillatory; in dim x = 1 they could define sharp fronts or peaks moving at constant or variable speeds, with possible collision or mutual repulsion. Target pat- 22Bernold Fiedler and Arnd Scheel terns or spirals can emerge in dim x = 2. Stacks of spirals, which Winfree called scroll waves, are possible in dim x = 3. For a first cursory orientation in such phenomena and their mathematical treatment we again refer to [Fi02] and the article of Fife in the present volume, as well as the many references there.For our present survey, we focus on the spatio-temporal dynamics of the following rather "simple", prototypical parabolic partial differential equation:(2.1)Here t ≥ 0 denotes time, x ∈ Ω ⊆ R m is space and u = u(t, x) ∈ R N is the solution vector. We consider C 1 -nonlinearities f and constant, positive, diagonal diffusion matrices D. This eliminates the beautiful pattern formation processes due to chemotaxis; see [JäLu92, St00], for example. Note that we will not always restrict f to be a pure reaction term, like f = f (u) or f = f (x, u). More general than that, we sometimes allow for a dependence of f on ∇u to include advection effects. The domain Ω will be assumed smooth and bounded, typically with Dirichlet or Neumann boundary c...
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