Abstract. This paper gives theoretical results on spinodal decomposition for the Cahn-Hillard equation. We prove a mechanism which explains why most solutions for the Cahn-Hilliard equation starting near a homogeneous equilibrium within the spinodal interval exhibit phase separation with a characteristic wavelength when exiting a ball of radius R. Namely, most solutions are driven into a region of phase space in which linear behavior dominates for much longer than expected.The Cahn-Hilliard equation depends on a small parameter ε, modeling the (atomic scale) interaction length; we quantify the behavior of solutions as ε → 0. Specifically, we show that most solutions starting close to the homogeneous equilibrium remain close to the corresponding solution of the linearized equation with relative distance O(ε 2−n/2 ) up to a ball of radius R in the H 2 (Ω)-norm, where R is proportional to ε −1+ +n/4 as ε → 0. Here, n ∈ {1, 2, 3} denotes the dimension of the considered domain, and > 0 can be chosen arbitrarily small. Not only does this approach significantly increase the radius of explanation for spinodal decomposition, but it also gives a clear picture of how the phenomenon occurs.While these results hold for the standard cubic nonlinearity, we also show that considerably better results can be obtained for similar higher order nonlinearities. In particular, we obtain R ∼ ε −2+ +n/2 for every > 0 by choosing a suitable nonlinearity.

The appearance of numerous period-doubling cascades is among the most prominent features of parametrized maps, that is, smooth one-parameter families of maps F : R × M → M, where M is a smooth locally compact manifold without boundary, typically R N . Each cascade has infinitely many period-doubling bifurcations, and it is typical to observe -such as in all the examples we investigate here -that whenever there are any cascades, there are infinitely many cascades. We develop a general theory of cascades for generic F . We illustrate this theory with several examples. We show that there is a close connection between the transition through infinitely many cascades and the creation of a horseshoe. 1 arXiv:0903.3613v3 [math.DS]

The Birkhoff Ergodic Theorem concludes that time averages, i.e., Birkhoff averages, B N (f ) ∶= Σ N −1 n=0 f (x n ) N of a function f along a length N ergodic trajectory (x n ) of a function T converge to the space average ∫ f dµ, where µ is the unique invariant probability measure. Convergence of the time average to the space average is slow. We use a modified average of f (x n ) by giving very small weights to the "end" terms when n is near 0 or N − 1. When (x n ) is a trajectory on a quasiperiodic torus and f and T are C ∞ , our Weighted Birkhoff average (denoted WB N (f )) converges "super" fast to ∫ f dµ with respect to the number of iterates N , i.e. with error decaying faster than N −m for every integer m. Our goal is to show that our Weighted Birkhoff average is a powerful computational tool, and this paper illustrates its use for several examples where the quasiperiodic set is one or two dimensional. In particular, we compute rotation numbers and conjugacies (i.e. changes of variables) and their Fourier series, often with 30-digit accuracy. d j=1 a j ρ j = 0, then every a j = 0. We then say such a ρ is irrational.Let T be a C ∞ quasiperiodic map. The quasiperiodicity persists for most small perturbations by the Kolmogorov-Arnold-Moser theory. We believe that quasiperiodicity is one of only three types of invariant sets with a dense trajectory that can occur in typical smooth maps. The other two types are periodic sets and chaotic sets. See [1] for the statement of our formal conjecture of this triumvirate. For example, quasiperiodicity occurs in a system of weakly coupled oscillators, in which there is an invariant smooth *

The use of rigorous verification methods is a powerful tool which permits progress in the analysis of dynamical processes that is not possible using purely analytical techniques. In this paper we develop a set of tools for branch validation, which allows for the rigorous verification of branch behavior, bifurcation, and solution index on branches generated through a saddle-node bifurcation. While the presented methodology can be applied in a variety of settings, we illustrate the use of these tools in the context of materials science. In particular, lattice models have been proposed as a more realistic reflection of the behavior of materials than traditional continuum models. For example, unlike their continuum counterparts, lattice models can account for phenomena such as pinning, and a significant body of work has been developed to study traveling waves. However, in a variety of other contexts such as bifurcation theory, questions about lattice dynamical systems are significantly harder to answer than those for continuum models. In the present paper, we show that computer-assisted proof techniques can be used to answer some of these questions. We apply these tools to the discrete Allen-Cahn equation, giving us results on the existence of branches of mosaic solutions and their robustness as it relates to grain size. We also demonstrate that there are situations in which classical continuation methods can fail to identify the correct branching behavior.

The appearance of infinitely-many period-doubling cascades is one of the most prominent features observed in the study of maps depending on a parameter. They are associated with chaotic behavior, since bifurcation diagrams of a map with a parameter often reveal a complicated intermingling of period-doubling cascades and chaos.Period doubling can be studied at three levels of complexity. The first is an individual period-doubling bifurcation. The second is an infinite collection of period doublings that are connected together by periodic orbits in a pattern called a cascade. It was first described by Myrberg and later in more detail by Feigenbaum. The third involves infinitely many cascades and a parameter value µ 2 of the map at which there is chaos. We show that often virtually all (i.e., all but finitely many) "regular" periodic orbits at µ 2 are each connected to exactly one cascade by a path of regular periodic orbits; and virtually all cascades are either paired -connected to exactly one other cascade, or solitary -connected to exactly one regular periodic orbit at µ 2 . The solitary cascades are robust to large perturbations. Hence the investigation of infinitely many cascades is essentially reduced to studying the regular periodic orbits of F (µ 2 , ·). Examples discussed include the forced-damped pendulum and the double-well Duffing equation.