# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

December 2006 , Volume 16 , Issue 4

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2006, 16(4): 721-743 doi: 10.3934/dcds.2006.16.721 +[Abstract](1948) +[PDF](234.5KB)
Abstract:
This paper is concerned with the homogenization of the nonlinear wave equation

$\partial_{t t} u_{\varepsilon} - \Delta u_{\varepsilon} + \partial_t F(u_{\varepsilon}) = 0$ in $\Omega_{\varepsilon}\times(0,+\infty),$

where $\Omega_{\varepsilon}$ is a domain containing holes with small capacity. In the context of optimal control, this semilinear hyperbolic equation was studied by Lions (1980) through a theory of ultra-weak solutions. Combining his arguments with the abstract framework proposed by Cioranescu and Murat (1982), for the homogenization of elliptic problems, a new approach is presented to solve the above nonlinear homogenization problem. In the linear case, one improves early classical results by Cionarescu, Donato, Murat and Zuazua (1991).

2006, 16(4): 745-756 doi: 10.3934/dcds.2006.16.745 +[Abstract](1514) +[PDF](160.5KB)
Abstract:
We describe the dynamics of a system of two reaction-diffusion equations on the circle. We show that the elements of the $\omega$-limit sets of every solution can be classified by the number of times which they wind around the circle line --- they look either flat or screw-shaped. We prove a Poincaré-Bendixson result. Furthermore, we give a criterion under which screw-shaped stationary or periodic solutions are unstable.
2006, 16(4): 757-781 doi: 10.3934/dcds.2006.16.757 +[Abstract](1519) +[PDF](1422.3KB)
Abstract:
We consider a dissipative standard map--like system, which is governed by two parameters measuring the strength of the dissipation and of the perturbation. In order to investigate the dynamics, we follow a numerical and an analytical approach. The numerical study relies on the frequency analysis and on the computation of the differential fast Lyapunov indicators. The analytical approach is based on the computation of a suitable normal form for dissipative systems, which allows us to derive an analytic expression of the frequency.
We explore different kinds of attractors (invariant curves, periodic orbits, strange attractors) and their relation with the choice of the perturbing function and of the main frequency of motion (i.e., the frequency of the invariant trajectory of the unperturbed system). In this context we also investigate the occurrence of periodic attractors by looking at the relationship between their periods and the parameters ruling the mapping. Particular attention is devoted to the investigation of the weakly chaotic regime and its transition to the conservative case.
2006, 16(4): 783-842 doi: 10.3934/dcds.2006.16.783 +[Abstract](2183) +[PDF](1673.3KB)
Abstract:
We study the gradient flow associated with the functional $F_\phi(u)$ := $\frac{1}{2}\int_{I} \phi(u_x)~dx$, where $\phi$ is non convex, and with its singular perturbation $F_\phi^\varepsilon(u)$:=$\frac{1}{2}\int_I (\varepsilon^2 (u_{x x})^2 + \phi(u_x))dx$. We discuss, with the support of numerical simulations, various aspects of the global dynamics of solutions $u^\varepsilon$ of the singularly perturbed equation $u_t = - \varepsilon^2 u_{x x x x} + \frac{1}{2} \phi''(u_x)u_{x x}$ for small values of $\varepsilon>0$. Our analysis leads to a reinterpretation of the unperturbed equation $u_t = \frac{1}{2} (\phi'(u_x))_x$, and to a well defined notion of a solution. We also examine the conjecture that this solution coincides with the limit of $u^\varepsilon$ as $\varepsilon\to 0^+$.
2006, 16(4): 843-856 doi: 10.3934/dcds.2006.16.843 +[Abstract](2078) +[PDF](207.7KB)
Abstract:
In the case of a $C^1$ self-map of $R^3$ we prove the Chow, Mallet-Paret and Yorke conjecture on the form of sequences of local fixed point indices of iterations and give a complete description of possible sequences of indices.
2006, 16(4): 857-869 doi: 10.3934/dcds.2006.16.857 +[Abstract](1931) +[PDF](174.4KB)
Abstract:
We study the multifractal spectrum for hyperbolic maps from the exponential family. We define a class of potentials for which we prove the existence of conformal measures. Next, we show that the multifractal spectrum of this conformal measure is the Legendre transform of the temperature function. We prove that the domain of the spectrum is unbounded and show that there are two possibilities for its shape.
2006, 16(4): 871-882 doi: 10.3934/dcds.2006.16.871 +[Abstract](1759) +[PDF](167.5KB)
Abstract:
Let Int$^1WS(M)$ be the $C^1$-interior of the set of diffeomorphisms of a smooth closed manifold $M$ having the weak shadowing property. The second author has shown that if $\dim M = 2$ and all of the sources and sinks of a diffeomorphism $f \in$ Int$^1WS(M)$ are trivial, then $f$ is structurally stable. In this paper, we show that there exist diffeomorphisms $f \in$ Int$^1WS(M)$, $\dim M = 2$, such that $(i)$ $f$ belongs to the $C^1$-interior of diffeomorphisms for which the $C^0$-transversality condition is not satisfied, $(ii)$ $f$ has a saddle connection. These results are based on the following theorem: if the phase diagram of an $\Omega$-stable diffeomorphism $f$ of a manifold $M$ of arbitrary dimension does not contain chains of length $m > 3$, then $f$ has the weak shadowing property.
2006, 16(4): 883-896 doi: 10.3934/dcds.2006.16.883 +[Abstract](1649) +[PDF](223.6KB)
Abstract:
Given a continuous $l$-modal map $g$ of the interval $[0,1]$ we prove the existence of a polynomial $P$ with modality $\leq l$ such that $g$ is strongly semi-conjugate to $P$ in $[0,1]$. This is an improvement of a result in [4]. We do a modification on the Thurston operator in order to control the semi-finite combinatorial case. It turns out that all the essential attractors of $P$ have the same local topological type as those of $g$. This allows to construct the strong semi-conjugacy. We also present some examples agreeing with the results.
2006, 16(4): 897-922 doi: 10.3934/dcds.2006.16.897 +[Abstract](2135) +[PDF](300.0KB)
Abstract:
Let us consider a real quadratic-like germ $f_$∗ which is infinitely renormalizable with tripling essentially bounded combinatorics and consider the lamination given by the hybrid classes in the space of quadratic-like germs, then its holonomy map is shown to be $C^1$ at $f_$∗ if the combinatorics of $f_$∗ satisfies a growth condition. As a consequence, a proof of the self-similarity of the Mandelbrot set for this type of combinatorics is given.
2006, 16(4): 923-960 doi: 10.3934/dcds.2006.16.923 +[Abstract](2100) +[PDF](370.8KB)
Abstract:
In this paper we apply the equivariant degree method to the Hopf bifurcation problem for a system of symmetric functional differential equations. Local Hopf bifurcation is classified by means of an equivariant topological invariant based on the symmetric properties of the characteristic operator. As examples, symmetric configurations of identical oscillators, with dihedral, tetrahedral, octahedral, and icosahedral symmetries, are analyzed.

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