# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

2008 , Volume 22 , Issue 3

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2008, 22(3): 465-508 doi: 10.3934/dcds.2008.22.465 +[Abstract](615) +[PDF](442.5KB)
Abstract:
We consider the equation $\varepsilon^2\Delta$ ũ-ũ+ũ$^p =0$ in a bounded, smooth domain $\Omega$ in $\R^2$ under homogeneous Neumann boundary conditions. Let $\Gamma$ be a segment contained in $\Omega$, connecting orthogonally the boundary, non-degenerate and non-minimal with respect to the curve length. For any given integer $N\ge 2$ and for small $\varepsilon$ away from certain critical numbers, we construct a solution exhibiting $N$ interior layers at mutual distances $O(\varepsilon|\ln\varepsilon|)$ whose center of mass collapse onto $\Gamma$ at speed $O(\varepsilon^{1+\mu})$ for small positive constant $\mu$ as $\varepsilon\to 0$. Asymptotic location of these layers is governed by a Toda system.
2008, 22(3): 509-528 doi: 10.3934/dcds.2008.22.509 +[Abstract](664) +[PDF](276.7KB)
Abstract:
We consider linear equations $v'=A(t)v$ that may exhibit different asymptotic behaviors in different directions. These can be thought of as stable, unstable and central behaviors, although here with respect to arbitrary asymptotic rates $e^{c \rho(t)}$ determined by a function $\rho(t)$, including the usual exponential behavior $\rho(t)=t$ as a very special case. In particular, we consider the notion of $\rho$-nonuniform exponential trichotomy, that combines simultaneously the nonuniformly hyperbolic behavior with arbitrary asymptotic rates. We show that for $\rho$ in a large class of rate functions, any linear equation in block form in a finite-dimensional space, with three blocks having asymptotic rates $e^{c \rho(t)}$ respectively with $c$ negative, zero, and positive, admits a $\rho$-nonuniform exponential trichotomy. We also give explicit examples that cannot be made uniform and for which one cannot take $\rho(t)=t$ without making all Lyapunov exponents infinite. Furthermore, we obtain sharp bounds for the constants that determine the exponential trichotomy. These are expressed in terms of appropriate Lyapunov exponents that measure the growth rate with respect to the function $\rho$.
2008, 22(3): 529-540 doi: 10.3934/dcds.2008.22.529 +[Abstract](458) +[PDF](183.4KB)
Abstract:
We give a deterministic representation for position dependent random maps and describe the structure of its set of invariant measures. Our construction generalizes the skew product representation of random maps with constant probabilities. In particular, we establish one-to-one correspondence between eigenfunctions corresponding to eigenvalues of unit modulus for the Frobenius-Perron (transfer) operator of the random map and for those of the skew. An immediate consequence is one-to-one correspondence between absolutely continuous invariant measures (acims) for the position dependent random map and acims for its deterministic representation.
2008, 22(3): 541-556 doi: 10.3934/dcds.2008.22.541 +[Abstract](434) +[PDF](208.5KB)
Abstract:
Let $T:X\to X$ be a dynamical system, and $\phi: X\to \mathbb{R}$ a function on $X$. A function $\theta:X\to \mathbb{R}$ is called a sub-action if $\theta$ satisfies the equation

$\phi \leq \theta \circ T - \theta + m(\phi, T)$

where $m(\phi, T)=$sup{$\int \phi d\mu:\mu$ is an invariant probability measure for $T$}. The existence and regularity of sub-actions are important for the study of optimizing measures. We prove the existence of Hölder sub-actions for Lipschitz functions on certain classes of Manneville-Pomeau type maps. We also construct locally Hölder sub-actions for Lipschitz functions on Young Towers. In some settings (uniform hyperbolicity and Manneville-Pomeau maps) this implies Hölder sub-actions for the underlying system modeled by the Tower.

2008, 22(3): 557-586 doi: 10.3934/dcds.2008.22.557 +[Abstract](689) +[PDF](389.0KB)
Abstract:
We prove the existence of a compact, finite dimensional, global attractor for a coupled PDE system comprising a nonlinearly damped semilinear wave equation and a nonlinear system of thermoelastic plate equations, without any mechanical (viscous or structural) dissipation in the plate component. The plate dynamics is modelled following Berger's approach; we investigate both cases when rotational inertia is included into the model and when it is not. A major part in the proof is played by an estimate--known as stabilizability estimate--which shows that the difference of any two trajectories can be exponentially stabilized to zero, modulo a compact perturbation. In particular, this inequality yields bounds for the attractor's fractal dimension which are independent of two key parameters, namely $\gamma$ and $\kappa$, the former related to the presence of rotational inertia in the plate model and the latter to the coupling terms. Finally, we show the upper semi-continuity of the attractor with respect to these parameters.
2008, 22(3): 587-604 doi: 10.3934/dcds.2008.22.587 +[Abstract](466) +[PDF](279.1KB)
Abstract:
In this paper, we study the topology of Bott integrable Hamiltonian flows on $S^{2}\times S^{1}$ in terms of some types of periodic orbits, called NMS periodic orbits. The set of these periodic orbits can be identified by means of some operations applied on global and local links. These operations come from the round handle decomposition of these systems on $S^{2}\times S^{1}.$ We apply the results to obtain a non-integrability criterium.
2008, 22(3): 605-628 doi: 10.3934/dcds.2008.22.605 +[Abstract](465) +[PDF](294.5KB)
Abstract:
The Krasnosel'skii type degree formula for the equation

$\dot u = - Au + F(u)$

where $A:D(A)\to E$ is a linear operator on a separable Banach space $E$ such that $-A$ is a generator of a $C_0$ semigroup of bounded linear operators of $E$ and $F:E\to E$ is a locally Lipschitz $k$-set contraction, is provided. Precisely, it is shown that if $V$ is an open bounded subset of $E$ such that $0$∉$(-A+F)(\partial V \cap D(A))$, then the topological degree of $-A+F$ with respect to $V$ is equal to the fixed point index of the operator of translation along trajectories for sufficiently small positive time. The obtained degree formula is crucial for the method of translation along trajectories. It is applied to the nonautonomous periodic problem and an average principle is derived. As an application a first order system of partial differential equations is considered.

2008, 22(3): 629-662 doi: 10.3934/dcds.2008.22.629 +[Abstract](435) +[PDF](301.6KB)
Abstract:
This is a continuation of our work on the nonlinear stability of traveling shock fronts arising in multidimensional conservation laws with fourth order regularization only. Our motivating example is the thin film equation for which planar waves correspond with fluid coating a pre-wetted surface. Under only the fourth order regularization, we established the nonlinear stability of compressive waves for dimensions $d\geq 2$, and of under-compressive waves for dimensions $d\geq 3$ under general spectral conditions. The case of stability for under-compressive waves in the thin film equations for the critical dimensions $d=1,2$ remained open. In this paper we study the nonlinear stability of under-compressive waves by assuming both the second and fourth order regularization. We present a step toward the open problem by establishing the nonlinear stability of under-compressive waves in dimensions $d\geq 2$ under general spectral conditions. We emphasize the above mentioned stability question remains still open.
2008, 22(3): 663-682 doi: 10.3934/dcds.2008.22.663 +[Abstract](576) +[PDF](277.7KB)
Abstract:
We give a combinatorial classification of postsingularly finite exponential maps in terms of external addresses starting with the entry $0$. This extends the classification results for critically preperiodic polynomials [2] to exponential maps. Our proof relies on the topological characterization of postsingularly finite exponential maps given recently in [14]. These results illustrate once again the fruitful interplay between combinatorics, topology and complex structure which has often been successful in complex dynamics.
2008, 22(3): 683-697 doi: 10.3934/dcds.2008.22.683 +[Abstract](653) +[PDF](260.5KB)
Abstract:
Let $f$ be a diffeomorphism of a closed $C^\infty$ manifold. In this paper, we define the notion of the $C^1$-stable shadowing property for a closed $f$-invariant set, and prove that $(i)$ the chain recurrent set $R(f)$ of $f$ has the $C^1$-stable shadowing property if and only if $f$ satisfies both Axiom A and the no-cycle condition, and $(ii)$ for the chain component $C_f(p)$ of $f$ containing a hyperbolic periodic point $p$, $C_f(p)$ has the $C^1$-stable shadowing property if and only if $C_f(p)$ is the hyperbolic homoclinic class of $p$.
2008, 22(3): 699-709 doi: 10.3934/dcds.2008.22.699 +[Abstract](489) +[PDF](199.0KB)
Abstract:
The main concern of this paper is to give lower bounds for the Hausdorff dimension of the Geometric Lorenz attractor in terms of the eigenvalues of the singularity and the symbolic dynamic associated with the geometrical distribution with the attractor.
2008, 22(3): 711-727 doi: 10.3934/dcds.2008.22.711 +[Abstract](467) +[PDF](302.4KB)
Abstract:
In this paper we prove a generalization of the iterated homogenization theorem for monotone operators, proved by Lions et al. in [20] and [21]. Our results enable us to homogenize more realistic models of multiscale structures.
2008, 22(3): 729-758 doi: 10.3934/dcds.2008.22.729 +[Abstract](568) +[PDF](389.5KB)
Abstract:
We consider the following pseudoparabolic regularization of a forward-backward quasilinear diffusion equation: $u_t=\Delta \phi(u)+\varepsilon\Delta u_t$ ($\varepsilon>0$). As suggested by several models of the applied sciences, the function $\phi$ is nonmonotone and vanishing at infinity. We investigate the limit points of the set of solutions to the associated Neumann problem as $\varepsilon\to 0$, proving existence of suitable weak solutions of the original ill-posed equation. Qualitative properties of such solutions are also addressed.
2008, 22(3): 759-789 doi: 10.3934/dcds.2008.22.759 +[Abstract](456) +[PDF](4469.9KB)
Abstract:
We obtain in this paper topological models of binary differential equation at local codimension 2 singularities where all the coefficients of the equation vanish at the singular point. We also study the bifurcations of these singularities when the equation is deformed in a generic 2-parameter families of equations.
2008, 22(3): 791-806 doi: 10.3934/dcds.2008.22.791 +[Abstract](405) +[PDF](298.8KB)
Abstract:
Planar piecewise isometries with convex polygonal atoms that are piecewise irrational rotations can naturally generate a packing of phase space given by periodic cells that are discs. We show that such packings cannot contain certain subpackings of Apollonian packings, namely those belonging to a family of Arbelos subpackings. We do this by showing that the unit complex numbers giving the directions of tangency within such an isometric-generated packing lie in a finitely generated subgroup of the circle group, whereas this is not the case for the Arbelos subpackings. In the opposite direction, we show that, given an arbitrary disc packing of a polygonal region, there is a piecewise isometry whose regular cells approximate the given packing to any specified precision.
2008, 22(3): 807-815 doi: 10.3934/dcds.2008.22.807 +[Abstract](383) +[PDF](163.1KB)
Abstract:
We introduce the Fibonacci bimodal maps on the interval and show that their two turning points are both in the same minimal invariant Cantor set. Two of these maps with the same orientation have the same kneading sequences and, among bimodal maps without central returns, they exhibit turning points with the strongest recurrence as possible.
2008, 22(3): 816-816 doi: 10.3934/dcds.2008.22.816 +[Abstract](348) +[PDF](79.0KB)
Abstract:
Erratum to "Canard cycles with two breaking parameters'' (Discrete and Continuous Dynamical Systems - Series A, Vol.17, no. 4, 2007, 787-806).

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