# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

May 2003 , Volume 9 , Issue 3

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2003, 9(3): 505-548 doi: 10.3934/dcds.2003.9.505 +[Abstract](1785) +[PDF](438.7KB)
Abstract:
We prove that, for an open class of unimodal maps unfolding a saddle-node bifurcation, chaotic behaviour is a prevalent phenomenon: for a set of parameters with positive Lebesgue density at the bifurcation value the maps exhibit a strange attractor.
2003, 9(3): 549-558 doi: 10.3934/dcds.2003.9.549 +[Abstract](2230) +[PDF](224.7KB)
Abstract:
We define the notion of fiber bundle via a twisted tensor product on the transition matrices. We define the notion of topological conjugacy and shift equivalence in this bundle context and show that topological conjugacy implies shift equivalence. We show that the "Ashley system" $\Sigma_A$ fits into our fiber bundle context. We introduce another system $\Sigma_W$, topologically conjugate to the full $2-$shift, which has the same base space and fiber as the Ashley system, but is constructed with a different twisting. We show that $\Sigma_A$ and $\Sigma_W$ are shift equivalent but not bundle isomorphic.
2003, 9(3): 559-576 doi: 10.3934/dcds.2003.9.559 +[Abstract](1946) +[PDF](711.7KB)
Abstract:
The subject of this paper is a demonstration of the accuracy and robustness of evolution Galerkin schemes applied to two-dimensional Riemann problems with finitely many constant states. In order to have a test case with known exact solution we consider a linear first order system for the wave equation and test evolution Galerkin methods as well as other commonly used schemes with respect to their accuracy in capturing important structural phenomena of the solution. For the two-dimensional Riemann problems with finitely many constant states some parts of the exact solution are constructed in the following three steps. Using a self-similar transformation we solve the Riemann problem outside a neighborhood of the origin and then work inwards. Next a Goursant-type problem has to be solved to describe the interaction of waves up to the sonic circle. Inside it a system of composite elliptic-hyperbolic type is obtained, which may not always be solvable exactly. There an interesting local maximum principle can be shown. Finally, an exact partial solution is used for numerical comparisons.
2003, 9(3): 577-584 doi: 10.3934/dcds.2003.9.577 +[Abstract](2287) +[PDF](142.3KB)
Abstract:
Consider the equation

$u''+|u|^{p-1}u=b|u'|^{q-1}u',\quad t\geq 0,\qquad$(E)

where $p$, $q>1$ and $b>0$ are real numbers. A detailed study of the large-time behavior of solutions of (E) was carried out in [5]. We here investigate the critical case $q=2p/(p+1)$, which is scale-invariant and was not covered in [5]. We prove that all nontrivial solutions blow-up in finite time and that the asymptotic behavior near blow-up exhibits a strong dependence upon the values of $b$. Namely,
(a) if $b\geq b_1(p):=(p+1)((p+1)/2p)^{p/(p+1)}$, then all solutions blow up with a sign, with the rate

$u(t)$~$\pm (T-t)^{-2/(p-1)}\quad$ as $t\to T;$

(b) if $b$<$b_1(p)$, then all solutions have oscillatory blow-up, with

$u(t)=(T-t)^{-2/(p-1)}w$(log$(T-t)+C$),

where $w(s)$ is a single sign-changing periodic function.
Our proofs rely on perturbed energy arguments, invariant regions and on the study of the equation for $w$ via Poincaré-Bendixson and index theory.

2003, 9(3): 585-602 doi: 10.3934/dcds.2003.9.585 +[Abstract](2132) +[PDF](203.5KB)
Abstract:
Homoclinic orbits of semilinear parabolic partial differential equations can split under time-periodic forcing as for ordinary differential equations. The stable and unstable manifold may intersect transverse at persisting homoclinic points. The size of the splitting is estimated to be exponentially small of order exp$(-c/\epsilon)$ in the period $\epsilon$ of the forcing with $\epsilon \rightarrow 0$.
2003, 9(3): 603-616 doi: 10.3934/dcds.2003.9.603 +[Abstract](2055) +[PDF](175.8KB)
Abstract:
We study the existence of periodic solutions of the first order Hamiltonian system

$\dot q = H_p (p,q),\quad \dot p=-H_q(p,q),$

such that

$H(p,q)= h,$

when the prescribed energy surface $S_h=${$(p,q)\in \mathbf R^N \times \mathbf R^N;H(p,q)=h$} is non-compact.
In our previous work, we have considered the class of singular Hamiltonians like

$H(p,q)$~$(|p|^\beta /\beta )-(1 /|q|^\alpha) \quad$ with $1 \leq\alpha<\beta$ and $\beta\geq 2.$

It has proven the existence of generalized (collision) solutions as a limit of approximate solutions corresponding to critical points of certain functionals. In this paper, we relate the Morse index of critical points with the number of collisions of the generalized solution via blow up arguments. In particular, we obtain the existence of a classical (non-collision) solution for $\alpha \in ]\beta /2,\beta$[ when $N \geq 4$ and for $\alpha \in ]2\beta/ 3 ,\beta$[ when $N=3$. As a consequence, we get for smooth Hamiltonians like

$H(p,q)$~$|q|^\alpha (|p|^\beta +1) \quad$ with $1< \alpha < \beta$ and $\beta \geq 2,$

the same existence results since the two classes of Hamiltonians have the same energy surfaces.

2003, 9(3): 617-632 doi: 10.3934/dcds.2003.9.617 +[Abstract](2050) +[PDF](198.9KB)
Abstract:
We relate the rotation interval $\rho(f)$ of a unimodal map $f$ of the interval with its kneading invariant $K(f)$. In particular, we show that for any $\mu \in (0,\frac{1}{2})$, there are kneading invariants $\nu_\mu$ and $\nu_{\mu, h o m}$ such that $\rho(f)=[\mu, \frac{1}{2}]$ if and only if $\nu_\mu \preceq K(f) \preceq \nu_{\mu, h o m}$.
2003, 9(3): 633-650 doi: 10.3934/dcds.2003.9.633 +[Abstract](2907) +[PDF](206.7KB)
Abstract:
A time-dependent system modeling the interaction between a Stokes fluid and an elastic structure is studied. A divergence-free weak formulation is introduced which does not involve the fluid pressure field. The existence and uniqueness of a weak solution is proved. Strong energy estimates are derived under additional assumptions on the data. The existence of an $L^2$ integrable pressure field is established after the verification of an inf-sup condition.
2003, 9(3): 651-662 doi: 10.3934/dcds.2003.9.651 +[Abstract](2919) +[PDF](167.1KB)
Abstract:
We study the global existence and asymptotic behavior of solutions to the Cauchy problem for the semilinear dissipative wave equations: $\square u + \partial_t u = |u|^{\alpha+1}$, $u|_{t=0}=\varepsilon u_0 \in H^1 \cap L^1$, $\partial_t u |_{t=0} = \varepsilon u_1 \in L^2 \cap L^1$ with a small parameter $\varepsilon>0$. When $N\le 3$ and $2/N<\alpha \le 2/[N-2]^+$, we show the global solvability and derive the sharp rates of the solutions.
2003, 9(3): 663-676 doi: 10.3934/dcds.2003.9.663 +[Abstract](2042) +[PDF](190.4KB)
Abstract:
The Maxwell-Bloch equations describing the propagation of electromagnetic waves in a gas of quantum mechanical systems with two energy levels is investigated. The system under consideration consists of a generally nonlinear second order system of differential equations for the dielectrical polarization and the density coupled with Maxwell's equations for the electromagnetic field. The goal is to show decay of the polarization field for $t\rightarrow\infty$.
2003, 9(3): 677-704 doi: 10.3934/dcds.2003.9.677 +[Abstract](2011) +[PDF](301.7KB)
Abstract:
Using methods of the theory of nonautonomous linear differential systems, namely exponential dichotomies and rotation numbers, we generalize some aspects of Yakubovich's Frequency Theorem from periodic control systems to systems with bounded uniformly continuous coefficients.
2003, 9(3): 705-726 doi: 10.3934/dcds.2003.9.705 +[Abstract](2235) +[PDF](230.4KB)
Abstract:
We study two models arising in phase transition dynamics. The state of the system is described by the pair $(\theta,\chi)$, where $\theta$ is the (relative) temperature and $\chi$ is the order parameter or phase field. The main difference between the two models relies on whether global constraints on $\chi$ are imposed or not: accordingly, the resulting models will be called conserved or nonconserved. Memory effects influencing both the heat flux and the dynamics of $\chi$ have been considered in a number of recent papers. Here we assume the Fourier law for the heat flux in the energy balance equation, while we consider memory effects in the order parameter dynamics. We analyze the well-posedness of corresponding Cauchy-Neumann problems for both conserved and nonconserved models. Various results are derived according to properties of the memory kernel involved.
2003, 9(3): 727-744 doi: 10.3934/dcds.2003.9.727 +[Abstract](2885) +[PDF](211.4KB)
Abstract:
Nonautonomous and random dynamical systems perturbed by impulses are considered. The impulses form a flow. Over this flow the perturbed system also has the structure of a new nonautonomous/random dynamical system. The long time behavior of this system is considered. In particular the existence of an attractor is proven. The result can be applied to a large class of dissipative systems given by partial or ordinary differential equations. As an example of this class of problems the Lorenz system is studied. For another problem given by a one-dimensional affine differential equation and perturbed by affine impulses, the attractor can be calculated explicitly.
2003, 9(3): 745-750 doi: 10.3934/dcds.2003.9.745 +[Abstract](2044) +[PDF](129.6KB)
Abstract:
Recently in [6] Y. Lacroix proved that any distribution function can be obtained as a limit law of return time for any ergodic aperiodic system. In this note we provide an alternative construction, based on Bratteli-Vershik representations of systems, which works for any minimal Cantor system having an infinite periodic spectrum. The construction is especially simple for odometers.
2003, 9(3): 751-770 doi: 10.3934/dcds.2003.9.751 +[Abstract](2401) +[PDF](217.6KB)
Abstract:
In this paper, we study the existence of periodic solutions of equations

$x''+a x^+ - b x^-$ $+ g(x')=p(t),$

$x''+a x^+ - b x^-$ $+ f(x)+g(x')=p(t),$

where $(a, b)$ lies on one of the Fučik spectrum curves. We provide sufficient conditions for the existence of periodic solutions for the given equations if the limits $\lim_{x\to+\infty}g(x)=g(+\infty), \lim_{x\to-\infty}g(x)=g(-\infty)$ and $\lim_{x\to+\infty}f(x)=f(+\infty)$, $\lim_{x\to-\infty}f(x)=f(-\infty)$ exist and are finite. We also prove that the former equation has at least one periodic solution if $g(x)$ satisfies sublinear condition and that the latter equation has at least one periodic solution if $g(x)$ is bounded and $f(x)$ satisfies subquadratic condition.

2003, 9(3): 771-787 doi: 10.3934/dcds.2003.9.771 +[Abstract](2239) +[PDF](212.8KB)
Abstract:
The $3n+1$ function is given by $T(n)=n/2$ for $n$ even, $T(n)=(3n+1)/2$ for $n$ odd. Given a positive integer $a$, another number $b$ is a called a predecessor of $a$ if some iterate $T^\nu(b)$ equals $a$. Here some ideas are described which may lead to a proof showing that the set of predecessors of $a$ has positive lower asymptotic density, for any positive integer $a\ne 0$ mod 3. Three unbridged gaps in the argument are formulated as conjectures.

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