# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

July 2005 , Volume 12 , Issue 4

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2005, 12(4): 569-594 doi: 10.3934/dcds.2005.12.569 +[Abstract](1476) +[PDF](418.1KB)
Abstract:
We consider the classical D'Alembert Hamiltonian model for a rotationally symmetric planet revolving on Keplerian ellipse around a fixed star in an almost exact "day/year" resonance and prove that, notwithstanding proper degeneracies, the system is stable for exponentially long times, provided the oblateness and the eccentricity are suitably small.
2005, 12(4): 595-606 doi: 10.3934/dcds.2005.12.595 +[Abstract](1902) +[PDF](216.1KB)
Abstract:
In this paper we investigate the existence and asymptotic behavior of positive solutions to a quasilinear elliptic problem with Neumann condition.
2005, 12(4): 607-628 doi: 10.3934/dcds.2005.12.607 +[Abstract](1968) +[PDF](279.8KB)
Abstract:
We study the properties of the transport density measure in the Monge-Kantorovich optimal mass transport problem in the presence of so-called Dirichlet constraint, i.e. when some closed set is given along which the cost of transportation is zero. The Hausdorff dimension estimates, as well as summability and higher regularity properties of the transport density are studied. The uniqueness of the transport density is proven in the case when the masses to be transported are represented by measures absolutely continuous with respect to the Lebesgue measure.
2005, 12(4): 629-638 doi: 10.3934/dcds.2005.12.629 +[Abstract](1513) +[PDF](224.6KB)
Abstract:
A nonsingular flow $\varphi_t$ on a $3$-manifold induces a flow on the plane bundle orthogonal to $\varphi_t$ by the derivative. This flow also induces a flow $\psi_t$ on its projectivized bundle $PX$, which is called the projective flow. In this paper, we will investigate this projective flow in order to understand the original flow $\varphi_t$, in particular, under the condition that $\varphi_t$ is minimal and $\psi_t$ has more than one minimal sets: If the projective flow $\psi_t$ has more than two minimal sets, then we will show that $\varphi_t$ is topologically equivalent to an irrational flow on the $3$-torus. In the case when $\psi_t$ has exactly two minimal sets, then we obtain several properties of the minimal sets of $\psi_t$. In particular, we construct two $C^\infty$ sections to $PX$ which separate these minimal sets (and hence $PX$ is a trivial bundle) if $\varphi_t$ is not topologically equivalent to an irrational flow on the $3$-torus. As an application of this characterization, the chain recurrent set of the projective flow is shown to be the whole $PX$.
2005, 12(4): 639-656 doi: 10.3934/dcds.2005.12.639 +[Abstract](1496) +[PDF](285.8KB)
Abstract:
We establish upper bounds on the rate of decay of correlations of tower systems with summable variation of the Jacobian and integrable return time. That is, we consider situations in which the Jacobian is not Hölder and the return time is only subexponentially decaying. We obtain a subexponential bound on the correlations, which is essentially the slowest of the decays of the variation of the Jacobian and of the return time.
2005, 12(4): 657-674 doi: 10.3934/dcds.2005.12.657 +[Abstract](1741) +[PDF](254.4KB)
Abstract:
In this work we show that the smooth classification of divergent diagrams of folds $(f_1, \ldots, f_s) : (\mathbb R^n,0) \to (\mathbb R^n \times \cdots \times \mathbb R^n,0)$ can be reduced to the classification of the $s$-tuples $(\varphi_1, \ldots, \varphi_s)$ of associated involutions. We apply the result to obtain normal forms when $s \leq n$ and $\{\varphi_1, \ldots, \varphi_s\}$ is a transversal set of linear involutions. A complete description is given when $s=2$ and $n\geq 2$. We also present a brief discussion on applications of our results to the study of discontinuous vector fields and discrete reversible dynamical systems.
2005, 12(4): 675-686 doi: 10.3934/dcds.2005.12.675 +[Abstract](1645) +[PDF](271.3KB)
Abstract:
This paper concerns with the number and distribution of limit cycles of a perturbed cubic Hamiltonian system which has 5 centers and 4 saddle points. The stability analysis and bifurcation methods of differential equations are applied to study the homoclinic loop bifurcation under $Z_2$-equivariant cubic perturbation. It is proved that the perturbed system can have 11 limit cycles with two different distributions, one of which is already known, the other is new.
2005, 12(4): 687-722 doi: 10.3934/dcds.2005.12.687 +[Abstract](1872) +[PDF](383.8KB)
Abstract:
The aim of the work is to study the stability of equilibrium points for some functional differential equations with state-dependent delay. As a preliminary step, existence, continuation, uniqueness and smoothness results have been shown for solutions. From a given functional differential equation with state-dependent delay, a linearized equation is constructed which gives sufficient conditions for asymptotic stability of equilibrium solutions. Also, a saddle point theorem is shown in the case where the equilibrium point is a hyperbolic equilibrium point for the linearized equation.
2005, 12(4): 723-736 doi: 10.3934/dcds.2005.12.723 +[Abstract](2093) +[PDF](268.0KB)
Abstract:
A right-sided, nearest neighbour cellular automaton (RNNCA) is a continuous transformation $\Phi:\mathcal A^{\mathbb Z} \rightarrow\mathcal A^{\mathbb Z}$ determined by a local rule $\phi:\mathcal A^{\{0,1\}}\rightarrow\mathcal A$ so that, for any $\mathbf a\in\mathcal A^{\mathbb Z}$ and any $z\in\mathbb Z$, $\Phi(\mathbf a)_z = \phi(a_z,a_{z+1})$. We say that $\Phi$ is bipermutative if, for any choice of $a\in\mathcal A$, the map $\mathcal A\ni b \mapsto \phi(a,b)\in\mathcal A$ is bijective, and also, for any choice of $b\in\mathcal A$, the map $\mathcal A\ni a \mapsto \phi(a,b)\in\mathcal A$ is bijective.
We characterize the invariant measures of bipermutative RNNCA. First we introduce the equivalent notion of a quasigroup CA. Then we characterize $\Phi$-invariant measures when $\mathcal A$ is a (nonabelian) group, and $\phi(a,b) = a\cdot b$. Then we show that, if $\Phi$ is any bipermutative RNNCA, and $\mu$ is $\Phi$-invariant, then $\Phi$ must be $\mu$-almost everywhere $K$-to-1, for some constant $K$. We then characterize invariant measures when $\mathcal \mathcal A^{\mathbb Z}$ is a group shift and $\Phi$ is an endomorphic CA.
2005, 12(4): 737-746 doi: 10.3934/dcds.2005.12.737 +[Abstract](1883) +[PDF](177.6KB)
Abstract:
We prove the existences of multiple sign-changing solutions for a semilinear elliptic eigenvalue problem with constraint by using variational methods under weaker conditions.
2005, 12(4): 747-760 doi: 10.3934/dcds.2005.12.747 +[Abstract](1681) +[PDF](228.0KB)
Abstract:
We prove existence of radial positive solutions for the equation

$-\Delta u + V(y) u =u^{\frac{N+2}{N-2}+\varepsilon} \quad$ in $\quad \mathbb R^N$

where the potential $V$ is a radial smooth function with $V(0)<0$.
In particular, we show that the solutions have the shape of a super-position of spikes blowing-up at the origin as $\varepsilon \to 0$, with different rates of concentration.

2005, 12(4): 761-772 doi: 10.3934/dcds.2005.12.761 +[Abstract](1548) +[PDF](206.8KB)
Abstract:
In this note we introduce a decoupling technique for operator matrices with "non-diagonal" domains on "coupled" spaces which greatly simplifies the study of Cauchy problems stemming from wave equations with dynamic boundary conditions.
2005, 12(4): 773-789 doi: 10.3934/dcds.2005.12.773 +[Abstract](1979) +[PDF](250.9KB)
Abstract:
We calculate the Ruelle operator of a transcendental entire function $f$ having only a finite set of algebraic singularities. Moreover, under certain topological conditions on the postcritical set we prove (i) if $f$ has a summable critical point, then $f$ is not structurally stable and (ii) if all critical points of $f$ belonging to Julia set are summable, then there do not exist invariant lines fields on the Julia set.

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