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Discrete & Continuous Dynamical Systems - A

2006 , Volume 14 , Issue 3

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Asymptotically stable equilibria for monotone semiflows
M. W. Hirsch and Hal L. Smith
2006, 14(3): 385-398 doi: 10.3934/dcds.2006.14.385 +[Abstract](572) +[PDF](240.5KB)
Abstract:
Conditions for the existence of a stable equilibrium and for the existence of an asymptotically stable equilibrium for a strongly order preserving semiflow are presented. Analyticity of the semiflow and the compactness of certain subsets of the set of equilibria are required for the latter and yield finiteness of the equilibrium set. Our results are applied to semilinear parabolic partial differential equations and to the classical Kolmogorov competition system with diffusion.
Vertex maps for trees: Algebra and periods of periodic orbits
Chris Bernhardt
2006, 14(3): 399-408 doi: 10.3934/dcds.2006.14.399 +[Abstract](563) +[PDF](199.6KB)
Abstract:
Let $T$ be a tree with $n$ vertices. Let $f: T \rightarrow T$ be continuous and suppose that the $n$ vertices form a periodic orbit under $f$. The combinatorial information that comes from possible permutations of the vertices gives rise to an irreducible representation of $S_n$. Using the algebraic information it is shown that $f$ must have periodic orbits of certain periods. Finally, a family of maps is defined which shows that the result about periods is best possible if $n=2^k+2^l$ for $k, l \geq 0$.
Necessary and sufficient conditions for semi-uniform ergodic theorems and their applications
Zuohuan Zheng, Jing Xia and Zhiming Zheng
2006, 14(3): 409-417 doi: 10.3934/dcds.2006.14.409 +[Abstract](546) +[PDF](223.7KB)
Abstract:
It has been established one-side uniform convergence in both the Birkhoff and sub-additive ergodic theorems under conditions on growth rates with respect to all the invariant measures. In this paper we show these conditions are both necessary and sufficient. These results are applied to study quasiperiodically forced systems. Some meaningful geometric properties of invariant sets of such systems are presented. We also show that any strange compact invariant set of a $\mathcal{C}^1$ quasiperiodically forced system must support an invariant measure with a non-negative normal Lyapunov exponent.
Unstable manifolds and Hölder structures associated with noninvertible maps
Eugen Mihailescu
2006, 14(3): 419-446 doi: 10.3934/dcds.2006.14.419 +[Abstract](555) +[PDF](392.6KB)
Abstract:
We study the case of a smooth noninvertible map $f$ with Axiom A, in higher dimension. In this paper, we look first at the unstable dimension (i.e the Hausdorff dimension of the intersection between local unstable manifolds and a basic set $\Lambda$), and prove that it is given by the zero of the pressure function of the unstable potential, considered on the natural extension $\hat\Lambda$ of the basic set $\Lambda$; as a consequence, the unstable dimension is independent of the prehistory $\hat x$. Then we take a closer look at the theorem of construction for the local unstable manifolds of a perturbation $g$ of $f$, and for the conjugacy $\Phi_g$ defined on $\hat \Lambda$. If the map $g$ is holomorphic, one can prove some special estimates of the Hölder exponent of $\Phi_g$ on the liftings of the local unstable manifolds. In this way we obtain a new estimate of the speed of convergence of the unstable dimension of $g$, when $g \rightarrow f$. Afterwards we prove the real analyticity of the unstable dimension when the map $f$ depends on a real analytic parameter. In the end we show that there exist Gibbs measures on the intersections between local unstable manifolds and basic sets, and that they are in fact geometric measures; using this, the unstable dimension turns out to be equal to the upper box dimension. We notice also that in the noninvertible case, the Hausdorff dimension of basic sets does not vary continuously with respect to the perturbation $g$ of $f$. In the case of noninvertible Axiom A maps on $\mathbb P^2$, there can exist an infinite number of local unstable manifolds passing through the same point $x$ of the basic set $\Lambda$, thus there is no unstable lamination. Therefore many of the methods used in the case of diffeomorphisms break down and new phenomena and methods of proof must appear. The results in this paper answer to some questions of Urbanski ([21]) about the extension of one dimensional theory of Hausdorff dimension of fractals to the higher dimensional case. They also improve some results and estimates from [7].
Fundamental semigroups for dynamical systems
Fritz Colonius and Marco Spadini
2006, 14(3): 447-463 doi: 10.3934/dcds.2006.14.447 +[Abstract](537) +[PDF](285.1KB)
Abstract:
Algebraic semigroups describing the dynamic behavior are associated to compact, locally maximal chain transitive subsets. The construction is based on perturbations and associated local control sets. The dependence on the perturbation structure is analyzed.
On $C^1$-persistently expansive homoclinic classes
Martín Sambarino and José L. Vieitez
2006, 14(3): 465-481 doi: 10.3934/dcds.2006.14.465 +[Abstract](514) +[PDF](315.6KB)
Abstract:
Let $f: M \to M$ be a diffeomorphism defined in a $d$-dimensional compact boundary-less manifold $M$. We prove that $C^1$-persistently expansive homoclinic classes $H(p)$, $p$ an $f$-hyperbolic periodic point, have a dominated splitting $E\oplus F$, $\dim(E)=\mbox{index}(p)$. Moreover, we prove that if the $H(p)$-germ of $f$ is expansive (in particular if $H(p)$ is an attractor, repeller or maximal invariant) then it is hyperbolic.
Optimal fusion of sensor data for Kalman filtering
Z. G. Feng, Kok Lay Teo, N. U. Ahmed, Yulin Zhao and W. Y. Yan
2006, 14(3): 483-503 doi: 10.3934/dcds.2006.14.483 +[Abstract](519) +[PDF](199.8KB)
Abstract:
In this paper we consider the question of optimal fusion of sensor data for Kalman filtering. The basic problem is to design a linear filter whose output provides an unbiased minimum variance estimate of a signal process whose noisy measurements from multiple sensors are available for input to the filter. The problem is to assign weights to each of the sources (sensor data) dynamically so as to minimize estimation errors. We formulate the problem as an optimal control problem where the weight given to each of the sensor data is considered as one of the control variables satisfying certain constraints. There are as many controls as there are sensors. Using the control parametrization enhancing transform technique (CPET), we develop an efficient method for determining the optimal fusion strategy. Some numerical results are presented for illustration.
A convergent numerical scheme for the Camassa--Holm equation based on multipeakons
Helge Holden and Xavier Raynaud
2006, 14(3): 505-523 doi: 10.3934/dcds.2006.14.505 +[Abstract](592) +[PDF](377.1KB)
Abstract:
The Camassa--Holm equation $u_t$$-$uxxt+3u$u_x-2u_x$uxx-uuxxx=0 enjoys special solutions of the form $u(x,t)=$Σi=1n$p_i(t)e^{-|x-q_i(t)|}$, denoted multipeakons, that interact in a way similar to that of solitons. We show that given initial data $u|_{t=0}=u_0$ in $H^1$(R) such that u-uxx is a positive Radon measure, one can construct a sequence of multipeakons that converges in Lloc(R, Hloc1(R)) to the unique global solution of the Camassa--Holm equation. The approach also provides a convergent, energy preserving nondissipative numerical method which is illustrated on several examples.
Decay rate of higher order derivatives for solutions to the 2-D dissipative quasi-geostrophic flows
Yong Zhou
2006, 14(3): 525-532 doi: 10.3934/dcds.2006.14.525 +[Abstract](595) +[PDF](203.5KB)
Abstract:
In this paper we derive a decay rate of higher order derivatives for solutions to the 2-D dissipative quasi-geostrophic flows under the condition that the $L^2$-norm itself decays. Moreover, under an additional assumption that the solution stays sufficiently close to that of the corresponding linear equation, then both lower bounds and upper bounds on the decay of higher derivatives are obtained.
One-dimensional attractor for a dissipative system with a cylindrical phase space
Rogério Martins
2006, 14(3): 533-547 doi: 10.3934/dcds.2006.14.533 +[Abstract](586) +[PDF](267.4KB)
Abstract:
Consider an attractor of a dissipative non-autonomous system with one angle coordinate. We give conditions for this attractor to be homeomorphic to the circle where we find connections with the work of R. A. Smith. Several applications are studied, such as: the forced pendulum, discretizations of the sine-Gordon equation, n'th order equations, among others.
Global attractivity, I/O monotone small-gain theorems, and biological delay systems
G. A. Enciso and E. D. Sontag
2006, 14(3): 549-578 doi: 10.3934/dcds.2006.14.549 +[Abstract](491) +[PDF](407.3KB)
Abstract:
This paper further develops a method, originally introduced by Angeli and the second author, for proving global attractivity of steady states in certain classes of dynamical systems. In this approach, one views the given system as a negative feedback loop of a monotone controlled system. An auxiliary discrete system, whose global attractivity implies that of the original system, plays a key role in the theory, which is presented in a general Banach space setting. Applications are given to delay systems, as well as to systems with multiple inputs and outputs, and the question of expressing a given system in the required negative feedback form is addressed.
Time-periodic solutions of the Boltzmann equation
Seiji Ukai
2006, 14(3): 579-596 doi: 10.3934/dcds.2006.14.579 +[Abstract](666) +[PDF](305.3KB)
Abstract:
The Boltzmann equation with a time-periodic inhomogeneous term is solved on the existence of a time-periodic solution that is close to an absolute Maxwellian and has the same period as the inhomogeneous term, under some smallness assumption on the inhomogeneous term and for the spatial dimension $n\ge 5$, and also for the case $n=3$ and $ 4$ with an additional assumption that the spatial integral of the macroscopic component of the inhomogeneous term vanishes. This solution is a unique time-periodic solution near the relevant Maxwellian and asymptotically stable in time. Similar results are established also with the space-periodic boundary condition. As a special case, our results cover the case where the inhomogeneous term is time-independent, proving the unique existence and asymptotic stability of stationary solutions. The proof is based on a combination of the contraction mapping principle and time-decay estimates of solutions to the linearized Boltzmann equation.
Minimum 'energy' approximations of invariant measures for nonsingular transformations
Christopher Bose and Rua Murray
2006, 14(3): 597-615 doi: 10.3934/dcds.2006.14.597 +[Abstract](540) +[PDF](317.0KB)
Abstract:
We study variational methods for rigorous approximation of invariant densities for a nonsingular map $T$ on a Borel measure space. The general method takes the form of a convergent sequence of optimization problems on $L^p$, $1 \leq p < \infty$ with a convex objective and finite moment constraints. Provided $T$ admits an invariant density in the appropriate $L^p$ space, weak convergence of the sequence of optimal solutions is observed; norm convergence can be obtained when the objective is a Kadec functional. No regularity or expansiveness assumptions on $T$ need to be made, and the method applies to maps on multidimensional domains. Objectives leading to norm convergence include Entropy, 'Energy' and 'Positively Constrained Energy'.
   Explicit solutions for the finite moment problems in the case of the 'Energy' functional are derived using duality - the optimality condition is then a linear algebra problem. Strong duality is obtained even though the dual functional may not be coercive and the set of moment test functions is not assumed to be pseudo-Haar. Finally, some numerical studies are presented for the case of moment test functions derived from a finite partition of the dynamical phase space and the results are compared with Ulam's method.

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