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1531-3492

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### Volume 10, 2008

## Discrete & Continuous Dynamical Systems - B

August 2005 , Volume 5 , Issue 3

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2005, 5(3): 513-528
doi: 10.3934/dcdsb.2005.5.513

*+*[Abstract](961)*+*[PDF](278.7KB)**Abstract:**

We consider a Lagrangian system on the d-dimensional torus, and the associated Hamilton-Jacobi equation. Assuming that the Aubry set of the system consists in a finite number of hyperbolic periodic orbits of the Euler-Lagrange flow, we study the vanishing-viscosity limit, from the viscous equation to the inviscid problem. Under suitable assumptions, we show that solutions of the viscous Hamilton-Jacobi equation converge to a unique solution of the inviscid problem.

2005, 5(3): 529-542
doi: 10.3934/dcdsb.2005.5.529

*+*[Abstract](666)*+*[PDF](255.5KB)**Abstract:**

By a dishonest process we understand a process in which, for some initial data, there occurs an unaccounted for loss of the described quan- tity throughout the evolution. Classical examples are offered by shattering fragmentation, where the total mass is decreasing faster than predicted by the formal conservation laws, or explosive birth-and-death processes which, being formally conservative, suffer from the loss of individuals in the course of evo- lution. In this note we shall show, for these two processes, that if dishonesty occurs for one initial datum, then it must occur for any of them.

2005, 5(3): 543-564
doi: 10.3934/dcdsb.2005.5.543

*+*[Abstract](442)*+*[PDF](356.5KB)**Abstract:**

We examine a simple discrete time Markov model of TCP congestion control, which contains congestion avoidance, fast retransmit and time-out, and we prove that it has a unique invariant measure. If the process is scaled by a factor $\sqrt{p}$, then the invariant measures converge as $p \to 0$, where $p$ is the probability of error in any given data packet. This is the $1/\sqrt{p}$-behavior of TCP throughput.

If the scaled process is transformed to continuous time, we show that it converges to a piecewise linear limit process. The unique invariant measure of the limit process coincides with the limit of the invariant measures above and can be easily computed.

Finally, we examine a slightly more sophisticated way of modelling time-outs.

2005, 5(3): 565-586
doi: 10.3934/dcdsb.2005.5.565

*+*[Abstract](467)*+*[PDF](339.3KB)**Abstract:**

Motivated by entropy estimation from chaotic time series, we pro- vide a comprehensive analysis of hitting times of cylinder sets in the setting of Gibbsian sources. We prove two strong approximation results from which we easily deduce pointwise convergence to entropy, lognormal fluctuations, precise large deviation estimates and an explicit formula for the hitting-time multifractal spectrum. It follows from our analysis that the hitting time of a n-cylinder fluctuates in the same way as the inverse measure of this n-cylinder at 'small scales', but in a different way at 'large scales'. In particular, the Rényi entropy differs from the hitting-time spectrum, contradicting a naive ansatz. This phenomenon was recently numerically observed for return times that are more di±cult to handle theoretically. The results we obtain for return times, though less precise than for hitting times, complete the available ones.

2005, 5(3): 587-598
doi: 10.3934/dcdsb.2005.5.587

*+*[Abstract](511)*+*[PDF](282.1KB)**Abstract:**

We extend the anti-integrability theory of Aubry to non-autonomous twist maps between symplectic spaces to show the shift dynamics can be embedded in a natural way. Examples are given to illustrate that the embedded shift can be a full shift, a subshift of finite type or of infinite type.

2005, 5(3): 599-630
doi: 10.3934/dcdsb.2005.5.599

*+*[Abstract](690)*+*[PDF](393.1KB)**Abstract:**

In this paper we address the following traffic regulation problem: given a junction with some incoming roads and some outgoing ones, is it preferable to regulate the flux via a traffic light or via a traffic circle on which the incoming traffic enters continuously? More precisely, assuming that drivers distribute on outgoing roads according to some known coefficients, our aim is to understand which solution performs better from the point of view of total amount of cars going through the junction.

To deal with this problem we consider a fluid dynamic model for traffic flow on a road network. The model is that proposed in [9] and is applied to the case of crossings with lights and with circles. For the first we consider timing of lights as control and determine the asymptotic fluxes. For the second we extend and complete the model of [9] introducing some right of way parameters. Also in this case we determine the asymptotic behavior.

We then compare the performances of the two solutions. Finally, we can indicate which choice is preferable, depending on traffic level and control necessity, and give indications on how to tune traffic light timing and traffic circle right of way parameters.

2005, 5(3): 631-658
doi: 10.3934/dcdsb.2005.5.631

*+*[Abstract](593)*+*[PDF](452.6KB)**Abstract:**

The aim of this work is to propose an efficient numerical approximation of high frequency pulses propagating in nonlinear dispersive optical media. We consider the nonlinear Maxwell's equations with instantaneous nonlinearity. We first derive a physically and asymptotically equivalent model that is semi-linear. Then, for a large class of semi-linear systems, we describe the solution in terms of profiles. These profiles are solution of a singular equation involving one more variable describing the phase of the solution. We introduce a discretization of this equation using finite differences in space and time and an appropriate Fourier basis (with few elements) for the phase. The main point is that accurate solution of the nonlinear Maxwell equation can be computed with a mesh size of order of the wave length. This approximation is asymptotic-preserving in the sense that a multi-scale expansion can be performed on the discrete solution and the result of this expansion is a discretization of the continuous limit. In order to improve the computational delay, computations are performed in a window moving at the group velocity of the pulse. The second harmonic generation is used as an example to illustrate the proposed methodology. However, the numerical method proposed for this benchmark study can be applied to many other cases of nonlinear optics with high frequency pulses.

2005, 5(3): 659-686
doi: 10.3934/dcdsb.2005.5.659

*+*[Abstract](581)*+*[PDF](531.1KB)**Abstract:**

The widely accepted theory of two-dimensional turbulence predicts a direct downscale enstrophy cascade with an energy spectrum behaving like $k^{-3}$ and an inverse upscale energy cascade with a $k^{-5/3}$ decay. Nevertheless, this theory is in fact an idealization valid only in an infinite domain in the limit of infinite Reynolds numbers, and is almost impossible to reproduce numerically. A more complete theoretical framework for the two-dimensional turbulence has been recently proposed by Tung

*et al*. This theory seems to be more consistent with experimental observations, and numerical simulations than the classical one developed by Kraichnan, Leith and Batchelor.

Multiresolution methods like the wavelet packets or the cosine packets, well known in signal decomposition, can be used for the 2D turbulence analysis. Wavelet or cosine decompositions are more and more used in physical applications and in particular in fluid mechanics. Following the works of M. Farge

*et al*, we present a numerical and qualitative study of a two-dimensional turbulence fluid using these methods. The decompositions allow to separate the fluid in two parts which are analyzed and the corresponding energy spectra are computed. In the first part of this paper, the methods are presented and the numerical results are briefly compared to the theoretical spectra predicted by the both theories. A more detailed study, using only wavelet packets decompositions and based on numerical and experimental data, will be carried out and the results will be reported in the second part of the paper. A tentative of physical interpretation of the different components of the flow will be also proposed.

2005, 5(3): 687-698
doi: 10.3934/dcdsb.2005.5.687

*+*[Abstract](598)*+*[PDF](2258.0KB)**Abstract:**

We provide numerical evidence of global diffusion occurring in slightly perturbed integrable Hamiltonian systems and symplectic maps. We show that even if a system is sufficiently close to be integrable, global diffusion occurs on a set with peculiar topology, the so-called Arnold web, and is qualitatively different from Chirikov diffusion, occurring in more perturbed systems.

2005, 5(3): 699-718
doi: 10.3934/dcdsb.2005.5.699

*+*[Abstract](499)*+*[PDF](301.4KB)**Abstract:**

A discrete-delayed model of plasmid-bearing, plasmid-free organisms competing for a single-limited nutrient in a chemostat is established. Rigorous mathematical analysis of the asymptotic behavior of this model is presented. An interesting method to analyze the local stability of interior equilibrium is developed. The argument is also applicable to a model of plasmid-bearing, plasmid-free organisms competing for two complementary nutrients in a chemostat.

2005, 5(3): 719-734
doi: 10.3934/dcdsb.2005.5.719

*+*[Abstract](650)*+*[PDF](547.2KB)**Abstract:**

For the Fermi-Pasta-Ulam chain, an effective Hamiltonian is constructed, describing the motion of approximate, weakly localized discrete breathers traveling along the chain. The velocity of these moving and localized vibrations can be estimated analytically as the group velocity of the corresponding wave packet. The Peierls-Nabarro barrier is estimated for strongly localized discrete breathers.

2005, 5(3): 735-752
doi: 10.3934/dcdsb.2005.5.735

*+*[Abstract](647)*+*[PDF](522.4KB)**Abstract:**

A logistic population model with a maturation delay stage for adults is investigated. The adult population is related to its previous life stage with a maturation delay $r$, and has a non-linear exponential birth rate $be^{-pr}$ with a birth decay coefficient $p$. As $r$ increases, the unique positive equilibrium solution may experience two stability switchings, that is, from stable to unstable, and then back to stable again. The decay coefficient $p$ can also qualitatively influence the stability property of the system. Hopf bifurcation and the stability of the bifurcating periodic solution are analyzed by means of the center manifold theory and the normal form technique. By applying the integral averaging theory, phase-locked and phase-shifting solutions induced by the external excitation are also investigated and verified by numerical simulations.

2005, 5(3): 753-768
doi: 10.3934/dcdsb.2005.5.753

*+*[Abstract](697)*+*[PDF](252.5KB)**Abstract:**

Our aim in this article is to derive models for nonisothermal phase separation. Starting from the two fundamental laws of thermodynamics, we consider the approach of Gurtin, based on a balance law for microforces, to derive nonisothermal Cahn-Hilliard type equations. These equations extend previous models derived by Alt and Pawłow based on an entropy principle to nonisotropic materials and to systems that are far from equilibrium. We also extend this approach to the Ginzburg-Landau (Allen-Cahn) equation, for which we recover, as particular cases, some models obtained by Frémond with a physically different approach.

2005, 5(3): 769-798
doi: 10.3934/dcdsb.2005.5.769

*+*[Abstract](692)*+*[PDF](819.0KB)**Abstract:**

The Polynomial-Preserving Recovery (PPR) technique is extended to recover continuous gradients from $C^0$ finite element solutions of an arbitrary order in 2D and 3D problems. The stability of the PPR is theoretically investigated in a general framework. In 2D, the stability is established under a simple geometric condition. The numerical experiments demonstrated that the PPR-recovered gradient enjoys superconvergence, and the Zienkiewicz-Zhu error estimator based on the PPR-recovered gradient is asymptotically exact.

2005, 5(3): 799-816
doi: 10.3934/dcdsb.2005.5.799

*+*[Abstract](483)*+*[PDF](362.7KB)**Abstract:**

In this work, our target is to analyze the dynamics around the $1:-1$ resonance which appears when a family of periodic orbits of a real analytic three-degree of freedom Hamiltonian system changes its stability from elliptic to a complex hyperbolic saddle passing through degenerate elliptic. Our analytical approach consists of computing, up to some given arbitrary order, the normal form around that resonant (or

*critical*) periodic orbit.

Hence, dealing with the normal form itself and the differential equations related to it, we derive the generic existence of a two-parameter family of invariant 2D tori which bifurcate from the critical periodic orbit. Moreover, the coefficient of the normal form that determines the stability of the bifurcated tori is identified. This allows us to show the Hopf-like character of the unfolding: elliptic tori unfold "around'' hyperbolic periodic orbits (case of

*direct*bifurcation) while normal hyperbolic tori appear "around'' elliptic periodic orbits (case of

*inverse*bifurcation). Further, the parametrization of the main invariant objects as well as a global description of the dynamics of the normal form are also given.

2005, 5(3): 817-840
doi: 10.3934/dcdsb.2005.5.817

*+*[Abstract](646)*+*[PDF](320.5KB)**Abstract:**

A rigorous normal mode error analysis is carried out for two second-order projection type methods. It is shown that although the two schemes provide second-order accuracy for the velocity in $\L^2$-norm, their accuracies for the velocity in $\H^1$-norm and for the pressure in $L^2$-norm are different, and only the consistent splitting scheme introduced in [6] provides full second-order accuracy for all variable in their natural norms. The advantages and disadvantages of the normal mode analysis vs. the energy method are also elaborated.

2005, 5(3): 841-860
doi: 10.3934/dcdsb.2005.5.841

*+*[Abstract](533)*+*[PDF](721.6KB)**Abstract:**

We analyzed the local dynamics of a three-dimensional Ricker type discrete-time competition model that is analogous to the May-Leonard (M-L) differential equation model. The symmetric discrete M-L model is mentioned by Hofbauer et al. [[7] J. Math. Biol., 25:553--570,1987] as "perhaps one of the most difficult three species problems''. Both of the discrete and the continuous M-L models have similar local dynamics. However, the discrete model is not dynamically consistent with the continuous model. Unlike the continuous M-L model, the discrete Hopf bifurcations (Neimark-Sacker bifurcations) of the discrete M-L model are not degenerate. The continuous M-L model is the limiting case of the discrete model.

2005, 5(3): 861-880
doi: 10.3934/dcdsb.2005.5.861

*+*[Abstract](617)*+*[PDF](212.7KB)**Abstract:**

In this work we study the structure of approximate solutions of a nonautonomous discrete-time control system in a compact metric space $X$ which is determined by a sequence of continuous functions $v_i: X \times X \to R^1$, $i=0,\pm 1,\pm 2,$.... The main result in this paper deals with the turnpike property of optimal control problems. To have this property means that the approximate solutions of the problems are determined mainly by the the sequence $\{v_i\}_{i=-\infty}^{\infty}$, and are essentially independent of the choice of interval and endpoint conditions, except in regions close to the endpoints.

2005, 5(3): 881-897
doi: 10.3934/dcdsb.2005.5.881

*+*[Abstract](633)*+*[PDF](386.3KB)**Abstract:**

This article deals with a two-parameter family of piecewise smooth unimodal maps with one break point. Using superstable cycles and their symbolic representation we describe the structure of the periodicity regions of the 2D bifurcation diagram. Particular attention is paid to the bistability regions corresponding to two coexisting attractors, and to the border-collision bifurcations.

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