ISSN:

1531-3492

eISSN:

1553-524X

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### Volume 10, 2008

## Discrete & Continuous Dynamical Systems - B

November 2005 , Volume 5 , Issue 4

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2005, 5(4): 899-916
doi: 10.3934/dcdsb.2005.5.899

*+*[Abstract](647)*+*[PDF](278.1KB)**Abstract:**

We study the existence and the upper semicontinuity of the global attractor for a second order lattice dynamical system.

2005, 5(4): 917-928
doi: 10.3934/dcdsb.2005.5.917

*+*[Abstract](556)*+*[PDF](225.6KB)**Abstract:**

In this paper we address the basic mathematical properties of a general population model having distributed growth and mortality rates. The problem considered generalizes previous efforts [3] in three ways. First, our model involves nonlinear growth and mortality terms. Second, the parameter space is assumed to be any compact subset of (0,∞) x (0, ∞), and third, the solutions of the rate distribution model are constructed in spaces of measures. The latter point is particularly appropriate for the asymptotic behavior, in which survival of the fittest manifests itself as a Dirac delta measure being the attractor of the dynamical system. As opposed to previous approaches to these problems, the measure space formulation allows the (weakly) stable equilibrium to be a point in the state space.

2005, 5(4): 929-956
doi: 10.3934/dcdsb.2005.5.929

*+*[Abstract](711)*+*[PDF](1194.2KB)**Abstract:**

This article deals with the transfer of a satellite between Keplerian orbits. We study the controllability properties of the system and make a preliminary analysis of the time optimal control using the maximum principle. Second order sufficient conditions are also given. Finally, the time optimal trajectory to transfer the system from an initial low orbit with large eccentricity to a terminal geostationary orbit is obtained numerically.

2005, 5(4): 957-990
doi: 10.3934/dcdsb.2005.5.957

*+*[Abstract](644)*+*[PDF](674.5KB)**Abstract:**

We apply techniques of subriemannian geometry on Lie groups and of optimal synthesis on 2-D manifolds to the population transfer problem in a three-level quantum system driven by two laser pulses, of arbitrary shape and frequency. In the rotating wave approximation, we consider a nonisotropic model, i.e., a model in which the two coupling constants of the lasers are different. The aim is to induce transitions from the first to the third level, minimizing 1) the time of the transition (with bounded laser amplitudes), 2) the energy transferred by lasers to the system (with fixed final time). After reducing the problem to real variables, for the purpose 1) we develop a theory of time optimal syntheses for distributional problem on 2-D manifolds, while for the purpose 2) we use techniques of subriemannian geometry on 3-D Lie groups. The complete optimal syntheses are computed.

2005, 5(4): 991-1004
doi: 10.3934/dcdsb.2005.5.991

*+*[Abstract](706)*+*[PDF](894.9KB)**Abstract:**

We discuss the use of the maximal Lyapunov Characteristic Number as a stochasticity indicator in connection with the persistence of the FPU paradox in the thermodynamic limit. We show that the positiveness of the LCN does not imply that the dynamic is ergodic in statistical sense. On the other hand, our numerical exploration suggests that the energy surface may be separated into different chaotic regions that may trap the orbit for a long time. This is compatible with the existence of exponentially long times of relaxation to statistical equilibrium in the sense of Nekhoroshev's theory. Thus, the relevance of the FPU phenomenon for large systems remains a still open problem.

2005, 5(4): 1005-1014
doi: 10.3934/dcdsb.2005.5.1005

*+*[Abstract](714)*+*[PDF](214.3KB)**Abstract:**

In this paper, we establish the existence of non-trivial nonnegative periodic solutions for a class of degenerate parabolic equations with nonlocal terms by using the theory of Leray-Schauder's degree.

2005, 5(4): 1015-1026
doi: 10.3934/dcdsb.2005.5.1015

*+*[Abstract](651)*+*[PDF](218.0KB)**Abstract:**

We study a nonlocal time-delayed reaction-diffusion population model on an infinite one-dimensional spatial domain. Depending on the model parameters, a non-trivial uniform equilibrium state may exist. We prove a comparison theorem for our equation for the case when the birth function is monotone, and then we use this to establish nonlinear stability of the non-trivial uniform equilibrium state when it exists. A certain class of non-monotone birth functions relevant to certain species is also considered, namely birth functions that are increasing at low densities but decreasing at high densities. In this case we prove that solutions still converge to the non-trivial equilibrium, provided the birth function is increasing at the equilibrium level.

2005, 5(4): 1027-1042
doi: 10.3934/dcdsb.2005.5.1027

*+*[Abstract](751)*+*[PDF](309.3KB)**Abstract:**

We study domain walls in stripe forming systems that are externally forced by a periodic pattern, which is close to spatial resonance of 2:1 (the period of the forcing being half of the internal wavelength) and moving relative to the internal pattern. Two transitions are identified: A transition where the pattern lags behind the forcing as the forcing becomes too fast and a spontaneous symmetry-breaking transition of walls (kinks). The departure from perfect resonance is found to render the kink bifurcation imperfect and causes the walls to drift. We study the velocity of the kinks, which behaves strongly nonlinear close to the transitions. A phase approximation is used to understand the behavior and is found to be valid in a large range of parameters. Results from the phase equation can be generalized to hold for different ratios n:1.

2005, 5(4): 1043-1056
doi: 10.3934/dcdsb.2005.5.1043

*+*[Abstract](736)*+*[PDF](451.5KB)**Abstract:**

By applying the theory of asymptotic speeds of spread and traveling waves to a nonlocal epidemic model, we established the existence of minimal wave speed for monotone traveling waves, and show that it coincides with the spreading speed for solutions with initial functions having compact supports. The numerical simulations are also presented.

2005, 5(4): 1057-1075
doi: 10.3934/dcdsb.2005.5.1057

*+*[Abstract](759)*+*[PDF](287.2KB)**Abstract:**

In this paper, we consider a population of animals that moves between different areas according to some Markovian rule. A continuous time capture-recapture sampling technique is used to monitor the distribution of the population between the different areas. Using measure change techniques finite-dimensional filters for the number of animals in each region are derived. Using the EM algorithm the parameters of the model are updated.

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