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Discrete & Continuous Dynamical Systems - B

2005 , Volume 5 , Issue 4

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Upper semicontinuity of the attractor for a second order lattice dynamical system
Ahmed Y. Abdallah
2005, 5(4): 899-916 doi: 10.3934/dcdsb.2005.5.899 +[Abstract](41) +[PDF](278.1KB)
We study the existence and the upper semicontinuity of the global attractor for a second order lattice dynamical system.
Rate distributions and survival of the fittest: a formulation on the space of measures
Azmy S. Ackleh , Ben G. Fitzpatrick and  Horst R. Thieme
2005, 5(4): 917-928 doi: 10.3934/dcdsb.2005.5.917 +[Abstract](62) +[PDF](225.6KB)
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.
Geometric optimal control of elliptic Keplerian orbits
B. Bonnard , J.-B. Caillau and  E. Trélat
2005, 5(4): 929-956 doi: 10.3934/dcdsb.2005.5.929 +[Abstract](45) +[PDF](1194.2KB)
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.
Nonisotropic 3-level quantum systems: complete solutions for minimum time and minimum energy
Ugo Boscain , Thomas Chambrion and  Grégoire Charlot
2005, 5(4): 957-990 doi: 10.3934/dcdsb.2005.5.957 +[Abstract](28) +[PDF](674.5KB)
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.
Local chaotic behaviour in the Fermi-Pasta-Ulam system
Antonio Giorgilli , Simone Paleari and  Tiziano Penati
2005, 5(4): 991-1004 doi: 10.3934/dcdsb.2005.5.991 +[Abstract](36) +[PDF](894.9KB)
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.
Existence of non-trivial nonnegative periodic solutions for a class of degenerate parabolic equations with nonlocal terms
Rui Huang , Yifu Wang and  Yuanyuan Ke
2005, 5(4): 1005-1014 doi: 10.3934/dcdsb.2005.5.1005 +[Abstract](44) +[PDF](214.3KB)
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.
Comparison and convergence to equilibrium in a nonlocal delayed reaction-diffusion model on an infinite domain
Michele V. Bartuccelli , S.A. Gourley and  Y. Kyrychko
2005, 5(4): 1015-1026 doi: 10.3934/dcdsb.2005.5.1015 +[Abstract](46) +[PDF](218.0KB)
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.
Kinks in stripe forming systems under traveling wave forcing
S. Rüdiger , J. Casademunt and  L. Kramer
2005, 5(4): 1027-1042 doi: 10.3934/dcdsb.2005.5.1027 +[Abstract](57) +[PDF](309.3KB)
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.
Asymptotic speed of spread and traveling waves for a nonlocal epidemic model
Dashun Xu and  Xiao-Qiang Zhao
2005, 5(4): 1043-1056 doi: 10.3934/dcdsb.2005.5.1043 +[Abstract](40) +[PDF](451.5KB)
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.
A Markov modulated continuous-time capture-recapture population estimation model
Lakhdar Aggoun and  Lakdere Benkherouf
2005, 5(4): 1057-1075 doi: 10.3934/dcdsb.2005.5.1057 +[Abstract](34) +[PDF](287.2KB)
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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