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Discrete & Continuous Dynamical Systems - B

2001 , Volume 1 , Issue 3

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Variational proof for hard Discrete breathers in some classes of Hamiltonian dynamical systems
S. Aubry , G. Kopidakis and  V. Kadelburg
2001, 1(3): 271-298 doi: 10.3934/dcdsb.2001.1.271 +[Abstract](98) +[PDF](2492.7KB)
We present a new method for proving the existence of Discrete Breathers in translationally invariant Hamiltonian systems describing massive particles interacting by a short range covex potential provided their frequency is above the linear phonon spectrum. The method holds for systems either with optical phonons (with a phonon gap) or with acoustic phonons (without phonon gap but with nonvanishing sound velocities), and does not use the concept of anticontinuous limit as most early methods. Discrete Breathers are obtained as loops in the phase space which maximize a certain average energy function for a fixed pseudoaction appropriately defined. It suffices to exhibit a trial loop with energy larger than the linear phonon energy at the same pseudoaction to prove the existence of a Discrete Breather with a frequency above the linear phonon spectrum. As a straightforward application of the method, Discrete Breathers are proven to exist at any energy (even small) in the quartic (or $\beta$) one-dimensional FPU model, which up to now was lacking a rigorous existence proof. The method can also work for piezoactive DBs in one or more dimensions and in many more complex models.
Accessible points in the Julia sets of stable exponentials
Ranjit Bhattacharjee , Robert L. Devaney , R.E. Lee Deville , Krešimir Josić and  Monica Moreno-Rocha
2001, 1(3): 299-318 doi: 10.3934/dcdsb.2001.1.299 +[Abstract](36) +[PDF](358.9KB)
In this paper we consider the question of accessibility of points in the Julia sets of complex exponential functions in the case where the exponential admits an attracting cycle. In the case of an attracting fixed point it is known that the Julia set is a Cantor bouquet and that the only points accessible from the basin are the endpoints of the bouquet. In case the cycle has period two or greater, there are many more restrictions on which points in the Julia set are accessible. In this paper we give precise conditions for a point to be accessible in the periodic point case in terms of the kneading sequence for the cycle.
On the asymptotic behaviour of the solution of parabolic problems in cylindrical domains of large size in some directions
M. Chipot and  A. Rougirel
2001, 1(3): 319-338 doi: 10.3934/dcdsb.2001.1.319 +[Abstract](48) +[PDF](224.2KB)
We study the asymptotic behaviour of the solution of linear and nonlinear parabolic problems in cylindrical domains becoming unbounded in one or several directions. In particular we show that if the data depend only of the cross section of the domains the solution converges toward the solution of problems set on this cross section. In the applications this allows for instance to reduce the computations to two dimensional cases.
Travelling waves in a nonlinear degenerate diffusion model for bacterial pattern formation
R.A. Satnoianu , Philip K. Maini , F.S. Garduno and  J.P. Armitage
2001, 1(3): 339-362 doi: 10.3934/dcdsb.2001.1.339 +[Abstract](46) +[PDF](270.2KB)
We study a reaction diffusion model recently proposed in [5] to describe the spatiotemporal evolution of the bacterium Bacillus subtilis on agar plates containing nutrient. An interesting mathematical feature of the model, which is a coupled pair of partial differential equations, is that the bacterial density satisfies a degenerate nonlinear diffusion equation. It was shown numerically that this model can exhibit quasi-one-dimensional constant speed travelling wave solutions. We present an analytic study of the existence and uniqueness problem for constant speed travelling wave solutions. We find that such solutions exist only for speeds greater than some threshold speed giving minimum speed waves which have a sharp profile. For speeds greater than this minimum speed the waves are smooth. We also characterise the dependence of the wave profile on the decay of the front of the initial perturbation in bacterial density. An investigation of the partial differential equation problem establishes, via a global existence and uniqueness argument, that these waves are the only long time solutions supported by the problem. Numerical solutions of the partial differential equation problem are presented and they confirm the results of the analysis.
Multi-bump patterns by a normal form approach
Vivi Rottschäfer
2001, 1(3): 363-386 doi: 10.3934/dcdsb.2001.1.363 +[Abstract](37) +[PDF](308.1KB)
In this paper the behaviour of small solutions in a reaction-diffusion model problem is studied near a co-dimension 2 point. The normal form theory for reversible vector fields is applied on the stationary part of the reaction- diffusion system. This normal form is reduced to a 3-dimensional ODE that is completely integrable. An explicit expression for the solutions to the ODE and therefore for the reaction-diffusion system is given under certain conditions. These solutions have the same multi-bump pattern as the asymptotically stable stationary multi-bump solutions that were found in the numerical simulations of the full reaction-diffusion system.
The prime number periodical cicada problem
G.F. Webb
2001, 1(3): 387-399 doi: 10.3934/dcdsb.2001.1.387 +[Abstract](87) +[PDF](243.4KB)
Mathematical models are presented to argue for the significance of prime number emergences of 13 year and 17 year periodical cicadas (Magicicada spp.). The prime number values arise as resonances of emergences with 2 and 3 year quasi-cycling predators. Predators with 2 and 3 year quasi-cycles are present due to their age dependent fecundity and mortality rates. Their quasi-cycles are enhanced by the predation of cicadas during emergences and thus exert significant influence on the cicada periodic life cycles.

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