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Discrete & Continuous Dynamical Systems - B

2010 , Volume 14 , Issue 3

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Minimum free energy in the frequency domain for a heat conductor with memory
Giovambattista Amendola, Mauro Fabrizio and John Murrough Golden
2010, 14(3): 793-816 doi: 10.3934/dcdsb.2010.14.793 +[Abstract](416) +[PDF](277.5KB)
Abstract:
An explicit expression is given in the frequency domain for the minimum free energy related to a particular state of a linear rigid heat conductor with memory effects in the constitutive equations, using the fact that this quantity coincides with the maximum recoverable work obtainable from that state. The constitutive equations for the internal energy and the heat flux are expressed as linear functionals of the histories of temperature and its gradient, respectively, together with the present value of the latter quantity. Another equivalent expression for the minimum free energy is also deduced and used to derive explicit formulae for a discrete spectrum model.
Existence of pulsating waves in a monostable reaction-diffusion system in solid combustion
Michaël Bages and Patrick Martinez
2010, 14(3): 817-869 doi: 10.3934/dcdsb.2010.14.817 +[Abstract](473) +[PDF](517.0KB)
Abstract:
In this paper, we study the solid combustion system with the monostable nonlinearity $f(T)=T$. Our goal is to prove the existence of pulsating waves.
   First, the specific form of the nonlinearity leads to transform the problem into a scalar reaction diffusion equation with nonusual infinite limits.
   Next, we prove that this scalar equation admits a family of pulsating waves, applying an easy fixed point argument; moreover we precise the asymptotic behaviour of the pulsating waves, developping them in Fourier series and studying the behaviour of its Fourier coefficients.
   Finally, all these informations on the equation let us prove that there exists a family of pulsating waves of the original SHS system, the family of admissible propagation speed being a precised half-line.
Chaos and quasi-periodicity in diffeomorphisms of the solid torus
Henk W. Broer, Carles Simó and Renato Vitolo
2010, 14(3): 871-905 doi: 10.3934/dcdsb.2010.14.871 +[Abstract](617) +[PDF](11395.1KB)
Abstract:
This paper focuses on the parametric abundance and the 'Cantorial' persistence under perturbations of a recently discovered class of strange attractors for diffeomorphisms, the so-called quasi-periodic Hénon-like. Such attractors were first detected in the Poincaré map of a periodically driven model of the atmospheric flow: they were characterised by marked quasi-periodic intermittency and by $\Lambda_1>0,\Lambda_2\approx0$, where $\Lambda_1$ and $\Lambda_2$ are the two largest Lyapunov exponents. It was also conjectured that these attractors coincide with the closure of the unstable manifold of a hyperbolic invariant circle of saddle-type.
    This type of attractor is here investigated in a model map of the solid torus, constructed by a skew coupling of the Hénon family of planar maps with the Arnol$'$d family of circle maps. It is proved that Hénon-like strange attractors occur in certain parameter domains. Numerical evidence is produced, suggesting that quasi-periodic circle attractors and quasi-periodic Hénon-like attractors persist in relatively large subsets of the parameter space. We also discuss two problems in the numerical identification of so-called strange nonchaotic attractors and the persistence of all these classes of attractors under perturbation of the skew product structure.
Evaporation law in kinetic gravitational systems described by simplified Landau models
Pierre Carcaud, Pierre-Henri Chavanis, Mohammed Lemou and Florian Méhats
2010, 14(3): 907-934 doi: 10.3934/dcdsb.2010.14.907 +[Abstract](471) +[PDF](339.8KB)
Abstract:
This paper is devoted to a mathematical and numerical study of a simplified kinetic model for evaporation phenomena in gravitational systems. This is a first step towards a mathematical understanding of more realistic kinetic models in this area. It is well known in the astrophysics literature that the appropriate kinetic model to describe escape (evaporation) from stars clusters is the so-called Vlasov-Landau-Poisson system with vanishing boundary condition at positive microscopic energies. Since collisions between stars and their self-consistent interactions are both taken into account in this model, its mathematical analysis is difficult, and so far not achieved. Here, as a first step, we focus on a simplified framework of this model and make the following assumptions: i) Only homogenous (space-independent) distributions functions are considered, leading to a collisional kinetic model with a vanishing boundary condition in velocity. ii) The interaction potential involved in the Landau collision operator is of Maxwellian type. iii) The escape velocity (or energy) is supposed to be constant. Using these assumptions, we first establish the well-posedness of the associated Cauchy problem. Then, we focus on the long time behavior of the solution and prove that the energy of the system decreases in time as $O(1/\log(t))$ (logarithmic evaporation), with convergence to a Dirac distribution in velocity when time goes to infinity. Finally, a suitable numerical scheme is constructed for this model and some simulations are performed to illustrate the theoretical study.
A gradient flow scheme for nonlinear fourth order equations
Bertram Düring, Daniel Matthes and Josipa Pina Milišić
2010, 14(3): 935-959 doi: 10.3934/dcdsb.2010.14.935 +[Abstract](746) +[PDF](910.1KB)
Abstract:
We propose a method for numerical integration of Wasserstein gradient flows based on the classical minimizing movement scheme. In each time step, the discrete approximation is obtained as the solution of a constrained quadratic minimization problem on a finite-dimensional function space. Our method is applied to the nonlinear fourth-order Derrida-Lebowitz-Speer-Spohn equation, which arises in quantum semiconductor theory. We prove well-posedness of the scheme and derive a priori estimates on the discrete solution. Furthermore, we present numerical results which indicate second-order convergence and unconditional stability of our scheme. Finally, we compare these results to those obtained from different semi- and fully implicit finite difference discretizations.
Period increment cascades in a discontinuous map with square-root singularity
Partha Sharathi Dutta and Soumitro Banerjee
2010, 14(3): 961-976 doi: 10.3934/dcdsb.2010.14.961 +[Abstract](598) +[PDF](683.7KB)
Abstract:
We consider a discontinuous map with square-root singularity, which is relevant to many physical systems. Such maps occur in modeling grazing-sliding bifurcations in switching dynamical systems, or if the Poincaré plane coincides with the switching plane. It is shown that there are notable differences in the bifurcation scenarios between this type of discontinuous map and a continuous map with square-root singularity. We determine the bifurcation structures and the scaling constant analytically. A different kind of period increment is observed, and the possibility of breakdown of period increment cascade is detected. Finally, we show that a system of piecewise smooth ordinary differential equations can exhibit the same type of bifurcation behavior.
On the high order approximation of the centre manifold for ODEs
Ariadna Farrés and Àngel Jorba
2010, 14(3): 977-1000 doi: 10.3934/dcdsb.2010.14.977 +[Abstract](547) +[PDF](1968.2KB)
Abstract:
Many times in dynamical systems one wants to understand the bounded motion around an equilibrium point. From a numerical point of view, we can take arbitrary initial conditions close to the equilibrium points, integrate the trajectories and plot them to have a rough idea of motion. If the dimension of the phase space is high, we can take suitable Poincaré sections and/or projections to visualise the dynamics. Of course, if the linear behaviour around the equilibrium point has an unstable direction, this procedure is useless as the trajectories will escape quickly. We need to get rid, in some way, of the instability of the system.
   Here we focus on equilibrium points whose linear dynamics is a cross product of one hyperbolic directions and several elliptic ones. We will compute a high order approximation of the centre manifold around the equilibrium point and use it to describe the behaviour of the system in an extended neighbourhood of this point. Our approach is based on the graph transform method. To derive an efficient algorithm we use recurrent expressions for the expansion of the non - linear terms on the equations of motion.
   Although this method does not require the system to be Hamiltonian, we have taken a Hamiltonian system as an example. We have compared its efficiency with a more classical approach for this type of systems, the Lie series method. It turns out that in this example the graph transform method is more efficient than the Lie series method. Finally, we have used this high order approximation of the centre manifold to describe the bounded motion of the system around and unstable equilibrium point.
Regular flows of weakly compressible viscoelastic fluids and the incompressible limit
Colette Guillopé, Zaynab Salloum and Raafat Talhouk
2010, 14(3): 1001-1028 doi: 10.3934/dcdsb.2010.14.1001 +[Abstract](638) +[PDF](355.6KB)
Abstract:
We consider compressible viscoelastic fluids satisfying the Oldroyd constitutive law. We prove the local existence (and uniqueness) of flows by a classical fixed point argument. We also prove some global properties of the solutions. In particular, we obtain some a priori estimates which are uniform in the Mach number and prove global existence of weakly compressible fluids flows. We show that weakly compressible flows with well-prepared initial data converge to incompressible ones when the Mach number converges to zero.
A spectral collocation method for solving initial value problems of first order ordinary differential equations
Ben-Yu Guo and Zhong-Qing Wang
2010, 14(3): 1029-1054 doi: 10.3934/dcdsb.2010.14.1029 +[Abstract](537) +[PDF](345.1KB)
Abstract:
We propose a spectral collocation method for solving initial value problems of first order ODEs, based on the Legendre-Gauss-Lobatto interpolation. This method is easy to be implemented and possesses the spectral accuracy. We also develop a multi-step version of this process, which is very available for long-time calculation. Numerical results demonstrate the high accuracy of suggested algorithms and coincide well with the theoretical analysis.
Mathematical analysis of a kinetic model for cell movement in network tissues
Thomas Hillen, Peter Hinow and Zhi-An Wang
2010, 14(3): 1055-1080 doi: 10.3934/dcdsb.2010.14.1055 +[Abstract](605) +[PDF](921.5KB)
Abstract:
Mesenchymal motion describes the movement of cells in biological tissues formed by fibre networks. An important example is the migration of tumour cells through collagen networks during the process of metastasis formation. We investigate the mesenchymal motion model proposed by T. Hillen in [14] in higher dimensions. We formulate the problem as an evolution equation in a Banach space of measure-valued functions and use methods from semigroup theory to show the global existence of mild and classical solutions. We investigate steady states of the model and show that patterns of network type exist as steady states. For the case of constant fibre distribution, we find an explicit solution and we prove the convergence to the parabolic limit.
Analysis of a predator-prey model with predators impulsively diffusing between two patches
Jianjun Jiao and Lansun Chen
2010, 14(3): 1081-1094 doi: 10.3934/dcdsb.2010.14.1081 +[Abstract](562) +[PDF](241.3KB)
Abstract:
In this work, a predator-prey model with predators impulsively diffusing between two patches is investigated. By the stroboscopic map of the discrete dynamical system, the prey-extinction periodic solution of the investigated system is proved to be globally asymptotically stable. By the theory of impulsive differential equation, the investigated system is also proved to be permanent. Finally, the numerical simulation is inserted to illustrate the results.
Global properties of a general predator-prey model with non-symmetric attack and consumption rate
Andrei Korobeinikov
2010, 14(3): 1095-1103 doi: 10.3934/dcdsb.2010.14.1095 +[Abstract](592) +[PDF](145.3KB)
Abstract:
Identification of conditions for stable coexistence of interacting populations is a problem of the highest priority in mathematical biology. This problem is usually considered under specific assumptions made regarding the functional forms of non-linear feedbacks. Apparently, such an approach is lacking generality. In this paper, we consider the dynamics of two species with interaction of predator--prey (consumer-supplier) type with the assumption that a part of the resource is neglected or wasted by the predator (consumer). This model describes, for instance, killing for fun; such behaviour is typical for many predators when the prey is abundant.
   We assume that the functional responses that are usually included in such models are given by unspecified functions. Using the direct Lyapunov method, we derive the conditions which ensure global asymptotic stability of this model. It is remarkable that these conditions impose much weaker constraints on the system properties than that are usually assumed.
Stationary patterns for an adsorbate-induced phase transition model I: Existence
Kousuke Kuto and Tohru Tsujikawa
2010, 14(3): 1105-1117 doi: 10.3934/dcdsb.2010.14.1105 +[Abstract](551) +[PDF](196.2KB)
Abstract:
We are concerned with a reaction-diffusion-advection system proposed by Hildebrand [4]. This system is a phase transition model arising in surface chemistry. For this model, several stationary patterns have been shown by the numerical simulations (e.g., [15]). In the present paper, we obtain sufficient conditions for the existence (or nonexistence) of nonconstant stationary solutions. Our proof is based on the Leray-Schauder degree theory. Some a priori estimates for solutions play an important role in the proof.
On the traveling wave solutions for a nonlinear diffusion-convection equation: Dynamical system approach
Jibin Li and Yi Zhang
2010, 14(3): 1119-1138 doi: 10.3934/dcdsb.2010.14.1119 +[Abstract](499) +[PDF](1306.1KB)
Abstract:
For a general nonlinear diffusion-convection equation, the existence of uncountably infinite many global monotonic wavefront solutions and semi-wavefront solutions with bounded support is proved. By using the method of planar dynamical systems, the dynamical behavior of the corresponding traveling wave system is discussed. For some concrete nonlinear diffusion-convection equations, more than thirty exact explicit parametric representations of the wavefront solutions, semi-wavefront solutions and unbounded traveling wave solutions are given.
An asymptotic analysis of the persistence threshold for the diffusive logistic model in spatial environments with localized patches
Alan E. Lindsay and Michael J. Ward
2010, 14(3): 1139-1179 doi: 10.3934/dcdsb.2010.14.1139 +[Abstract](490) +[PDF](577.4KB)
Abstract:
An indefinite weight eigenvalue problem characterizing the threshold condition for extinction of a population based on the single-species diffusive logistic model in a spatially heterogeneous environment is analyzed in a bounded two-dimensional domain with no-flux boundary conditions. In this eigenvalue problem, the spatial heterogeneity of the environment is reflected in the growth rate function, which is assumed to be concentrated in $n$ small circular disks, or portions of small circular disks, that are contained inside the domain. The constant bulk or background growth rate is assumed to be spatially uniform. The disks, or patches, represent either strongly favorable or strongly unfavorable local habitats. For this class of piecewise constant bang-bang growth rate function, an asymptotic expansion for the persistence threshold λ1, representing the positive principal eigenvalue for this indefinite weight eigenvalue problem, is calculated in the limit of small patch radii by using the method of matched asymptotic expansions. By analytically optimizing the coefficient of the leading-order term in the asymptotic expansion of λ1, general qualitative principles regarding the effect of habitat fragmentation are derived. In certain degenerate situations, it is shown that the optimum spatial arrangement of the favorable habit is determined by a higher-order coefficient in the asymptotic expansion of the persistence threshold.
Existence of transversal homoclinic orbits in higher dimensional discrete dynamical systems
Chen-Chang Peng and Kuan-Ju Chen
2010, 14(3): 1181-1197 doi: 10.3934/dcdsb.2010.14.1181 +[Abstract](418) +[PDF](205.7KB)
Abstract:
A rigorous numerical proof for establishing existence of a transversal homoclinic orbit for a saddle fixed point with higher dimensional unstable eigenspaces is presented. As the first component of this method, a shadowing theorem that guarantees the existence of such a homoclinic orbit near a suitable pseudo orbit given the invertibility of a certain Jacobian is proved. The second component consists of a refinement procedure for numerically computing a pseudo homoclinic orbit with sufficiently small local errors so as to satisfy the hypothesis of the theorem. The third component verifies that the homoclinic orbit is transversal. In [6], they proved the existence of transversal homoclinic orbits near anti-integrable limits and near singularities for the Arneodo-Coullet-Tresser maps. In this paper, the existence of transversal homoclinic orbits were proved far away from anti-integrable limits and singularities for these maps.
A class of doubly degenerate parabolic equations with periodic sources
Jiebao Sun, Boying Wu, Jing Li and Dazhi Zhang
2010, 14(3): 1199-1210 doi: 10.3934/dcdsb.2010.14.1199 +[Abstract](549) +[PDF](189.1KB)
Abstract:
In this paper, we investigate a class of doubly degenerate parabolic equations with periodic sources subject to homogeneous Dirichlet boundary conditions. By means of the theory of Leray-Schauder degree, we establish the existence of non-trivial nonnegative periodic solutions. The key step is how to establish the uniform bound estimate of approximate solutions, for this purpose we will make use of Moser iteration and some results of the eigenvalue problem for the $p$-Laplacian equation.
Anti-shifting phenomenon of a convective nonlinear diffusion equation
Chunpeng Wang, Jingxue Yin and Bibo Lu
2010, 14(3): 1211-1236 doi: 10.3934/dcdsb.2010.14.1211 +[Abstract](614) +[PDF](301.5KB)
Abstract:
This paper concerns a convective nonlinear diffusion equation which is strongly degenerate. The existence and uniqueness of the $BV$ solution to the initial-boundary problem are proved. Then we deal with the anti-shifting phenomenon by investigating the corresponding free boundary problem. As a consequence, it is possible to find a suitable convection such that the discontinuous point of the solution remains unmoved.
Persistence of lower dimensional elliptic invariant tori for a class of nearly integrable reversible systems
Xiaocai Wang, Junxiang Xu and Dongfeng Zhang
2010, 14(3): 1237-1249 doi: 10.3934/dcdsb.2010.14.1237 +[Abstract](499) +[PDF](202.0KB)
Abstract:
In this paper we consider the persistence of lower dimensional elliptic invariant tori with prescribed frequencies in reversible systems, and prove that if the frequency mapping has non-zero Brouwer's degree at a certain point that satisfies Melnikov's non-resonance conditions, then the invariant torus with given frequency persists under small perturbations.
Dynamics of domain wall in thin film driven by spin current
Lei Yang and Xiao-Ping Wang
2010, 14(3): 1251-1263 doi: 10.3934/dcdsb.2010.14.1251 +[Abstract](534) +[PDF](193.7KB)
Abstract:
The dynamics of magnetization under the applied spin current is modeled by the generalized Landau-Lifshitz-Gilbert equation with a spin transfer torque term. Using matched asymptotic expansion with the domain wall thickness $\epsilon$ as the small parameter, we derive analytically the dynamic law for the domain wall motion induced by the spin current. We show that the domain wall driven by adiabatic current spin-transfer torque moves with a decreasing velocity and eventually stops. With a pinning potential, the domain wall motion is a damped oscillation around the pinning site with an intrinsic frequency which is independent of the strength of the current. When the AC current is applied, the dynamic law shows that the frequency of the applied current can be turned to maximize the amplitude of the oscillation. The results obtained are consistent with the recent experimental and numerical results.

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