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Discrete & Continuous Dynamical Systems - B
July 2011 , Volume 16 , Issue 1
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2011, 16(1): 1-14
doi: 10.3934/dcdsb.2011.16.1
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Abstract:
The CEV model is given by the stochastic differential equation $X_t=X_0+\int_0^t\mu X_s ds+\int_0^t\sigma (X^+_s)^p dW_s$, $\frac{1}{2}\le p<1$. It features a non-Lipschitz diffusion coefficient and gets absorbed at zero with a positive probability. We show the weak convergence of Euler-Maruyama approximations $X_t^n$ to the process $X_t$, $0 \le t \le T$, in the Skorokhod metric, by giving a new approximation by continuous processes. We calculate ruin probabilities as an example of such approximation. The ruin probability evaluated by simulations is not guaranteed to converge to the theoretical one, because the limiting distribution is discontinuous at zero. To approximate the size of the jump at zero we use the Levy metric, and also confirm the convergence numerically.
The CEV model is given by the stochastic differential equation $X_t=X_0+\int_0^t\mu X_s ds+\int_0^t\sigma (X^+_s)^p dW_s$, $\frac{1}{2}\le p<1$. It features a non-Lipschitz diffusion coefficient and gets absorbed at zero with a positive probability. We show the weak convergence of Euler-Maruyama approximations $X_t^n$ to the process $X_t$, $0 \le t \le T$, in the Skorokhod metric, by giving a new approximation by continuous processes. We calculate ruin probabilities as an example of such approximation. The ruin probability evaluated by simulations is not guaranteed to converge to the theoretical one, because the limiting distribution is discontinuous at zero. To approximate the size of the jump at zero we use the Levy metric, and also confirm the convergence numerically.
2011, 16(1): 15-29
doi: 10.3934/dcdsb.2011.16.15
+[Abstract](1341)
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Abstract:
We investigate the singular limit, as $\varepsilon \to 0$, of the Fisher equation $\partial _t u=\varepsilon \Delta u + \varepsilon ^{-1}u(1-u)$ in the whole space. We consider initial data with compact support plus perturbations with slow exponential decay, therefore completing the analysis presented in [1]. We prove that the sharp interface limit moves with a constant speed depending dramatically on the tails of the initial data. We make a thorough analysis of both the generation and motion of interface, thus providing a new estimate of the thickness of the transition layers.
We investigate the singular limit, as $\varepsilon \to 0$, of the Fisher equation $\partial _t u=\varepsilon \Delta u + \varepsilon ^{-1}u(1-u)$ in the whole space. We consider initial data with compact support plus perturbations with slow exponential decay, therefore completing the analysis presented in [1]. We prove that the sharp interface limit moves with a constant speed depending dramatically on the tails of the initial data. We make a thorough analysis of both the generation and motion of interface, thus providing a new estimate of the thickness of the transition layers.
2011, 16(1): 31-55
doi: 10.3934/dcdsb.2011.16.31
+[Abstract](1454)
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Abstract:
We consider a generalized Stochastic Cahn-Hilliard equation with multiplicative white noise posed on bounded convex domains in $R^d$, $d=1,2,3$, with piece-wise smooth boundary, and introduce an additive time dependent white noise term in the chemical potential. Since the Green's function of the problem is induced by a convolution semigroup, we present the equation in a weak stochastic integral formulation and prove existence of solution when $d\leq 2$ for general domains, and for $d=3$ for domains with minimum eigenfunction growth, without making use of any explicit expression of the spectrum and the eigenfunctions. The analysis is based on stochastic integral calculus, Galerkin approximations and the asymptotic spectral properties of the Neumann Laplacian operator. Existence is also derived for some non-convex cases when the boundary is smooth.
We consider a generalized Stochastic Cahn-Hilliard equation with multiplicative white noise posed on bounded convex domains in $R^d$, $d=1,2,3$, with piece-wise smooth boundary, and introduce an additive time dependent white noise term in the chemical potential. Since the Green's function of the problem is induced by a convolution semigroup, we present the equation in a weak stochastic integral formulation and prove existence of solution when $d\leq 2$ for general domains, and for $d=3$ for domains with minimum eigenfunction growth, without making use of any explicit expression of the spectrum and the eigenfunctions. The analysis is based on stochastic integral calculus, Galerkin approximations and the asymptotic spectral properties of the Neumann Laplacian operator. Existence is also derived for some non-convex cases when the boundary is smooth.
2011, 16(1): 57-71
doi: 10.3934/dcdsb.2011.16.57
+[Abstract](1652)
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Abstract:
We consider a bistable integral equation which governs the stationary solutions of a convolution model of solid--solid phase transitions on a circle. We study the bifurcations of the set of the stationary solutions as the diffusion coefficient is increased to examine the transition from an uncountably infinite number of steady states to three for the continuum limit of the semi--discretised system. We show how the symmetry of the problem is responsible for the generation and stabilisation of equilibria and comment on the puzzling connection between continuity and stability that exists in this problem.
We consider a bistable integral equation which governs the stationary solutions of a convolution model of solid--solid phase transitions on a circle. We study the bifurcations of the set of the stationary solutions as the diffusion coefficient is increased to examine the transition from an uncountably infinite number of steady states to three for the continuum limit of the semi--discretised system. We show how the symmetry of the problem is responsible for the generation and stabilisation of equilibria and comment on the puzzling connection between continuity and stability that exists in this problem.
2011, 16(1): 73-107
doi: 10.3934/dcdsb.2011.16.73
+[Abstract](1534)
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Abstract:
Observational and model based studies provide ample evidence for the presence of multidecadal variability in the North Atlantic sea-surface temperature known as the Atlantic Multidecadal Oscillation (AMO). This variability is characterised by a multidecadal time scale, a westward propagation of temperature anomalies, and a phase difference between the anomalous meridional and zonal overturning circulations.
We study the AMO in a low-order model obtained by projecting a model for thermally driven ocean flows onto a 27-dimensional function space. We study bifurcations of attractors by varying the equator-to-pole temperature gradient ($\Delta T$) and a damping parameter ($\gamma$).
For $\Delta T = 20^\circ$C and $\gamma = 0$ the low-order model has a stable equilibrium corresponding to a steady ocean flow. By increasing $\gamma$ to 1 a supercritical Hopf bifurcation gives birth to a periodic attractor with the spatio-temporal signature of the AMO. Through a period doubling cascade this periodic orbit gives birth to Hénon-like strange attractors. Finally, we study the effects of annual modulation by introducing a time-periodic forcing. Then the AMO appears through a Hopf-Neĭmark-Sacker bifurcation. For $\Delta T = 24^\circ$C we detected at least 11 quasi-periodic doublings of the invariant torus. After these doublings we find quasi-periodic Hénon-like strange attractors.
Observational and model based studies provide ample evidence for the presence of multidecadal variability in the North Atlantic sea-surface temperature known as the Atlantic Multidecadal Oscillation (AMO). This variability is characterised by a multidecadal time scale, a westward propagation of temperature anomalies, and a phase difference between the anomalous meridional and zonal overturning circulations.
We study the AMO in a low-order model obtained by projecting a model for thermally driven ocean flows onto a 27-dimensional function space. We study bifurcations of attractors by varying the equator-to-pole temperature gradient ($\Delta T$) and a damping parameter ($\gamma$).
For $\Delta T = 20^\circ$C and $\gamma = 0$ the low-order model has a stable equilibrium corresponding to a steady ocean flow. By increasing $\gamma$ to 1 a supercritical Hopf bifurcation gives birth to a periodic attractor with the spatio-temporal signature of the AMO. Through a period doubling cascade this periodic orbit gives birth to Hénon-like strange attractors. Finally, we study the effects of annual modulation by introducing a time-periodic forcing. Then the AMO appears through a Hopf-Neĭmark-Sacker bifurcation. For $\Delta T = 24^\circ$C we detected at least 11 quasi-periodic doublings of the invariant torus. After these doublings we find quasi-periodic Hénon-like strange attractors.
2011, 16(1): 109-150
doi: 10.3934/dcdsb.2011.16.109
+[Abstract](1611)
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Abstract:
We consider a two-group asset flow model of a financial instrument with one group focused on price trend, the other on value. We prove the existence of both stable and unstable regions for the system of differential equations and show that a strong motivation based on (particularly recent) price trend is associated with instability. Numerical computations using a set of typical parameters describe precise regions of stability and instability. A precise limiting connection between the discrete and differential equations is also established.
We consider a two-group asset flow model of a financial instrument with one group focused on price trend, the other on value. We prove the existence of both stable and unstable regions for the system of differential equations and show that a strong motivation based on (particularly recent) price trend is associated with instability. Numerical computations using a set of typical parameters describe precise regions of stability and instability. A precise limiting connection between the discrete and differential equations is also established.
2011, 16(1): 151-171
doi: 10.3934/dcdsb.2011.16.151
+[Abstract](1228)
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Abstract:
We present analytical and numerical investigations of the dynamics of the dissipative standard map. We first study the existence of periodic orbits by using a constructive version of the implicit function theorem; then, we introduce a parametric representation, which provides the interval of the drift parameter ensuring the existence of a periodic orbit with a given period. The determination of quasi--periodic attractors is efficiently obtained using the parametric representation combined with a Newton's procedure, aimed to reduce the error of the approximate solution provided by the parametric representation. These methods allow us to relate the drift parameter of the periodic orbits to that of the invariant attractors, as well as to constrain the drift of a periodic orbit within Arnold's tongues in the parameter space.
We present analytical and numerical investigations of the dynamics of the dissipative standard map. We first study the existence of periodic orbits by using a constructive version of the implicit function theorem; then, we introduce a parametric representation, which provides the interval of the drift parameter ensuring the existence of a periodic orbit with a given period. The determination of quasi--periodic attractors is efficiently obtained using the parametric representation combined with a Newton's procedure, aimed to reduce the error of the approximate solution provided by the parametric representation. These methods allow us to relate the drift parameter of the periodic orbits to that of the invariant attractors, as well as to constrain the drift of a periodic orbit within Arnold's tongues in the parameter space.
2011, 16(1): 173-188
doi: 10.3934/dcdsb.2011.16.173
+[Abstract](1371)
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Abstract:
In this article, we present a dynamic phase transition and stability analysis for the Cahn-Hilliard equations in cylindrical geometry. Two types of phase transitions (the continuous type and the jump type) are determined explicitly in terms of relevant physical and geometric parameters.
In this article, we present a dynamic phase transition and stability analysis for the Cahn-Hilliard equations in cylindrical geometry. Two types of phase transitions (the continuous type and the jump type) are determined explicitly in terms of relevant physical and geometric parameters.
2011, 16(1): 189-196
doi: 10.3934/dcdsb.2011.16.189
+[Abstract](1514)
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Abstract:
In this paper we consider a reaction-diffusion system which describes the acidic nitrate-ferroin reaction. We first show that there exists a minimum speed travelling wave solution. Then some estimates of the minimum speed(s) are derived. Finally, we find that the set of admissible wave speed is $[c_{m i n},\infty)$ under certain condition.
In this paper we consider a reaction-diffusion system which describes the acidic nitrate-ferroin reaction. We first show that there exists a minimum speed travelling wave solution. Then some estimates of the minimum speed(s) are derived. Finally, we find that the set of admissible wave speed is $[c_{m i n},\infty)$ under certain condition.
2011, 16(1): 197-224
doi: 10.3934/dcdsb.2011.16.197
+[Abstract](1163)
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Abstract:
In this article, we consider the dryland vegetation model proposed by Gilad et al [6, 7]. This model consists of three nonlinear parabolic partial differential equations, one of which is degenerate parabolic of the family of the porous media equation [3, 7], and we prove the existence of its weak solutions. Our approach based on the classical Galerkin methods combines and makes use of techniques, parabolic regularization, truncation, maximum principle, compactness. We observe in this way various properties and regularity results of the solutions.
In this article, we consider the dryland vegetation model proposed by Gilad et al [6, 7]. This model consists of three nonlinear parabolic partial differential equations, one of which is degenerate parabolic of the family of the porous media equation [3, 7], and we prove the existence of its weak solutions. Our approach based on the classical Galerkin methods combines and makes use of techniques, parabolic regularization, truncation, maximum principle, compactness. We observe in this way various properties and regularity results of the solutions.
2011, 16(1): 225-238
doi: 10.3934/dcdsb.2011.16.225
+[Abstract](1397)
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Abstract:
In this paper we consider a three dimensional Navier-Stokes type equations with delay terms. We discuss the existence of weak and strong solutions and we study the asymptotic behavior of the strong solutions. Moreover, under suitable assumptions, we show the exponential stability of stationary solutions.
In this paper we consider a three dimensional Navier-Stokes type equations with delay terms. We discuss the existence of weak and strong solutions and we study the asymptotic behavior of the strong solutions. Moreover, under suitable assumptions, we show the exponential stability of stationary solutions.
2011, 16(1): 239-263
doi: 10.3934/dcdsb.2011.16.239
+[Abstract](1510)
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Abstract:
With a view to stabilization issues of hybrid systems exhibiting a regular structure in terms of symmetry, we introduce the concept of symmetry switching and relate symmetry-induced switching strategies to the asymptotic stability of switched linear systems. To this end, a general notion of hybrid symmetries for switched systems is established whereupon orbital switching is treated which builds on the existence of hybrid symmetries. In the main part, we formulate and prove sufficient conditions for asymptotic stability under slow symmetry switching. As an example of both theoretical and practical interest, we examine time-varying networks of dynamical systems and perform stabilization by means of orbital switching. Behind all that, this work is meant to provide the groundwork for the treatment of equivariant bifurcation phenomena of hybrid systems.
With a view to stabilization issues of hybrid systems exhibiting a regular structure in terms of symmetry, we introduce the concept of symmetry switching and relate symmetry-induced switching strategies to the asymptotic stability of switched linear systems. To this end, a general notion of hybrid symmetries for switched systems is established whereupon orbital switching is treated which builds on the existence of hybrid symmetries. In the main part, we formulate and prove sufficient conditions for asymptotic stability under slow symmetry switching. As an example of both theoretical and practical interest, we examine time-varying networks of dynamical systems and perform stabilization by means of orbital switching. Behind all that, this work is meant to provide the groundwork for the treatment of equivariant bifurcation phenomena of hybrid systems.
2011, 16(1): 265-282
doi: 10.3934/dcdsb.2011.16.265
+[Abstract](1174)
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Abstract:
We consider a class of nonlinearly hyperelastic rods with helical symmetry, cf. [7]. Such a rod is mechanically invariant under the symmetries of a circular-cylindrical helix. Examples include idealized DNA molecules, wire ropes and cables. We examine the limit as the pitch of the helix characterizing the symmetry approaches zero and show that the resulting model is a hemitropic rod. The former is mechanically invariant under all proper rotations about its centerline and generally possesses chirality or handedness in its mechanical response, cf. [7]. An isotropic rod is also rotationally invariant but, in addition, enjoys certain reflection symmetries, which rule out chirality. Isotropy implies hemitropy, but the converse is not generally true. We employ both averaging methods and methods of gamma convergence to obtain the effective or homogenized (hemitropic) problem, the latter not corresponding to a naïve average.
We consider a class of nonlinearly hyperelastic rods with helical symmetry, cf. [7]. Such a rod is mechanically invariant under the symmetries of a circular-cylindrical helix. Examples include idealized DNA molecules, wire ropes and cables. We examine the limit as the pitch of the helix characterizing the symmetry approaches zero and show that the resulting model is a hemitropic rod. The former is mechanically invariant under all proper rotations about its centerline and generally possesses chirality or handedness in its mechanical response, cf. [7]. An isotropic rod is also rotationally invariant but, in addition, enjoys certain reflection symmetries, which rule out chirality. Isotropy implies hemitropy, but the converse is not generally true. We employ both averaging methods and methods of gamma convergence to obtain the effective or homogenized (hemitropic) problem, the latter not corresponding to a naïve average.
2011, 16(1): 283-317
doi: 10.3934/dcdsb.2011.16.283
+[Abstract](1341)
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Abstract:
In this work we study the problem of step size selection for numerical schemes, which guarantees that the numerical solution presents the same qualitative behavior as the original system of ordinary differential equations. We apply tools from nonlinear control theory, specifically Lyapunov function and small-gain based feedback stabilization methods for systems with a globally asymptotically stable equilibrium point. Proceeding this way, we derive conditions under which the step size selection problem is solvable (including a nonlinear generalization of the well-known A-stability property for the implicit Euler scheme) as well as step size selection strategies for several applications.
In this work we study the problem of step size selection for numerical schemes, which guarantees that the numerical solution presents the same qualitative behavior as the original system of ordinary differential equations. We apply tools from nonlinear control theory, specifically Lyapunov function and small-gain based feedback stabilization methods for systems with a globally asymptotically stable equilibrium point. Proceeding this way, we derive conditions under which the step size selection problem is solvable (including a nonlinear generalization of the well-known A-stability property for the implicit Euler scheme) as well as step size selection strategies for several applications.
2011, 16(1): 319-332
doi: 10.3934/dcdsb.2011.16.319
+[Abstract](1344)
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Abstract:
It is known that the Jacobian of the discrete-time map of an impact oscillator in the neighborhood of a grazing orbit depends on the square-root of the distance the mass would have gone beyond the position of the wall if the wall were not there. This results in an infinite stretching of the phase space, known as the square-root singularity. In this paper we look closer into the Jacobian matrix and find out the behavior of its two parameters---the trace and the determinant, across the grazing event. We show that the determinant of the matrix remains invariant in the neighborhood of a grazing orbit, and that the singularity appears only in the trace of the matrix. Investigating the character of the trace, we show that the singularity disappears if the damped frequency of the oscillator is an integral multiple of half of the forcing frequency.
It is known that the Jacobian of the discrete-time map of an impact oscillator in the neighborhood of a grazing orbit depends on the square-root of the distance the mass would have gone beyond the position of the wall if the wall were not there. This results in an infinite stretching of the phase space, known as the square-root singularity. In this paper we look closer into the Jacobian matrix and find out the behavior of its two parameters---the trace and the determinant, across the grazing event. We show that the determinant of the matrix remains invariant in the neighborhood of a grazing orbit, and that the singularity appears only in the trace of the matrix. Investigating the character of the trace, we show that the singularity disappears if the damped frequency of the oscillator is an integral multiple of half of the forcing frequency.
2011, 16(1): 333-344
doi: 10.3934/dcdsb.2011.16.333
+[Abstract](1425)
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Abstract:
This paper aims at an initial-boundary value problem on bounded domains for a one-dimensional quasilinear hyperbolic model of blood flow with viscous damping. It is shown that, for given smooth initial data close to a constant equilibrium state, there exists a unique global smooth solution to the model. Time asymptotically, it is shown that the solution converges to the constant equilibrium state exponentially fast as time goes to infinity due to viscous damping and boundary effects.
This paper aims at an initial-boundary value problem on bounded domains for a one-dimensional quasilinear hyperbolic model of blood flow with viscous damping. It is shown that, for given smooth initial data close to a constant equilibrium state, there exists a unique global smooth solution to the model. Time asymptotically, it is shown that the solution converges to the constant equilibrium state exponentially fast as time goes to infinity due to viscous damping and boundary effects.
2011, 16(1): 345-360
doi: 10.3934/dcdsb.2011.16.345
+[Abstract](1481)
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Abstract:
In this paper, we study a two- and three-dimensional bipolar Euler-Poisson system (hydrodynamic model). The system arises in mathematical modeling for semiconductors and plasmas. We are interested in the steady state isentropic case supplemented by the proper boundary conditions. We first show the existence and uniqueness of irrotational subsonic stationary solutions for the two- and three-dimensional hydrodynamic model. Next, we investigate the zero-electron-mass limit, the zero-relaxation-time limit and the Debye-length (quasi-neutral) limit for above stationary solutions, respectively. For each limit, we show the strong convergence of the sequence of solutions and give the associated convergence rates.
In this paper, we study a two- and three-dimensional bipolar Euler-Poisson system (hydrodynamic model). The system arises in mathematical modeling for semiconductors and plasmas. We are interested in the steady state isentropic case supplemented by the proper boundary conditions. We first show the existence and uniqueness of irrotational subsonic stationary solutions for the two- and three-dimensional hydrodynamic model. Next, we investigate the zero-electron-mass limit, the zero-relaxation-time limit and the Debye-length (quasi-neutral) limit for above stationary solutions, respectively. For each limit, we show the strong convergence of the sequence of solutions and give the associated convergence rates.
2011, 16(1): 361-383
doi: 10.3934/dcdsb.2011.16.361
+[Abstract](1237)
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Abstract:
In this paper we present the derivation of a bilayer shallow water model with rigid lid hypothesis. We start from the incompressible Navier-Stokes equations, we introduce a small parameter $\varepsilon$ which is the ratio between the characteristic height and the characteristic length of the fluids domain. We use a formal asymptotic expansion then we resort to averaging to obtain the model. We also prove the stability of the model, in the following sense, up to a subsequence, every sequence of weak solutions converges to a solution of the model.
In this paper we present the derivation of a bilayer shallow water model with rigid lid hypothesis. We start from the incompressible Navier-Stokes equations, we introduce a small parameter $\varepsilon$ which is the ratio between the characteristic height and the characteristic length of the fluids domain. We use a formal asymptotic expansion then we resort to averaging to obtain the model. We also prove the stability of the model, in the following sense, up to a subsequence, every sequence of weak solutions converges to a solution of the model.
2011, 16(1): 385-392
doi: 10.3934/dcdsb.2011.16.385
+[Abstract](1402)
+[PDF](328.3KB)
Abstract:
An experimental conjecture on the existence of positive periodic solutions for the Brillouin electron beam focusing system $x''+a(1+\cos2t)x=\frac{1}{x}$ for $0 < a < 1$ is proved, using a topological degree theorem by Mawhin.
An experimental conjecture on the existence of positive periodic solutions for the Brillouin electron beam focusing system $x''+a(1+\cos2t)x=\frac{1}{x}$ for $0 < a < 1$ is proved, using a topological degree theorem by Mawhin.
2011, 16(1): 393-408
doi: 10.3934/dcdsb.2011.16.393
+[Abstract](1665)
+[PDF](209.9KB)
Abstract:
In this paper, we consider a generalized complex network possessing general topology, in which the coupling may be nonlinear, time-varying, nonsymmetric and the elements of each node have different time-varying delays. Some criteria on local and global exponential synchronization are derived in form of linear matrix inequalities (LMIs) for the complex network by constructing suitable Lyapunov functionals. Our results show that the obtained sufficient conditions are less conservative than ones in previous publications. Finally, two numerical examples and their simulation results are given to illustrate the effectiveness of the derived results.
In this paper, we consider a generalized complex network possessing general topology, in which the coupling may be nonlinear, time-varying, nonsymmetric and the elements of each node have different time-varying delays. Some criteria on local and global exponential synchronization are derived in form of linear matrix inequalities (LMIs) for the complex network by constructing suitable Lyapunov functionals. Our results show that the obtained sufficient conditions are less conservative than ones in previous publications. Finally, two numerical examples and their simulation results are given to illustrate the effectiveness of the derived results.
2011, 16(1): 409-421
doi: 10.3934/dcdsb.2011.16.409
+[Abstract](1263)
+[PDF](346.9KB)
Abstract:
In this paper, we consider the existence of unbounded solutions and periodic solutions for the perturbed asymmetric oscillator with damping
In this paper, we consider the existence of unbounded solutions and periodic solutions for the perturbed asymmetric oscillator with damping
$x'' + f(x )x' + ax^+ - bx^-$ $+ g(x)=p(t), $
where $x^+ =\max\{x,0\}, x^-$ $=\max\{-x,0\}$, $a$ and $b$ are two positive constants, $f(x)$ is a continuous function and $ p(t)$ is a $2\pi $-periodic continuous function, $g(x)$ is locally Lipschitz continuous and bounded. We discuss the existence of periodic solutions and unbounded solutions under two classes of conditions: the resonance case $\frac{1}{\sqrt{a}}+\frac{1}{\sqrt{b}}\in Q$ and the nonresonance case $\frac{1}{\sqrt{a}}+\frac{1}{\sqrt{b}} \notin Q$. Unlike many existing results in the literature where the function $g(x)$ is required to have asymptotic limits at infinity, our main results here allow $g(x)$ be oscillatory without asymptotic limits.
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