ISSN:
1531-3492
eISSN:
1553-524X
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Discrete & Continuous Dynamical Systems - B
March 2010 , Volume 13 , Issue 2
Special issue
dedicated to Peter E. Kloeden on the occasion of his 60th birthday
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2010, 13(2): 249-267
doi: 10.3934/dcdsb.2010.13.249
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Abstract:
Several discrete-time dynamic models are ultimately expressed in the form of iterated piecewise linear functions, in one- or two- dimensional spaces. In this paper we study a one-dimensional map made up of three linear pieces which are separated by two discontinuity points, motivated by a dynamic model arising in social sciences. Starting from the bifurcation structure associated with one-dimensional maps with only one discontinuity point, we show how this is modified by the introduction of a second discontinuity point, and we give the analytic expressions of the bifurcation curves of the principal tongues (or tongues of first degree) for the family of maps considered, which depends on five parameters.
Several discrete-time dynamic models are ultimately expressed in the form of iterated piecewise linear functions, in one- or two- dimensional spaces. In this paper we study a one-dimensional map made up of three linear pieces which are separated by two discontinuity points, motivated by a dynamic model arising in social sciences. Starting from the bifurcation structure associated with one-dimensional maps with only one discontinuity point, we show how this is modified by the introduction of a second discontinuity point, and we give the analytic expressions of the bifurcation curves of the principal tongues (or tongues of first degree) for the family of maps considered, which depends on five parameters.
2010, 13(2): 269-278
doi: 10.3934/dcdsb.2010.13.269
+[Abstract](2432)
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Abstract:
This paper considers a Cournot duopoly model assuming isoelastic demand and smooth cost functions with built-in capacity limits. When the firms cannot obtain positive profits they are assumed to choose small "stand-by" outputs rather than closing down, in order to avoid substantial fitting up costs when market conditions turn out more favorable. It is shown that the model provides chaotic behavior. In particular, the system has positive topological entropy and hence the map is chaotic in the Li-Yorke sense. Moreover, chaos is not only topological but also physically observable.
This paper considers a Cournot duopoly model assuming isoelastic demand and smooth cost functions with built-in capacity limits. When the firms cannot obtain positive profits they are assumed to choose small "stand-by" outputs rather than closing down, in order to avoid substantial fitting up costs when market conditions turn out more favorable. It is shown that the model provides chaotic behavior. In particular, the system has positive topological entropy and hence the map is chaotic in the Li-Yorke sense. Moreover, chaos is not only topological but also physically observable.
2010, 13(2): 279-303
doi: 10.3934/dcdsb.2010.13.279
+[Abstract](2737)
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Abstract:
We consider a semi-discrete car-following model and the macroscopic Aw-Rascle model for traffic flow given in Lagrangian form. The solution of the car-following model converges to a weak entropy solution of the system of hyperbolic balance laws with Cauchy initial data. For the homogeneous system, we allow vacuum in the initial data. By using properties of the semi-discrete model, we show that this solution of the hyperbolic system is stable in the $L^1$-norm.
We consider a semi-discrete car-following model and the macroscopic Aw-Rascle model for traffic flow given in Lagrangian form. The solution of the car-following model converges to a weak entropy solution of the system of hyperbolic balance laws with Cauchy initial data. For the homogeneous system, we allow vacuum in the initial data. By using properties of the semi-discrete model, we show that this solution of the hyperbolic system is stable in the $L^1$-norm.
2010, 13(2): 305-325
doi: 10.3934/dcdsb.2010.13.305
+[Abstract](2253)
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Abstract:
A terminal-state tracking optimal control problem for linear hyperbolic equations with distributed control is studied in this paper. An analytic solution formula for the optimal control problem is derived in the form of eigenseries. We show that the optimal solution satisfies the approximate controllability property. An explicit solution formula for the exact controllability problem is also expressed by the eigenseries formula when the target state and the controlled state have matching boundary conditions. We demonstrate by numerical simulations that the optimal solutions expressed by the series formula approach the target functions.
A terminal-state tracking optimal control problem for linear hyperbolic equations with distributed control is studied in this paper. An analytic solution formula for the optimal control problem is derived in the form of eigenseries. We show that the optimal solution satisfies the approximate controllability property. An explicit solution formula for the exact controllability problem is also expressed by the eigenseries formula when the target state and the controlled state have matching boundary conditions. We demonstrate by numerical simulations that the optimal solutions expressed by the series formula approach the target functions.
2010, 13(2): 327-345
doi: 10.3934/dcdsb.2010.13.327
+[Abstract](3440)
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Abstract:
In application areas, such as biology, physics and engineering, delays arise naturally because of the time it takes for the system to react to internal or external events. Often the associated mathematical model features more than one delay that are then weighted by some distribution function. This paper considers the effect of delay distribution on the asymptotic stability of the zero solution of functional differential equations - the corresponding mathematical models. We first show that the asymptotic stability of the zero solution of a first-order scalar equation with symmetrically distributed delays follows from the stability of the corresponding equation where the delay is fixed and given by the mean of the distribution. This result completes a proof of a stability condition in [Bernard, S., Bélair, J. and Mackey, M. C. Sufficient conditions for stability of linear differential equations with distributed delay. Discrete Contin. Dyn. Syst. Ser. B, 1(2):233-256, 2001], which was motivated in turn by an application from biology. We also discuss the corresponding case of second-order scalar delay differential equations, because they arise in physical systems that involve oscillating components. An example shows that it is not possible to give a general result for the second-order case. Namely, the boundaries of the stability regions of the distributed-delay equation and of the mean-delay equation may intersect, even if the distribution is symmetric.
In application areas, such as biology, physics and engineering, delays arise naturally because of the time it takes for the system to react to internal or external events. Often the associated mathematical model features more than one delay that are then weighted by some distribution function. This paper considers the effect of delay distribution on the asymptotic stability of the zero solution of functional differential equations - the corresponding mathematical models. We first show that the asymptotic stability of the zero solution of a first-order scalar equation with symmetrically distributed delays follows from the stability of the corresponding equation where the delay is fixed and given by the mean of the distribution. This result completes a proof of a stability condition in [Bernard, S., Bélair, J. and Mackey, M. C. Sufficient conditions for stability of linear differential equations with distributed delay. Discrete Contin. Dyn. Syst. Ser. B, 1(2):233-256, 2001], which was motivated in turn by an application from biology. We also discuss the corresponding case of second-order scalar delay differential equations, because they arise in physical systems that involve oscillating components. An example shows that it is not possible to give a general result for the second-order case. Namely, the boundaries of the stability regions of the distributed-delay equation and of the mean-delay equation may intersect, even if the distribution is symmetric.
2010, 13(2): 347-373
doi: 10.3934/dcdsb.2010.13.347
+[Abstract](2361)
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Abstract:
We introduce a systematic way to build symplectic schemes for the numerical integration of a large class of highly oscillatory Hamiltonian systems. The bottom line of our construction is to consider the Hamilton-Jacobi form of the Newton equations of motion, and to perform a two-scale expansion of the solution, for small times and high frequencies. The approximation obtained for the solution is then used as a generating function, from which the numerical scheme is derived. Several options for the derivation are presented. The various integrators obtained are tested and compared to several existing algorithms. The numerical results demonstrate their efficiency.
We introduce a systematic way to build symplectic schemes for the numerical integration of a large class of highly oscillatory Hamiltonian systems. The bottom line of our construction is to consider the Hamilton-Jacobi form of the Newton equations of motion, and to perform a two-scale expansion of the solution, for small times and high frequencies. The approximation obtained for the solution is then used as a generating function, from which the numerical scheme is derived. Several options for the derivation are presented. The various integrators obtained are tested and compared to several existing algorithms. The numerical results demonstrate their efficiency.
2010, 13(2): 375-391
doi: 10.3934/dcdsb.2010.13.375
+[Abstract](2484)
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Abstract:
In this paper we investigate the asymptotic behaviour of the solutions of the linear theory of thermo-porous-elasticity. That is, we consider the theory of elastic materials with voids when the heat conduction is of type II. We assume that the only dissipation mechanism is the porous dissipation. First we prove that, generically, the solutions are exponentially stable on time or, in other words, the decay of solutions can be controlled by a negative exponential for a generic class of materials. The reason lies in the fact that the temperature is strongly coupled with both the microscopic and macroscopic structures of the materials and plays the role of a "driving belt" between the dissipation at the microscopic structure and the macroscopic one. Later we note that the decay of solutions cannot be fast enough to make the solutions be zero in a finite period of time. Finally, we show that when the coupling term between the microscopic (or macroscopic) structure and the thermal variable vanishes, the solutions do not decay exponentially (generically).
In this paper we investigate the asymptotic behaviour of the solutions of the linear theory of thermo-porous-elasticity. That is, we consider the theory of elastic materials with voids when the heat conduction is of type II. We assume that the only dissipation mechanism is the porous dissipation. First we prove that, generically, the solutions are exponentially stable on time or, in other words, the decay of solutions can be controlled by a negative exponential for a generic class of materials. The reason lies in the fact that the temperature is strongly coupled with both the microscopic and macroscopic structures of the materials and plays the role of a "driving belt" between the dissipation at the microscopic structure and the macroscopic one. Later we note that the decay of solutions cannot be fast enough to make the solutions be zero in a finite period of time. Finally, we show that when the coupling term between the microscopic (or macroscopic) structure and the thermal variable vanishes, the solutions do not decay exponentially (generically).
2010, 13(2): 393-414
doi: 10.3934/dcdsb.2010.13.393
+[Abstract](3046)
+[PDF](319.1KB)
Abstract:
This paper is concerned with the existence of traveling wave solutions in delayed reaction diffusion systems which at least contain multi-species competition, cooperation and predator-prey models with diffusion and delays. By introducing the mixed quasimonotone condition and the exponentially mixed quasimonotone condition, we reduce the existence of traveling wave solutions to the existence of a pair of admissible upper-lower solutions. To illustrate our main results, the existence of traveling wave solutions of multi-species competition, cooperation and predator-prey Lotka-Volterra systems with delays is considered. In particular, we show the precisely asymptotic behavior of the traveling wave solutions of these Lotka-Volterra systems.
This paper is concerned with the existence of traveling wave solutions in delayed reaction diffusion systems which at least contain multi-species competition, cooperation and predator-prey models with diffusion and delays. By introducing the mixed quasimonotone condition and the exponentially mixed quasimonotone condition, we reduce the existence of traveling wave solutions to the existence of a pair of admissible upper-lower solutions. To illustrate our main results, the existence of traveling wave solutions of multi-species competition, cooperation and predator-prey Lotka-Volterra systems with delays is considered. In particular, we show the precisely asymptotic behavior of the traveling wave solutions of these Lotka-Volterra systems.
2010, 13(2): 415-433
doi: 10.3934/dcdsb.2010.13.415
+[Abstract](2655)
+[PDF](249.2KB)
Abstract:
This paper is concerned with the existence, uniqueness and asymptotically stability of traveling wave fronts of discrete quasi-linear equations with delay. We first establish the existence of traveling wave fronts by using the super-sub solution and monotone iteration technique. Then we show that the traveling wave front is unique up to a translation. At last, we employ the comparison principle and the squeezing technique to prove that the traveling wave front is globally asymptotic stable with phase shift.
This paper is concerned with the existence, uniqueness and asymptotically stability of traveling wave fronts of discrete quasi-linear equations with delay. We first establish the existence of traveling wave fronts by using the super-sub solution and monotone iteration technique. Then we show that the traveling wave front is unique up to a translation. At last, we employ the comparison principle and the squeezing technique to prove that the traveling wave front is globally asymptotic stable with phase shift.
2010, 13(2): 435-454
doi: 10.3934/dcdsb.2010.13.435
+[Abstract](2319)
+[PDF](576.3KB)
Abstract:
We study a time-difference scheme for a nonlinear degenerate parabolic equation with a transport term. The model generally describes diffusion in porous media with the formation of a free boundary, this being expressed by the presence of a multivalued function in the equation. We consider singular boundary conditions which contain the multivalued function as well, and prove the stability and the convergence of the scheme, emphasizing the precise nature of the convergence. This approach is aimed to be a mathematical background which justifies the correctness of the numerical algorithm for computing the solution to this type of equations by avoiding the approximation of the multivalued function. The theory is illustrated by numerical results which put into evidence both the effects due to the equation degeneration and the formation and advance of the free boundary.
We study a time-difference scheme for a nonlinear degenerate parabolic equation with a transport term. The model generally describes diffusion in porous media with the formation of a free boundary, this being expressed by the presence of a multivalued function in the equation. We consider singular boundary conditions which contain the multivalued function as well, and prove the stability and the convergence of the scheme, emphasizing the precise nature of the convergence. This approach is aimed to be a mathematical background which justifies the correctness of the numerical algorithm for computing the solution to this type of equations by avoiding the approximation of the multivalued function. The theory is illustrated by numerical results which put into evidence both the effects due to the equation degeneration and the formation and advance of the free boundary.
2010, 13(2): 455-487
doi: 10.3934/dcdsb.2010.13.455
+[Abstract](2809)
+[PDF](783.3KB)
Abstract:
The gregarious behavior of individuals of populations is an important factor in avoiding predators or for reproduction. Here, by using a random biased walk approach, we build a model which, after a transformation, takes the general form $u_{t}=[D(u)u_{x}]_{x}+g(u)$. The model involves a density-dependent non-linear diffusion coefficient $D$ whose sign changes as the population density $u$ increases. For negative values of $D$ aggregation occurs, while dispersion occurs for positive values of $D$. We deal with a family of degenerate negative diffusion equations with logistic-like growth rate $g$. We study the one-dimensional traveling wave dynamics for these equations and illustrate our results with a couple of examples. A discussion of the ill-posedness of the partial differential equation problem is included.
The gregarious behavior of individuals of populations is an important factor in avoiding predators or for reproduction. Here, by using a random biased walk approach, we build a model which, after a transformation, takes the general form $u_{t}=[D(u)u_{x}]_{x}+g(u)$. The model involves a density-dependent non-linear diffusion coefficient $D$ whose sign changes as the population density $u$ increases. For negative values of $D$ aggregation occurs, while dispersion occurs for positive values of $D$. We deal with a family of degenerate negative diffusion equations with logistic-like growth rate $g$. We study the one-dimensional traveling wave dynamics for these equations and illustrate our results with a couple of examples. A discussion of the ill-posedness of the partial differential equation problem is included.
2010, 13(2): 489-501
doi: 10.3934/dcdsb.2010.13.489
+[Abstract](2524)
+[PDF](282.6KB)
Abstract:
We show that the coupled continuum pipe flow model (CCPF) for flows in karst aquifers is ill-posed in the sense that no reasonable solution exists. We also demonstrate that Hua's modified CCPF model is ill-posed in 3D although it is well-posed in two spatial dimensions. A new modification of the original CCPF model that is consistent with basic physics is proposed and its well-posedness is proved here. We believe that this is the first physically relevant well-posed CCPF type model in 3D.
We show that the coupled continuum pipe flow model (CCPF) for flows in karst aquifers is ill-posed in the sense that no reasonable solution exists. We also demonstrate that Hua's modified CCPF model is ill-posed in 3D although it is well-posed in two spatial dimensions. A new modification of the original CCPF model that is consistent with basic physics is proposed and its well-posedness is proved here. We believe that this is the first physically relevant well-posed CCPF type model in 3D.
Hopf bifurcation of a class of two coupled relaxation oscillators of the van der Pol type with delay
2010, 13(2): 503-516
doi: 10.3934/dcdsb.2010.13.503
+[Abstract](2503)
+[PDF](240.5KB)
Abstract:
In this paper, we study the following system of two coupled relaxation oscillators of the van der Pol type with delay
In this paper, we study the following system of two coupled relaxation oscillators of the van der Pol type with delay
ε$\ddot{x}_1-(1-x_1^2)\dot{x}_1+x_1=h_1(x_2(t-\tau)-x_1(t-\tau)),$
ε$\ddot{x}_2-(1-x_2^2)\dot{x}_2+x_2=h_2(x_1(t-\tau)-x_2(t-\tau)),$
where $h_1$ and $h_2$ are nonlinear functions. It is shown that this system can exhibit Hopf bifurcation as the time delay $\tau$ passes certain critical values. The distribution of the eigenvalues of the linearized system is studied thoroughly in terms of the parameter $\ep$ and the linear parts of functions $h_1$ and $h_2$. The normal form theory for general retarded functional equations developed by Faria and Magalhães is applied to perform center manifold reduction and hence to obtain the explicit normal form Hopf bifurcation which can be used to determine the stability of the bifurcating periodic solutions and and the direction of Hopf bifurcation. Examples are given to confirm the theoretical results.
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