ISSN:
1531-3492
eISSN:
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Discrete & Continuous Dynamical Systems - B
May 2006 , Volume 6 , Issue 3
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2006, 6(3): 427-448
doi: 10.3934/dcdsb.2006.6.427
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Abstract:
New computation algorithms for a fluid-dynamic mathematical model of flows on networks are proposed, described and tested. First we improve the classical Godunov scheme (G) for a special flux function, thus obtaining a more efficient method, the Fast Godunov scheme (FG) which reduces the number of evaluations for the numerical flux. Then a new method, namely the Fast Shock Fitting method (FSF), based on good theorical properties of the solution of the problem is introduced. Numerical results and efficiency tests are presented in order to show the behaviour of FSF in comparison with G, FG and a conservative scheme of second order.
New computation algorithms for a fluid-dynamic mathematical model of flows on networks are proposed, described and tested. First we improve the classical Godunov scheme (G) for a special flux function, thus obtaining a more efficient method, the Fast Godunov scheme (FG) which reduces the number of evaluations for the numerical flux. Then a new method, namely the Fast Shock Fitting method (FSF), based on good theorical properties of the solution of the problem is introduced. Numerical results and efficiency tests are presented in order to show the behaviour of FSF in comparison with G, FG and a conservative scheme of second order.
2006, 6(3): 449-470
doi: 10.3934/dcdsb.2006.6.449
+[Abstract](1893)
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Abstract:
We consider a multiscale model describing the flow of a concentrated suspension. The model couples the macroscopic equation of conservation of momentum with a nonlinear nonlocal kinetic equation describing at the microscopic level the rheological behaviour of the fluid. We study the long-time limit of the time-dependent solution. For this purpose, we use the entropy method to prove the convergence to equilibrium of the kinetic equation.
We consider a multiscale model describing the flow of a concentrated suspension. The model couples the macroscopic equation of conservation of momentum with a nonlinear nonlocal kinetic equation describing at the microscopic level the rheological behaviour of the fluid. We study the long-time limit of the time-dependent solution. For this purpose, we use the entropy method to prove the convergence to equilibrium of the kinetic equation.
2006, 6(3): 471-480
doi: 10.3934/dcdsb.2006.6.471
+[Abstract](1803)
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Abstract:
It is well known that, in the presence of an attractive force having a Coulomb singularity, scattering solutions of the nonrelativistic Abraham--Lorentz--Dirac equation having nonrunaway character do not exist, for the case of motions on the line. By numerical computations on the full three dimensional case, we give indications that indeed there exists a full tube of initial data for which nonrunay solutions of scatterig type do not exist. We also give a heuristic argument which allows to estimate the size of such a tube of initial data. The numerical computations also show that in a thin region beyond such a tube one has the nonuniqueness phenomenon, i.e. the "mechanical'' data of position and velocity do not uniquely determine the nonrunaway trajectory.
It is well known that, in the presence of an attractive force having a Coulomb singularity, scattering solutions of the nonrelativistic Abraham--Lorentz--Dirac equation having nonrunaway character do not exist, for the case of motions on the line. By numerical computations on the full three dimensional case, we give indications that indeed there exists a full tube of initial data for which nonrunay solutions of scatterig type do not exist. We also give a heuristic argument which allows to estimate the size of such a tube of initial data. The numerical computations also show that in a thin region beyond such a tube one has the nonuniqueness phenomenon, i.e. the "mechanical'' data of position and velocity do not uniquely determine the nonrunaway trajectory.
2006, 6(3): 481-492
doi: 10.3934/dcdsb.2006.6.481
+[Abstract](1796)
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Abstract:
In this paper we obtain Meyers type regularity estimates for approximate solutions of nonlinear elliptic equations. These estimates are used in the analysis of a numerical scheme obtained from a numerical homogenization of nonlinear elliptic equations. Numerical homogenization of nonlinear elliptic equations results in discretization schemes that require additional integrability of the approximate solutions. The latter motivates our work.
In this paper we obtain Meyers type regularity estimates for approximate solutions of nonlinear elliptic equations. These estimates are used in the analysis of a numerical scheme obtained from a numerical homogenization of nonlinear elliptic equations. Numerical homogenization of nonlinear elliptic equations results in discretization schemes that require additional integrability of the approximate solutions. The latter motivates our work.
2006, 6(3): 493-523
doi: 10.3934/dcdsb.2006.6.493
+[Abstract](1963)
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Abstract:
We study the dispersive evolution of modulated pulses in a nonlinear oscillator chain embedded in a background field. The atoms of the chain interact pairwise with an arbitrary but finite number of neighbors. The pulses are modeled as macroscopic modulations of the exact spatiotemporally periodic solutions of the linearized model. The scaling of amplitude, space and time is chosen in such a way that we can describe how the envelope changes in time due to dispersive effects. By this multiscale ansatz we find that the macroscopic evolution of the amplitude is given by the nonlinear Schrödinger equation. The main part of the work is focused on the justification of the formally derived equation: We show that solutions which have initially the form of the assumed ansatz preserve this form over time-intervals with a positive macroscopic length. The proof is based on a normal-form transformation constructed in Fourier space, and the results depend on the validity of suitable nonresonance conditions.
We study the dispersive evolution of modulated pulses in a nonlinear oscillator chain embedded in a background field. The atoms of the chain interact pairwise with an arbitrary but finite number of neighbors. The pulses are modeled as macroscopic modulations of the exact spatiotemporally periodic solutions of the linearized model. The scaling of amplitude, space and time is chosen in such a way that we can describe how the envelope changes in time due to dispersive effects. By this multiscale ansatz we find that the macroscopic evolution of the amplitude is given by the nonlinear Schrödinger equation. The main part of the work is focused on the justification of the formally derived equation: We show that solutions which have initially the form of the assumed ansatz preserve this form over time-intervals with a positive macroscopic length. The proof is based on a normal-form transformation constructed in Fourier space, and the results depend on the validity of suitable nonresonance conditions.
2006, 6(3): 525-534
doi: 10.3934/dcdsb.2006.6.525
+[Abstract](2171)
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Abstract:
In this work we analyze a Gause type predator-prey model with a non-monotonic functional response and we show that it has two limit cycles encircling an unique singularity at the interior of the first quadrant, the innermost unstable and the outermost stable, completing the results obtained in previous paper [12, 17, 26, 28].
Moreover, using the Poisson bracket we give a proof, shorter than the ones found in the literature, for determining the type of a cusp point of a singularity at the first quadrant.
In this work we analyze a Gause type predator-prey model with a non-monotonic functional response and we show that it has two limit cycles encircling an unique singularity at the interior of the first quadrant, the innermost unstable and the outermost stable, completing the results obtained in previous paper [12, 17, 26, 28].
Moreover, using the Poisson bracket we give a proof, shorter than the ones found in the literature, for determining the type of a cusp point of a singularity at the first quadrant.
2006, 6(3): 535-558
doi: 10.3934/dcdsb.2006.6.535
+[Abstract](2127)
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Abstract:
In this paper, we present a new model for optimal control of discrete event systems (DESs) with an arbitrary control pattern. Here, a discrete event system is defined as a collection of event sets that depend on strings. When the system generates a string, the next event that may occur should be in the corresponding event set. In the optimal control model, there are rewards for choosing control inputs at strings and the sets of available control inputs also depend on strings. The performance measure is to find a policy under the condition where the discounted total reward among strings from the initial state is maximized. By applying ideas from Markov decision processes, we divide the problem into three sub-cases where the optimal value is respectively finite, positive infinite and negative infinite. For the case with finite optimal values, the optimality equation is shown and further characterized with its solutions. We also characterize the structure of the set of all optimal policies. Moreover, we discuss invariance and closeness of several languages. We present a new supervisory control problem of DESs with the control pattern being dependent on strings. We study the problem in both the event feedback control and the state feedback control by generalizing concepts of invariant and closed languages/predicates. Finally, we apply the above model and results to a job-matching problem.
In this paper, we present a new model for optimal control of discrete event systems (DESs) with an arbitrary control pattern. Here, a discrete event system is defined as a collection of event sets that depend on strings. When the system generates a string, the next event that may occur should be in the corresponding event set. In the optimal control model, there are rewards for choosing control inputs at strings and the sets of available control inputs also depend on strings. The performance measure is to find a policy under the condition where the discounted total reward among strings from the initial state is maximized. By applying ideas from Markov decision processes, we divide the problem into three sub-cases where the optimal value is respectively finite, positive infinite and negative infinite. For the case with finite optimal values, the optimality equation is shown and further characterized with its solutions. We also characterize the structure of the set of all optimal policies. Moreover, we discuss invariance and closeness of several languages. We present a new supervisory control problem of DESs with the control pattern being dependent on strings. We study the problem in both the event feedback control and the state feedback control by generalizing concepts of invariant and closed languages/predicates. Finally, we apply the above model and results to a job-matching problem.
2006, 6(3): 559-572
doi: 10.3934/dcdsb.2006.6.559
+[Abstract](2751)
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Abstract:
In this paper, a discrete-time system, derived from a predator-prey system by Euler's method with step one, is investigated in the closed first quadrant $R_+^2$. It is shown that the discrete-time system undergoes fold bifurcation, flip bifurcation and Neimark-Sacker bifurcation, and the discrete-time system has a stable invariant cycle in the interior of $R_+^2$ for some parameter values. Numerical simulations are provided to verify the theoretical analysis and show the complicated dynamical behavior. These results reveal far richer dynamics of the discrete model compared with the same type continuous model.
In this paper, a discrete-time system, derived from a predator-prey system by Euler's method with step one, is investigated in the closed first quadrant $R_+^2$. It is shown that the discrete-time system undergoes fold bifurcation, flip bifurcation and Neimark-Sacker bifurcation, and the discrete-time system has a stable invariant cycle in the interior of $R_+^2$ for some parameter values. Numerical simulations are provided to verify the theoretical analysis and show the complicated dynamical behavior. These results reveal far richer dynamics of the discrete model compared with the same type continuous model.
2006, 6(3): 573-590
doi: 10.3934/dcdsb.2006.6.573
+[Abstract](2040)
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Abstract:
This paper proposes a novel neural network model for associative memory using dynamical systems. The proposed model is based on synthesizing the external input vector, which is different from the conventional approach where the design is based on synthesizing the connection matrix. It is shown that this new neural network (a) stores the desired prototype patterns as asymptotically stable equilibrium points, (b) has no spurious states, and (c) has learning and forgetting capabilities. Moreover, new learning and forgetting algorithms are also developed via a novel operation on the matrix space. Numerical examples are presented to illustrate the effectiveness of the proposed neural network for associative memory. Indeed, results of simulation experiments demonstrate that the neural network is effective and can be implemented easily.
This paper proposes a novel neural network model for associative memory using dynamical systems. The proposed model is based on synthesizing the external input vector, which is different from the conventional approach where the design is based on synthesizing the connection matrix. It is shown that this new neural network (a) stores the desired prototype patterns as asymptotically stable equilibrium points, (b) has no spurious states, and (c) has learning and forgetting capabilities. Moreover, new learning and forgetting algorithms are also developed via a novel operation on the matrix space. Numerical examples are presented to illustrate the effectiveness of the proposed neural network for associative memory. Indeed, results of simulation experiments demonstrate that the neural network is effective and can be implemented easily.
2006, 6(3): 591-604
doi: 10.3934/dcdsb.2006.6.591
+[Abstract](2503)
+[PDF](257.0KB)
Abstract:
We study in this paper the bifurcation and stability of the solutions of the Rayleigh-Bénard convection which has the infinite Prandtl number, using a notion of bifurcation called attractor bifurcation. We prove that the problem bifurcates from the trivial solution to an attractor $\A_R$ when the Rayleigh number $R$ crosses the critical Rayleigh number $R_c$. As a special case, we also prove another result which corresponds to the classical pitchfork bifurcation, that this bifurcated attractor $\A_R$ consists of only two stable steady states when the first eigenvalue $R_1$ is simple.
We study in this paper the bifurcation and stability of the solutions of the Rayleigh-Bénard convection which has the infinite Prandtl number, using a notion of bifurcation called attractor bifurcation. We prove that the problem bifurcates from the trivial solution to an attractor $\A_R$ when the Rayleigh number $R$ crosses the critical Rayleigh number $R_c$. As a special case, we also prove another result which corresponds to the classical pitchfork bifurcation, that this bifurcated attractor $\A_R$ consists of only two stable steady states when the first eigenvalue $R_1$ is simple.
2006, 6(3): 605-622
doi: 10.3934/dcdsb.2006.6.605
+[Abstract](2138)
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Non-linear difference equation models are employed in biology to describe the dynamics of certain populations and their interaction with the environment. In this paper we analyze a non-linear system describing community intervention in mosquito control through management of their habitats. The system takes the general form:
Non-linear difference equation models are employed in biology to describe the dynamics of certain populations and their interaction with the environment. In this paper we analyze a non-linear system describing community intervention in mosquito control through management of their habitats. The system takes the general form:
$x_{n+1}= a x_{n}h(p y_{n})+b h(q y_{n})$
n=0,1,...
$y_{n+1}= c x_{n}+d y_{n}$
where the function $h\in C^{1}$ ( [ $0,\infty$) $\to $ [$0,1$] ) satisfying certain properties, will denote either $h(t)=h_{1}(t)=e^{-t}$ and/or $h(t)=h_{2}(t)=1/(1+t).$ We give conditions in terms of parameters for boundedness and stability. This enables us to explore the dynamics of prevalence/community-activity systems as affected by the range of parameters.
2006, 6(3): 623-640
doi: 10.3934/dcdsb.2006.6.623
+[Abstract](2442)
+[PDF](3512.3KB)
Abstract:
This paper extends Runge-Kutta discontinuous Galerkin (RKDG) methods to a nonlinear Dirac (NLD) model in relativistic quantum physics, and investigates interaction dynamics of corresponding solitary wave solutions. Weak inelastic interaction in ternary collisions is first observed by using high-order accurate schemes on finer meshes. A long-lived oscillating state is formed with an approximate constant frequency in collisions of two standing waves; another is with an increasing frequency in collisions of two moving solitons. We also prove three continuum conservation laws of the NLD model and an entropy inequality, i.e. the total charge non-increasing, of the semi-discrete RKDG methods, which are demonstrated by various numerical examples.
This paper extends Runge-Kutta discontinuous Galerkin (RKDG) methods to a nonlinear Dirac (NLD) model in relativistic quantum physics, and investigates interaction dynamics of corresponding solitary wave solutions. Weak inelastic interaction in ternary collisions is first observed by using high-order accurate schemes on finer meshes. A long-lived oscillating state is formed with an approximate constant frequency in collisions of two standing waves; another is with an increasing frequency in collisions of two moving solitons. We also prove three continuum conservation laws of the NLD model and an entropy inequality, i.e. the total charge non-increasing, of the semi-discrete RKDG methods, which are demonstrated by various numerical examples.
2006, 6(3): 641-649
doi: 10.3934/dcdsb.2006.6.641
+[Abstract](2126)
+[PDF](208.6KB)
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This paper deals with the behavior of symmetric discrete--time systems with delays. The influence of the delay over these systems is analyzed in the stabilization problem. Furthermore, conditions on the system are given in order to solve the pole--assignment problem. Finally, some examples are shown with the aim to clarify the obtained results.
This paper deals with the behavior of symmetric discrete--time systems with delays. The influence of the delay over these systems is analyzed in the stabilization problem. Furthermore, conditions on the system are given in order to solve the pole--assignment problem. Finally, some examples are shown with the aim to clarify the obtained results.
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