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1078-0947
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Discrete and Continuous Dynamical Systems
October 2011 , Volume 29 , Issue 4
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Trends and Developments in DE/Dynamics
Part III
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2011, 29(4): 1309-1344
doi: 10.3934/dcds.2011.29.1309
+[Abstract](3561)
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Abstract:
We consider a homoclinic bifurcation of a vector field in $\R^3$, where a one-dimensional unstable manifold of an equilibrium is contained in the two-dimensional stable manifold of this same equilibrium. How such one-dimensional connecting orbits arise is well understood, and software packages exist to detect and follow them in parameters.
In this paper we address an issue that it is far less well understood: how does the associated two-dimensional stable manifold change geometrically during the given homoclinic bifurcation? This question can be answered with the help of advanced numerical techniques. More specifically, we compute two-dimensional manifolds, and their one-dimensional intersection curves with a suitable cross-section, via the numerical continuation of orbit segments as solutions of a boundary value problem. In this way, we are able to explain how homoclinic bifurcations may lead to quite dramatic changes of the overall dynamics. This is demonstrated with two examples. We first consider a Shilnikov bifurcation in a semiconductor laser model, and show how the associated change of the two-dimensional stable manifold results in the creation of a new basin of attraction. We then investigate how the basins of the two symmetrically related attracting equilibria change to give rise to preturbulence in the first homoclinic explosion of the Lorenz system.
We consider a homoclinic bifurcation of a vector field in $\R^3$, where a one-dimensional unstable manifold of an equilibrium is contained in the two-dimensional stable manifold of this same equilibrium. How such one-dimensional connecting orbits arise is well understood, and software packages exist to detect and follow them in parameters.
In this paper we address an issue that it is far less well understood: how does the associated two-dimensional stable manifold change geometrically during the given homoclinic bifurcation? This question can be answered with the help of advanced numerical techniques. More specifically, we compute two-dimensional manifolds, and their one-dimensional intersection curves with a suitable cross-section, via the numerical continuation of orbit segments as solutions of a boundary value problem. In this way, we are able to explain how homoclinic bifurcations may lead to quite dramatic changes of the overall dynamics. This is demonstrated with two examples. We first consider a Shilnikov bifurcation in a semiconductor laser model, and show how the associated change of the two-dimensional stable manifold results in the creation of a new basin of attraction. We then investigate how the basins of the two symmetrically related attracting equilibria change to give rise to preturbulence in the first homoclinic explosion of the Lorenz system.
2011, 29(4): 1345-1365
doi: 10.3934/dcds.2011.29.1345
+[Abstract](3198)
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Abstract:
In our previous works we studied a one-dimensional free-boundary model related to the aggressive penetration of gaseous carbon dioxide in unsaturated concrete. Essentially, global existence and uniqueness of weak solutions to the model were obtained when the initial functions are bounded on the domain. In this paper we investigate the well-posedness of the problem for the case when the initial functions belong to a $L^2-$ class. Specifically, the uniqueness of weak solutions is proved by applying the dual equation method.
In our previous works we studied a one-dimensional free-boundary model related to the aggressive penetration of gaseous carbon dioxide in unsaturated concrete. Essentially, global existence and uniqueness of weak solutions to the model were obtained when the initial functions are bounded on the domain. In this paper we investigate the well-posedness of the problem for the case when the initial functions belong to a $L^2-$ class. Specifically, the uniqueness of weak solutions is proved by applying the dual equation method.
2011, 29(4): 1367-1391
doi: 10.3934/dcds.2011.29.1367
+[Abstract](3240)
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Abstract:
We consider a class of splitting schemes for fourth order nonlinear diffusion equations. Standard backward-time differencing requires the solution of a higher order elliptic problem, which can be both computationally expensive and work-intensive to code, in higher space dimensions. Recent papers in the literature provide computational evidence that a biharmonic-modified, forward time-stepping method, can provide good results for these problems. We provide a theoretical explanation of the results. For a basic nonlinear 'thin film' type equation we prove $H^1$ stability of the method given very simple boundedness constraints of the numerical solution. For a more general class of long-wave unstable problems, we prove stability and convergence, using only constraints on the smooth solution. Computational examples include both the model of 'thin film' type problems and a quantitative model for electrowetting in a Hele-Shaw cell (Lu et al J. Fluid Mech. 2007). The methods considered here are related to 'convexity splitting' methods for gradient flows with nonconvex energies.
We consider a class of splitting schemes for fourth order nonlinear diffusion equations. Standard backward-time differencing requires the solution of a higher order elliptic problem, which can be both computationally expensive and work-intensive to code, in higher space dimensions. Recent papers in the literature provide computational evidence that a biharmonic-modified, forward time-stepping method, can provide good results for these problems. We provide a theoretical explanation of the results. For a basic nonlinear 'thin film' type equation we prove $H^1$ stability of the method given very simple boundedness constraints of the numerical solution. For a more general class of long-wave unstable problems, we prove stability and convergence, using only constraints on the smooth solution. Computational examples include both the model of 'thin film' type problems and a quantitative model for electrowetting in a Hele-Shaw cell (Lu et al J. Fluid Mech. 2007). The methods considered here are related to 'convexity splitting' methods for gradient flows with nonconvex energies.
2011, 29(4): 1393-1404
doi: 10.3934/dcds.2011.29.1393
+[Abstract](4154)
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Abstract:
We consider a porous medium equation with a nonlocal diffusion effect given by an inverse fractional Laplacian operator. The equation is posed in the whole space $\mathbb{R}^n$. In a previous paper we have found mass-preserving, nonnegative weak solutions of the equation satisfying energy estimates. Here we establish the large-time behaviour. We first find selfsimilar nonnegative solutions by solving an elliptic obstacle problem for the pair pressure-density involving the Laplacian, obtaining what we call obstacle Barenblatt solutions. The theory for elliptic fractional problems with obstacles has been recently established. We then use entropy methods to show that the asymptotic behavior of general finite-mass solutions is described after renormalization by these special solutions, which represent a surprising variation of the Barenblatt profiles of the standard porous medium model.
We consider a porous medium equation with a nonlocal diffusion effect given by an inverse fractional Laplacian operator. The equation is posed in the whole space $\mathbb{R}^n$. In a previous paper we have found mass-preserving, nonnegative weak solutions of the equation satisfying energy estimates. Here we establish the large-time behaviour. We first find selfsimilar nonnegative solutions by solving an elliptic obstacle problem for the pair pressure-density involving the Laplacian, obtaining what we call obstacle Barenblatt solutions. The theory for elliptic fractional problems with obstacles has been recently established. We then use entropy methods to show that the asymptotic behavior of general finite-mass solutions is described after renormalization by these special solutions, which represent a surprising variation of the Barenblatt profiles of the standard porous medium model.
2011, 29(4): 1405-1417
doi: 10.3934/dcds.2011.29.1405
+[Abstract](2734)
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Abstract:
We study dynamical equivalence relations on the moduli space $\MP_d$ of complex polynomial dynamical systems. Our main result is that the critical-heights quotient $\MP_d \to \cT_d$* of [4] is the Hausdorffization of a relation based on the twisting deformation of the basin of infinity. We also study relations of topological conjugacy and the Branner-Hubbard wringing deformation.
We study dynamical equivalence relations on the moduli space $\MP_d$ of complex polynomial dynamical systems. Our main result is that the critical-heights quotient $\MP_d \to \cT_d$* of [4] is the Hausdorffization of a relation based on the twisting deformation of the basin of infinity. We also study relations of topological conjugacy and the Branner-Hubbard wringing deformation.
2011, 29(4): 1419-1441
doi: 10.3934/dcds.2011.29.1419
+[Abstract](3415)
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Abstract:
We show there are no symbolic extensions $C^1$-generically among diffeomorphisms containing nonhyperbolic robustly transitive sets with a center indecomposable bundle of dimension at least 2. Similarly, $C^1$-generically homoclinic classes with a center indecomposable bundle of dimension at least 2 that satisfy a technical assumption called index adaptation have no symbolic extensions.
We show there are no symbolic extensions $C^1$-generically among diffeomorphisms containing nonhyperbolic robustly transitive sets with a center indecomposable bundle of dimension at least 2. Similarly, $C^1$-generically homoclinic classes with a center indecomposable bundle of dimension at least 2 that satisfy a technical assumption called index adaptation have no symbolic extensions.
2011, 29(4): 1443-1461
doi: 10.3934/dcds.2011.29.1443
+[Abstract](3340)
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Abstract:
We present some asymptotic analysis of a diffuse interface relaxation to a nonlocal optimal domain partition problem and the associated nonlocal interfacial motion when the interfacial width is approaching to zero. Motivated by careful numerical calculations, we first discuss several assumptions on the steady state solutions of the coupled system of differential equations which are supported by numerical results. These assumptions allow us to construct a suitable ansatz to the solutions which not only captures the leading order behavior but also provides sufficient estimates on the next order behavior so that more accurate estimates can be shown for interesting physical quantities such as energies and eigenvalues. When adopted to the gradient flow system, the ansatz gives an estimate of the asymptotic convergence rate in time to the equilibrium partitions.
We present some asymptotic analysis of a diffuse interface relaxation to a nonlocal optimal domain partition problem and the associated nonlocal interfacial motion when the interfacial width is approaching to zero. Motivated by careful numerical calculations, we first discuss several assumptions on the steady state solutions of the coupled system of differential equations which are supported by numerical results. These assumptions allow us to construct a suitable ansatz to the solutions which not only captures the leading order behavior but also provides sufficient estimates on the next order behavior so that more accurate estimates can be shown for interesting physical quantities such as energies and eigenvalues. When adopted to the gradient flow system, the ansatz gives an estimate of the asymptotic convergence rate in time to the equilibrium partitions.
2011, 29(4): 1463-1470
doi: 10.3934/dcds.2011.29.1463
+[Abstract](3521)
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Abstract:
We study the motion of noncompact hypersurfaces moved by their mean curvature obtained by a rotation around $x$-axis of the graph a function $y=u(x,t)$ (defined for all $x\in \mathbb{R}$). We are interested to estimate its profile when the hypersurface closes open ends at the quenching (pinching) time $T$. We estimate its profile at the quenching time from above and below. We in particular prove that $u(x,T)$ ~ $|x|^{-a}$ as $|x|\to\infty$ if $u(x,0)$ tends to its infimum with algebraic rate $|x|^{-2a} $ (as $|x| \to \infty $ with $a>0$).
We study the motion of noncompact hypersurfaces moved by their mean curvature obtained by a rotation around $x$-axis of the graph a function $y=u(x,t)$ (defined for all $x\in \mathbb{R}$). We are interested to estimate its profile when the hypersurface closes open ends at the quenching (pinching) time $T$. We estimate its profile at the quenching time from above and below. We in particular prove that $u(x,T)$ ~ $|x|^{-a}$ as $|x|\to\infty$ if $u(x,0)$ tends to its infimum with algebraic rate $|x|^{-2a} $ (as $|x| \to \infty $ with $a>0$).
2011, 29(4): 1471-1495
doi: 10.3934/dcds.2011.29.1471
+[Abstract](3734)
+[PDF](552.1KB)
Abstract:
We consider generalized linear transient convection-diffusion problems for differential forms on bounded domains in $\mathbb{R}^{n}$. These involve Lie derivatives with respect to a prescribed smooth vector field. We construct both new Eulerian and semi-Lagrangian approaches to the discretization of the Lie derivatives in the context of a Galerkin approximation based on discrete differential forms. Our focus is on derivations of the schemes, details of implementation, as well as on application to the discretization of eddy current equations in moving media.
We consider generalized linear transient convection-diffusion problems for differential forms on bounded domains in $\mathbb{R}^{n}$. These involve Lie derivatives with respect to a prescribed smooth vector field. We construct both new Eulerian and semi-Lagrangian approaches to the discretization of the Lie derivatives in the context of a Galerkin approximation based on discrete differential forms. Our focus is on derivations of the schemes, details of implementation, as well as on application to the discretization of eddy current equations in moving media.
2011, 29(4): 1497-1516
doi: 10.3934/dcds.2011.29.1497
+[Abstract](2771)
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Abstract:
Domino tilings have been studied extensively for both their statistical properties [5], [12], [15] and their dynamical properties [3]. We construct a subshift of finite type using matching rules for several types of dominos. We combine the previous results about domino tilings to show that our subshift of finite type has a measure of maximal entropy with which the subshift has completely positive entropy but is not isomorphic to a Bernoulli shift.
Domino tilings have been studied extensively for both their statistical properties [5], [12], [15] and their dynamical properties [3]. We construct a subshift of finite type using matching rules for several types of dominos. We combine the previous results about domino tilings to show that our subshift of finite type has a measure of maximal entropy with which the subshift has completely positive entropy but is not isomorphic to a Bernoulli shift.
2011, 29(4): 1517-1552
doi: 10.3934/dcds.2011.29.1517
+[Abstract](3664)
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Abstract:
The main purpose of this paper is to approximate several non-local evolution equations by zero-sum repeated games in the spirit of the previous works of Kohn and the second author (2006 and 2009): general fully non-linear parabolic integro-differential equations on the one hand, and the integral curvature flow of an on the other hand. In order to do so, we start by constructing such a game for eikonal equations whose speed has a non-constant sign. This provides a (discrete) deterministic control interpretation of these evolution equations.
In all our games, two players choose positions successively, and their final payoff is determined by their positions and additional parameters of choice. Because of the non-locality of the problems approximated, by contrast with local problems, their choices have to "collect" information far from their current position. For parabolic integro-differential equations, players choose smooth functions on the whole space. For integral curvature flows, players choose hypersurfaces in the whole space and positions on these hypersurfaces.
The main purpose of this paper is to approximate several non-local evolution equations by zero-sum repeated games in the spirit of the previous works of Kohn and the second author (2006 and 2009): general fully non-linear parabolic integro-differential equations on the one hand, and the integral curvature flow of an on the other hand. In order to do so, we start by constructing such a game for eikonal equations whose speed has a non-constant sign. This provides a (discrete) deterministic control interpretation of these evolution equations.
In all our games, two players choose positions successively, and their final payoff is determined by their positions and additional parameters of choice. Because of the non-locality of the problems approximated, by contrast with local problems, their choices have to "collect" information far from their current position. For parabolic integro-differential equations, players choose smooth functions on the whole space. For integral curvature flows, players choose hypersurfaces in the whole space and positions on these hypersurfaces.
2011, 29(4): 1553-1571
doi: 10.3934/dcds.2011.29.1553
+[Abstract](3609)
+[PDF](450.7KB)
Abstract:
We study measures on the real line and present various versions of what it means for such a measure to take only finitely many values. We then study perturbations of the Laplacian by such measures. Using Kotani-Remling theory, we show that the resulting operators have empty absolutely continuous spectrum if the measures are not periodic. When combined with Gordon type arguments this allows us to prove purely singular continuous spectrum for some continuum models of quasicrystals.
We study measures on the real line and present various versions of what it means for such a measure to take only finitely many values. We then study perturbations of the Laplacian by such measures. Using Kotani-Remling theory, we show that the resulting operators have empty absolutely continuous spectrum if the measures are not periodic. When combined with Gordon type arguments this allows us to prove purely singular continuous spectrum for some continuum models of quasicrystals.
2011, 29(4): 1573-1636
doi: 10.3934/dcds.2011.29.1573
+[Abstract](2959)
+[PDF](909.1KB)
Abstract:
We study a three-dimensional model of cellular electrical activity, which is written as a pseudodifferential equation on a closed surface $\Gamma$ in $\R^3$ coupled with a system of ordinary differential equations on $\Gamma$. Previously the existence of a global classical solution was not known, due mainly to the lack of a uniform $L^\infty$ bound. The main difficulty lies in the fact that, unlike the Laplace operator that appears in traditional models, the pseudodifferential operator in the present model does not satisfy the maximum principle. We overcome this difficulty by introducing the notion of "quasipositivity principle" and prove a uniform $L^\infty$ bound of solutions -- hence the existence of global classical solutions -- for a large class of nonlinearities including the FitzHugh-Nagumo and the Hodgkin-Huxley kinetics. We then study the asymptotic behavior of solutions to show that the system possesses a finite dimensional global attractor consisting entirely of smooth functions despite the fact that the system is only partially dissipative. We also show that ordinary differential equation models without spatial extent, often used in modeling studies, can be obtained from the present model in the small-cell-size limit.
We study a three-dimensional model of cellular electrical activity, which is written as a pseudodifferential equation on a closed surface $\Gamma$ in $\R^3$ coupled with a system of ordinary differential equations on $\Gamma$. Previously the existence of a global classical solution was not known, due mainly to the lack of a uniform $L^\infty$ bound. The main difficulty lies in the fact that, unlike the Laplace operator that appears in traditional models, the pseudodifferential operator in the present model does not satisfy the maximum principle. We overcome this difficulty by introducing the notion of "quasipositivity principle" and prove a uniform $L^\infty$ bound of solutions -- hence the existence of global classical solutions -- for a large class of nonlinearities including the FitzHugh-Nagumo and the Hodgkin-Huxley kinetics. We then study the asymptotic behavior of solutions to show that the system possesses a finite dimensional global attractor consisting entirely of smooth functions despite the fact that the system is only partially dissipative. We also show that ordinary differential equation models without spatial extent, often used in modeling studies, can be obtained from the present model in the small-cell-size limit.
2011, 29(4): 1637-1649
doi: 10.3934/dcds.2011.29.1637
+[Abstract](3191)
+[PDF](432.0KB)
Abstract:
In a recent paper [4], we showed that the phenomenon of resonant tunneling, well known in linear quantum mechanical scattering theory, takes place for fast solitons of the Nonlinear Schrödinger (NLS) equation in the presence of certain large potentials. Here, we illustrate numerically this situation for the one dimensional cubic NLS equation with two classes of potentials, namely the 'box' potential and a repulsive 2-delta potential. In particular, under the resonant condition, we show that the transmitted wave is close to a soliton, calculate the transmitted mass of the solution and show that it converges to the total mass of the solution as the velocity of the soliton is increased.
In a recent paper [4], we showed that the phenomenon of resonant tunneling, well known in linear quantum mechanical scattering theory, takes place for fast solitons of the Nonlinear Schrödinger (NLS) equation in the presence of certain large potentials. Here, we illustrate numerically this situation for the one dimensional cubic NLS equation with two classes of potentials, namely the 'box' potential and a repulsive 2-delta potential. In particular, under the resonant condition, we show that the transmitted wave is close to a soliton, calculate the transmitted mass of the solution and show that it converges to the total mass of the solution as the velocity of the soliton is increased.
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