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1078-0947
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Discrete & Continuous Dynamical Systems
July 1998 , Volume 4 , Issue 3
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1998, 4(3): 393-403
doi: 10.3934/dcds.1998.4.393
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Abstract:
This paper deals with a class of second order dynamical systems with slowly oscillating coefficients, see $(1)$. Using variational methods, perturbative in nature, we show that $(1)$ has multi-bump homoclinics and a complex dynamics.
This paper deals with a class of second order dynamical systems with slowly oscillating coefficients, see $(1)$. Using variational methods, perturbative in nature, we show that $(1)$ has multi-bump homoclinics and a complex dynamics.
1998, 4(3): 405-430
doi: 10.3934/dcds.1998.4.405
+[Abstract](2780)
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In the first part of the paper we show that a hyperbolic area preserving Hénon map has a unique Gibbs measure whose Hausdorff dimension is equal to the Hausdorff dimension of its nonwandering (Julia) set. In the second part we introduce the notion of strong hyperbolicity for diffeomorphisms of compact manifolds. It is a foliation of the tangent space over a hyperbolic set to one dimensional contracting and expanding subspaces with different rates of contractions and expansions. We show that strong hyperbolicity is structurally stable. For a Strong Axiom A diffeomorphism $f$ we state a conjectured variational characterization of the Hausdorff dimension of the nonwandering set of $f$. In the third part we study the dynamics of polynomial maps $f:\mathbb{C}^2 \rightarrow \mathbb{C}^2$ which lift to holomorphic maps of $\mathbb C\mathbb P^2$. Let $J(f)$ be the closure of repelling periodic points of $f$. Using the structural stability results we exhibit open set of $f$ for which $J(f)$ behaves like the Julia set of one dimensional polynomial map.
In the first part of the paper we show that a hyperbolic area preserving Hénon map has a unique Gibbs measure whose Hausdorff dimension is equal to the Hausdorff dimension of its nonwandering (Julia) set. In the second part we introduce the notion of strong hyperbolicity for diffeomorphisms of compact manifolds. It is a foliation of the tangent space over a hyperbolic set to one dimensional contracting and expanding subspaces with different rates of contractions and expansions. We show that strong hyperbolicity is structurally stable. For a Strong Axiom A diffeomorphism $f$ we state a conjectured variational characterization of the Hausdorff dimension of the nonwandering set of $f$. In the third part we study the dynamics of polynomial maps $f:\mathbb{C}^2 \rightarrow \mathbb{C}^2$ which lift to holomorphic maps of $\mathbb C\mathbb P^2$. Let $J(f)$ be the closure of repelling periodic points of $f$. Using the structural stability results we exhibit open set of $f$ for which $J(f)$ behaves like the Julia set of one dimensional polynomial map.
1998, 4(3): 431-444
doi: 10.3934/dcds.1998.4.431
+[Abstract](3286)
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Abstract:
The first initial-boundary value problem for the damped Boussinesq equation $ u_{t t}-2bu_{t x x}=-\alpha u_{x x x x}+u_{x x}+\beta (u^2)_{x x}, x\in (0,\pi ),\quad t>0,$ with $\alpha, b=const>0,\quad \beta =const\in R^1,$ is considered with small initial data. For the most interesting case $\alpha >b^2$ corresponding to an infinite number of damped oscillations its solution is constructed in the form of a Fourier series which coefficients in their own turn are represented as series in small parameter present in the initial conditions. The solution of the corresponding problem for the classical Boussinesq equation on $[0,T],\quad T<+\infty,$ is obtained by means of passing to the limit $b\rightarrow +0.$ Long-time asymptotics of the solution in question is calculated which shows the presence of the damped oscillations decaying exponentially in time. This is in contrast with the long time behavior of the solution of the periodic problem studied in [30] which major term increases linearly with time.
The first initial-boundary value problem for the damped Boussinesq equation $ u_{t t}-2bu_{t x x}=-\alpha u_{x x x x}+u_{x x}+\beta (u^2)_{x x}, x\in (0,\pi ),\quad t>0,$ with $\alpha, b=const>0,\quad \beta =const\in R^1,$ is considered with small initial data. For the most interesting case $\alpha >b^2$ corresponding to an infinite number of damped oscillations its solution is constructed in the form of a Fourier series which coefficients in their own turn are represented as series in small parameter present in the initial conditions. The solution of the corresponding problem for the classical Boussinesq equation on $[0,T],\quad T<+\infty,$ is obtained by means of passing to the limit $b\rightarrow +0.$ Long-time asymptotics of the solution in question is calculated which shows the presence of the damped oscillations decaying exponentially in time. This is in contrast with the long time behavior of the solution of the periodic problem studied in [30] which major term increases linearly with time.
1998, 4(3): 445-454
doi: 10.3934/dcds.1998.4.445
+[Abstract](2463)
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Abstract:
We state a result concerning the limit of a class of minimization problems. This result is applied to describe the asymptotic behaviour of the solutions of an elliptic Dirichlet problem in exterior domains $\Omega$ of $\mathbb{R}^N$, when $\mathbb{R}^N$ \ $Omega$ becomes larger and larger.
We state a result concerning the limit of a class of minimization problems. This result is applied to describe the asymptotic behaviour of the solutions of an elliptic Dirichlet problem in exterior domains $\Omega$ of $\mathbb{R}^N$, when $\mathbb{R}^N$ \ $Omega$ becomes larger and larger.
1998, 4(3): 455-466
doi: 10.3934/dcds.1998.4.455
+[Abstract](2087)
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Abstract:
A general theory for the study of families of processes in the weak topology of some Banach space is suggested: sufficient conditions for the existence and connectedness of attractors are proved. The results apply to (nonlinear) nonautonomous evolution partial differential equations for which the behavior of the corresponding processes is better described when the phase space is endowed with its weak topology.
A general theory for the study of families of processes in the weak topology of some Banach space is suggested: sufficient conditions for the existence and connectedness of attractors are proved. The results apply to (nonlinear) nonautonomous evolution partial differential equations for which the behavior of the corresponding processes is better described when the phase space is endowed with its weak topology.
1998, 4(3): 467-474
doi: 10.3934/dcds.1998.4.467
+[Abstract](2370)
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Abstract:
In this paper, we are interested in the existence of subharmonic solutions for the problem $ u_{t t} + G'(u) = f(t), $ where $G:R^{ N} \rightarrow R$ is not necessarily convex and $f:R \rightarrow R^N$ is periodic with minimal period $T > 0$.
In this paper, we are interested in the existence of subharmonic solutions for the problem $ u_{t t} + G'(u) = f(t), $ where $G:R^{ N} \rightarrow R$ is not necessarily convex and $f:R \rightarrow R^N$ is periodic with minimal period $T > 0$.
1998, 4(3): 475-484
doi: 10.3934/dcds.1998.4.475
+[Abstract](2087)
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Abstract:
Flow equivalence of irreducible nontrivial square nonnegative integer matrices is completely determined by two computable invariants, the Parry-Sullivan number and the Bowen-Franks group. Twist-wise flow equivalence is a natural generalization that takes account of twisting in the local stable manifold of the orbits of a flow. Two new invariants in this category are established.
Flow equivalence of irreducible nontrivial square nonnegative integer matrices is completely determined by two computable invariants, the Parry-Sullivan number and the Bowen-Franks group. Twist-wise flow equivalence is a natural generalization that takes account of twisting in the local stable manifold of the orbits of a flow. Two new invariants in this category are established.
1998, 4(3): 485-496
doi: 10.3934/dcds.1998.4.485
+[Abstract](2507)
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Abstract:
In this paper we prove the existence of multiple solutions for two-point boundary value problem, which is resonant type both at origin and at infinity, and with higher eigenvalue and unbounded nonlinear terms. Our main ingredients are Morse theory and the computations of critical groups.
In this paper we prove the existence of multiple solutions for two-point boundary value problem, which is resonant type both at origin and at infinity, and with higher eigenvalue and unbounded nonlinear terms. Our main ingredients are Morse theory and the computations of critical groups.
1998, 4(3): 497-506
doi: 10.3934/dcds.1998.4.497
+[Abstract](3098)
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Abstract:
We analyze numerical approximations of positive solutions of a heat equation with a nonlinear flux condition which produces blow up of the solutions. By a semidiscretization using finite elements in the space variable we obtain a system of ordinary differential equations which is expected to be an approximation of the original problem. Our objective is to analyze whether this system has a similar behaviour than the original problem. We find a necessary and sufficient condition for blow up of this system. However, this condition is slightly different than the one known for the original problem, in particular, there are cases in which the continuous problem has blow up while its semidiscrete approximation does not.
Under certain assumptions we also prove that the numerical blow up time converges to the real blow-up time when the meshsize goes to zero. Our proofs are given in one space dimension. Similar arguments could be applied for higher dimensions but a further analysis is required.
We analyze numerical approximations of positive solutions of a heat equation with a nonlinear flux condition which produces blow up of the solutions. By a semidiscretization using finite elements in the space variable we obtain a system of ordinary differential equations which is expected to be an approximation of the original problem. Our objective is to analyze whether this system has a similar behaviour than the original problem. We find a necessary and sufficient condition for blow up of this system. However, this condition is slightly different than the one known for the original problem, in particular, there are cases in which the continuous problem has blow up while its semidiscrete approximation does not.
Under certain assumptions we also prove that the numerical blow up time converges to the real blow-up time when the meshsize goes to zero. Our proofs are given in one space dimension. Similar arguments could be applied for higher dimensions but a further analysis is required.
1998, 4(3): 507-522
doi: 10.3934/dcds.1998.4.507
+[Abstract](2422)
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Abstract:
We prove in this article the Gevrey class regularity and time-analyticity of the global (in time) strong solutions obtained by Tang and Wang (1995) for the time-dependent Ginzburg-Landau (TDGL) equations of superconductivity with an applied field.
We prove in this article the Gevrey class regularity and time-analyticity of the global (in time) strong solutions obtained by Tang and Wang (1995) for the time-dependent Ginzburg-Landau (TDGL) equations of superconductivity with an applied field.
1998, 4(3): 523-534
doi: 10.3934/dcds.1998.4.523
+[Abstract](2404)
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Abstract:
The goal of this paper is to study Banach algebra valued cocycles over Anosov actions. The study of cohomological equations over Anosov diffeomorphisms and flows was started in two influential papers by Livsic ([L1], [L2]), and experienced later a tremendous development. This paper is a continuation of [NT1] and [NT2]. We show here that the techniques used to study cocycles with values in Lie groups, and with values in diffeomorphism groups, can be adapted to Banach algebra valued cocycles. Results of this nature are necessary for the study of extensions of Anosov group actions on infinite dimensional manifolds.
The goal of this paper is to study Banach algebra valued cocycles over Anosov actions. The study of cohomological equations over Anosov diffeomorphisms and flows was started in two influential papers by Livsic ([L1], [L2]), and experienced later a tremendous development. This paper is a continuation of [NT1] and [NT2]. We show here that the techniques used to study cocycles with values in Lie groups, and with values in diffeomorphism groups, can be adapted to Banach algebra valued cocycles. Results of this nature are necessary for the study of extensions of Anosov group actions on infinite dimensional manifolds.
1998, 4(3): 535-557
doi: 10.3934/dcds.1998.4.535
+[Abstract](2135)
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Abstract:
Let $T:[0,1]\rightarrow[0,1]$ be an expanding piecewise-onto map with bounded distortion and countably many intervals of monotonicity. We prove a uniform bound on the rate of exponential convergence to equilibrium for iterates of the Perron-Frobenius operator. The quantitative information thus obtained is applied to prove explicit error bounds for Ulam's method for approximating invariant measures. The approach also yields rates of mixing for the matrix representations of Ulam approximation. "Monte-Carlo" type simulations of the scheme are discussed. The method of proof is applicable to multi-dimensional transformations, although the only generalisations presented here are to a limited class of "non-onto" one-dimensional transformations.
Let $T:[0,1]\rightarrow[0,1]$ be an expanding piecewise-onto map with bounded distortion and countably many intervals of monotonicity. We prove a uniform bound on the rate of exponential convergence to equilibrium for iterates of the Perron-Frobenius operator. The quantitative information thus obtained is applied to prove explicit error bounds for Ulam's method for approximating invariant measures. The approach also yields rates of mixing for the matrix representations of Ulam approximation. "Monte-Carlo" type simulations of the scheme are discussed. The method of proof is applicable to multi-dimensional transformations, although the only generalisations presented here are to a limited class of "non-onto" one-dimensional transformations.
1998, 4(3): 559-580
doi: 10.3934/dcds.1998.4.559
+[Abstract](2128)
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Abstract:
We study one parameter families of vector fields that are defined on three dimensional manifolds and whose nonwandering sets are structurally stable. As families, these families may not be structurally stable; heteroclinic bifurcations that give rise to moduli can occur. Some but not all moduli are related to the geometry of stable and unstable manifolds. We study a notion of stability, weaker then structural stability, in which geometry and dynamics on stable and unstable manifolds are reflected. We classify the families from the above mentioned class of families that are stable in this sense.
We study one parameter families of vector fields that are defined on three dimensional manifolds and whose nonwandering sets are structurally stable. As families, these families may not be structurally stable; heteroclinic bifurcations that give rise to moduli can occur. Some but not all moduli are related to the geometry of stable and unstable manifolds. We study a notion of stability, weaker then structural stability, in which geometry and dynamics on stable and unstable manifolds are reflected. We classify the families from the above mentioned class of families that are stable in this sense.
1998, 4(3): 581-591
doi: 10.3934/dcds.1998.4.581
+[Abstract](2425)
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Abstract:
In this paper, the $C^1$ interior of the set of all diffeomorphisms satisfying the OE-property is characterized as the set of all diffeomorphisms satisfying Axiom A and the strong transversality condition. Thus the $C^1$ interior of the set of all diffeomorphisms satisfying the OE-property is equal to the $C^1$ interior of the set of all diffeomorphisms satisfying the shadowing property.
In this paper, the $C^1$ interior of the set of all diffeomorphisms satisfying the OE-property is characterized as the set of all diffeomorphisms satisfying Axiom A and the strong transversality condition. Thus the $C^1$ interior of the set of all diffeomorphisms satisfying the OE-property is equal to the $C^1$ interior of the set of all diffeomorphisms satisfying the shadowing property.
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