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Discrete & Continuous Dynamical Systems - A
August 2010 , Volume 27 , Issue 3
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2010, 27(3): 863-906
doi: 10.3934/dcds.2010.27.863
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Abstract:
The Decoration Conjecture describes the structure of the set of braid types of Smale's horseshoe map ordered by forcing, providing information about the order in which periodic orbits can appear when a horseshoe is created. A proof of this conjecture is given for the class of so-called lone decorations, and it is explained how to calculate associated braid conjugacy invariants which provide additional information about forcing for horseshoe braids.
The Decoration Conjecture describes the structure of the set of braid types of Smale's horseshoe map ordered by forcing, providing information about the order in which periodic orbits can appear when a horseshoe is created. A proof of this conjecture is given for the class of so-called lone decorations, and it is explained how to calculate associated braid conjugacy invariants which provide additional information about forcing for horseshoe braids.
2010, 27(3): 907-917
doi: 10.3934/dcds.2010.27.907
+[Abstract](2747)
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Abstract:
Consistently fitting vanilla option surfaces is an important issue when it comes to modeling in finance. As far as local and stochastic volatility models are concerned, this problem boils down to the resolution of a nonlinear integro-differential pde. The non-locality of this equation stems from the quotient of two integral terms and is not defined for all bounded continuous functions. In this paper, we use a fixed point argument and suitable a priori estimates to prove short-time existence of solutions for this equation.
Consistently fitting vanilla option surfaces is an important issue when it comes to modeling in finance. As far as local and stochastic volatility models are concerned, this problem boils down to the resolution of a nonlinear integro-differential pde. The non-locality of this equation stems from the quotient of two integral terms and is not defined for all bounded continuous functions. In this paper, we use a fixed point argument and suitable a priori estimates to prove short-time existence of solutions for this equation.
2010, 27(3): 919-934
doi: 10.3934/dcds.2010.27.919
+[Abstract](2204)
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Abstract:
In this paper we complete our works on the local energy decay for the evolution damping problem in exterior domains. We consider the wave and Schrödinger equations in an exterior domain with dissipative boundary condition. We study the distribution of resonances under some natural assumptions on the behavior of the geodesics in order to deduce the uniform local energy decay.
In this paper we complete our works on the local energy decay for the evolution damping problem in exterior domains. We consider the wave and Schrödinger equations in an exterior domain with dissipative boundary condition. We study the distribution of resonances under some natural assumptions on the behavior of the geodesics in order to deduce the uniform local energy decay.
2010, 27(3): 935-943
doi: 10.3934/dcds.2010.27.935
+[Abstract](1848)
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Abstract:
We prove that if $K$ is the invariant set of an IFS in $\ R^{d}$ satisfying the Strong Open Set Condition, then the set of extremely non-normal points of $K$ is a comeagre subset of $K$.
We prove that if $K$ is the invariant set of an IFS in $\ R^{d}$ satisfying the Strong Open Set Condition, then the set of extremely non-normal points of $K$ is a comeagre subset of $K$.
2010, 27(3): 945-961
doi: 10.3934/dcds.2010.27.945
+[Abstract](2052)
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Abstract:
In this paper we provide a few recipes how to construct a topologically transitive cocycle over an arbitrary odometer possessing discrete orbits. It is shown that for every odometer, there exists a topologically transitive cocycle such that the set of points with discrete orbits starting form zero level has the cardinality of the continuum.
In this paper we provide a few recipes how to construct a topologically transitive cocycle over an arbitrary odometer possessing discrete orbits. It is shown that for every odometer, there exists a topologically transitive cocycle such that the set of points with discrete orbits starting form zero level has the cardinality of the continuum.
2010, 27(3): 963-980
doi: 10.3934/dcds.2010.27.963
+[Abstract](2309)
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Abstract:
The paper deals with the cyclicity of unbounded semi-hyperbolic 2-saddle cycles in polynomial Liénard systems of type $(m,n)$ with $m<2n+1$, $m$ and $n$ odd. We generalize the results in [1] (case $m=1$), providing a substantially simpler and more transparant proof than the one used in [1].
The paper deals with the cyclicity of unbounded semi-hyperbolic 2-saddle cycles in polynomial Liénard systems of type $(m,n)$ with $m<2n+1$, $m$ and $n$ odd. We generalize the results in [1] (case $m=1$), providing a substantially simpler and more transparant proof than the one used in [1].
2010, 27(3): 981-993
doi: 10.3934/dcds.2010.27.981
+[Abstract](2412)
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Abstract:
We consider interior regularity for weak solutions of nonlinear elliptic systems with subquadratic under controllable growth condition. By $\mathcal{A}$-harmonic approximation technique, we obtain a general criterion for a weak solution to be regular in the neighborhood of a given point. In particularly, the regular result is optimal.
We consider interior regularity for weak solutions of nonlinear elliptic systems with subquadratic under controllable growth condition. By $\mathcal{A}$-harmonic approximation technique, we obtain a general criterion for a weak solution to be regular in the neighborhood of a given point. In particularly, the regular result is optimal.
2010, 27(3): 995-1005
doi: 10.3934/dcds.2010.27.995
+[Abstract](2699)
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Abstract:
Inducing schemes provide a means of using symbolic dynamics to study equilibrium states of non-uniformly hyperbolic maps, but necessitate a solution to the liftability problem. One approach, due to Pesin and Senti, places conditions on the induced potential under which a unique equilibrium state exists among liftable measures, and then solves the liftability problem separately. Another approach, due to Bruin and Todd, places conditions on the original potential under which both problems may be solved simultaneously. These conditions include a bounded range condition, first introduced by Hofbauer and Keller. We compare these two sets of conditions and show that for many inducing schemes of interest, the conditions from the second approach are strictly stronger than the conditions from the first. We also show that the bounded range condition can be used to obtain Pesin and Senti's conditions for any inducing scheme with sufficiently slow growth of basic elements.
Inducing schemes provide a means of using symbolic dynamics to study equilibrium states of non-uniformly hyperbolic maps, but necessitate a solution to the liftability problem. One approach, due to Pesin and Senti, places conditions on the induced potential under which a unique equilibrium state exists among liftable measures, and then solves the liftability problem separately. Another approach, due to Bruin and Todd, places conditions on the original potential under which both problems may be solved simultaneously. These conditions include a bounded range condition, first introduced by Hofbauer and Keller. We compare these two sets of conditions and show that for many inducing schemes of interest, the conditions from the second approach are strictly stronger than the conditions from the first. We also show that the bounded range condition can be used to obtain Pesin and Senti's conditions for any inducing scheme with sufficiently slow growth of basic elements.
2010, 27(3): 1007-1024
doi: 10.3934/dcds.2010.27.1007
+[Abstract](2371)
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Abstract:
We investigate a family of maps that arises from a model in economics and game theory. It has some features similar to renormalization and some similar to intermittency. In a one-parameter family of maps in dimension 2, when the parameter goes to 0, the maps converge to the identity. Nevertheless, after a linear rescaling of both space and time, we get maps with attracting invariant closed curves. As the parameter goes to 0, those curves converge in a strong sense to a certain circle. We call those phenomena microdynamics. The model can be also understood as a family of discrete time approximations to a Brown-von Neumann differential equation.
We investigate a family of maps that arises from a model in economics and game theory. It has some features similar to renormalization and some similar to intermittency. In a one-parameter family of maps in dimension 2, when the parameter goes to 0, the maps converge to the identity. Nevertheless, after a linear rescaling of both space and time, we get maps with attracting invariant closed curves. As the parameter goes to 0, those curves converge in a strong sense to a certain circle. We call those phenomena microdynamics. The model can be also understood as a family of discrete time approximations to a Brown-von Neumann differential equation.
2010, 27(3): 1025-1035
doi: 10.3934/dcds.2010.27.1025
+[Abstract](2527)
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Abstract:
In this paper, the well-posedness and blow up phenomena for the interacting system of the Camassa-Holm and Degasperis-Procesi equations are studied. We first establish the local well-posedness of strong solutions for the system. Then the precise blow-up scenarios for the strong solutions to the system are derived.
In this paper, the well-posedness and blow up phenomena for the interacting system of the Camassa-Holm and Degasperis-Procesi equations are studied. We first establish the local well-posedness of strong solutions for the system. Then the precise blow-up scenarios for the strong solutions to the system are derived.
2010, 27(3): 1037-1058
doi: 10.3934/dcds.2010.27.1037
+[Abstract](2461)
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Abstract:
We prove the existence of an appropriate function (very weak solution) $u$ in the Lorentz space $L^{N',\infty}(\Omega), \ N'=\frac N{N-1}$ satisfying $Lu-Vu+g(x,u,\nabla u)=\mu$ in $\Omega$ an open bounded set of $\R^N$, and $u=0$ on $\partial\Omega$ in the sense that
We prove the existence of an appropriate function (very weak solution) $u$ in the Lorentz space $L^{N',\infty}(\Omega), \ N'=\frac N{N-1}$ satisfying $Lu-Vu+g(x,u,\nabla u)=\mu$ in $\Omega$ an open bounded set of $\R^N$, and $u=0$ on $\partial\Omega$ in the sense that
$(u,L\varphi)_0-(Vu,\varphi)_0+(g(\cdot,u,\nabla u),\varphi)_0=\mu(\varphi),\quad\forall\varphi\in C^2_c(\Omega).$
The potential $V \le \lambda < \lambda_1$ is assumed to be in the
weighted Lorentz space $L^{N,1}(\Omega,\delta)$, where
$\delta(x)= dist(x,\partial\Omega),\ \mu\in
M^1(\Omega,\delta)$, the set of weighted Radon measures
containing $L^1(\Omega,\delta)$, $L$ is an elliptic linear self
adjoint second order operator, and $\lambda_1$ is the first
eigenvalue of $L$ with zero Dirichlet boundary conditions.
If $\mu\in L^1(\Omega,\delta)$ we only assume that for the potential $V$ is in
L1loc$(\Omega)$, $V \le \lambda<\lambda_1$. If $\mu\in M^1(\Omega,\delta^\alpha),\
\alpha\in$[$0,1[$[, then we prove that the very weak solution $|\nabla u|$ is in the
Lorentz space $L^{\frac N{N-1+\alpha},\infty}(\Omega)$. We apply those results
to the existence of the so called large solutions with a right hand side data in
$L^1(\Omega,\delta)$. Finally, we prove some rearrangement comparison results.
2010, 27(3): 1059-1078
doi: 10.3934/dcds.2010.27.1059
+[Abstract](2197)
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Abstract:
Let $f$ be a unimodal map of the interval with critical point $c$. If the orbit of $c$ is not dense then most points in lim←{[0, 1], f} have neighborhoods that are homeomorphic with the product of a Cantor set and an open arc. The points without this property are called inhomogeneities, and the set, I, of inhomogeneities is equal to lim← {w(c), f|w(c)}. In this paper we consider the relationship between the limit complexity of $w(c)$ and the limit complexity of I. We show that if $w(c)$ is more complicated than a finite collection of convergent sequences then I can have arbitrarily high limit complexity. We give a complete description of the limit complexity of I for any possible $\w(c)$.
Let $f$ be a unimodal map of the interval with critical point $c$. If the orbit of $c$ is not dense then most points in lim←{[0, 1], f} have neighborhoods that are homeomorphic with the product of a Cantor set and an open arc. The points without this property are called inhomogeneities, and the set, I, of inhomogeneities is equal to lim← {w(c), f|w(c)}. In this paper we consider the relationship between the limit complexity of $w(c)$ and the limit complexity of I. We show that if $w(c)$ is more complicated than a finite collection of convergent sequences then I can have arbitrarily high limit complexity. We give a complete description of the limit complexity of I for any possible $\w(c)$.
2010, 27(3): 1079-1091
doi: 10.3934/dcds.2010.27.1079
+[Abstract](2352)
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Abstract:
In this paper, we prove a criterion for the existence of continuous non constant eigenfunctions for interval exchange transformations which are non topologically weakly mixing. We first construct, for any $m>3$, uniquely ergodic interval exchange transformations of Q-rank $2$ with irrational eigenvalues associated to continuous eigenfunctions which are not topologically weakly mixing; this answers a question of Ferenczi and Zamboni [5]. Moreover we construct, for any even integer $m \geq 4$, interval exchange transformations of Q-rank $2$ with both irrational eigenvalues (associated to continuous eigenfunctions) and non trivial rational eigenvalues (associated to piecewise continuous eigenfunctions); these examples can be chosen to be either uniquely ergodic or non minimal.
In this paper, we prove a criterion for the existence of continuous non constant eigenfunctions for interval exchange transformations which are non topologically weakly mixing. We first construct, for any $m>3$, uniquely ergodic interval exchange transformations of Q-rank $2$ with irrational eigenvalues associated to continuous eigenfunctions which are not topologically weakly mixing; this answers a question of Ferenczi and Zamboni [5]. Moreover we construct, for any even integer $m \geq 4$, interval exchange transformations of Q-rank $2$ with both irrational eigenvalues (associated to continuous eigenfunctions) and non trivial rational eigenvalues (associated to piecewise continuous eigenfunctions); these examples can be chosen to be either uniquely ergodic or non minimal.
2010, 27(3): 1093-1105
doi: 10.3934/dcds.2010.27.1093
+[Abstract](2243)
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Abstract:
We consider the fourth order nonlinear Schrödinger type equation (4NLS). The first purpose is to revisit the well-posedness theory of (4NLS). In [8], [9], [20] and [21], they proved the time-local well-posedness of (4NLS) in H *(R) with $s>1/2$ by using the Fourier restriction method. In this paper we give another proof of above result by using simpler approach than the Fourier restriction method. The second purpose is to construct the exact standing wave solution to (4NLS).
We consider the fourth order nonlinear Schrödinger type equation (4NLS). The first purpose is to revisit the well-posedness theory of (4NLS). In [8], [9], [20] and [21], they proved the time-local well-posedness of (4NLS) in H *(R) with $s>1/2$ by using the Fourier restriction method. In this paper we give another proof of above result by using simpler approach than the Fourier restriction method. The second purpose is to construct the exact standing wave solution to (4NLS).
2010, 27(3): 1107-1121
doi: 10.3934/dcds.2010.27.1107
+[Abstract](1949)
+[PDF](259.1KB)
Abstract:
Let $T$ be a piecewise expanding interval map and $T_H$ be an abstract perturbation of $T$ into an interval map with a hole. Given a number ℓ, 0 < ℓ < l, we compute an upper-bound on the size of a hole needed for the existence of an absolutely continuous conditionally invariant measure (accim) with escape rate not greater than -ln(1-ℓ). The two main ingredients of our approach are Ulam's method and an abstract perturbation result of Keller and Liverani.
Let $T$ be a piecewise expanding interval map and $T_H$ be an abstract perturbation of $T$ into an interval map with a hole. Given a number ℓ, 0 < ℓ < l, we compute an upper-bound on the size of a hole needed for the existence of an absolutely continuous conditionally invariant measure (accim) with escape rate not greater than -ln(1-ℓ). The two main ingredients of our approach are Ulam's method and an abstract perturbation result of Keller and Liverani.
2010, 27(3): 1123-1132
doi: 10.3934/dcds.2010.27.1123
+[Abstract](2173)
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Abstract:
We consider the positive solutions to classes of $pq-$Laplacian semipositone systems with Dirichlet boundary conditions, in particular, we study strongly coupled reaction terms which tend to $-\infty$ at the origin and satisfy a combined sublinear condition at $\infty.$ By using the method of sub-super solutions we establish our results.
We consider the positive solutions to classes of $pq-$Laplacian semipositone systems with Dirichlet boundary conditions, in particular, we study strongly coupled reaction terms which tend to $-\infty$ at the origin and satisfy a combined sublinear condition at $\infty.$ By using the method of sub-super solutions we establish our results.
2010, 27(3): 1133-1145
doi: 10.3934/dcds.2010.27.1133
+[Abstract](2230)
+[PDF](190.5KB)
Abstract:
Let $f$ be a diffeomorphism of a compact $C^\infty$ manifold, and let $p$ be a hyperbolic periodic point of $f$. In this paper we introduce the notion of $C^1$-stable expansivity for a closed $f$-invariant set, and prove that $(i)$ the chain recurrent set $\mathcal {R}(f)$ of $f$ is $C^1$-stably expansive if and only if $f$ satisfies both Axiom A and no-cycle condition, $(ii)$ the homoclinic class $H_f(p)$ of $f$ associated to $p$ is $C^1$-stably expansive if and only if $H_f(p)$ is hyperbolic, and $(iii)$ $C^1$-generically, the homoclinic class $H_f(p)$ is $C^1$-stably expansive if and only if $H_f(p)$ is $C^1$-persistently expansive.
Let $f$ be a diffeomorphism of a compact $C^\infty$ manifold, and let $p$ be a hyperbolic periodic point of $f$. In this paper we introduce the notion of $C^1$-stable expansivity for a closed $f$-invariant set, and prove that $(i)$ the chain recurrent set $\mathcal {R}(f)$ of $f$ is $C^1$-stably expansive if and only if $f$ satisfies both Axiom A and no-cycle condition, $(ii)$ the homoclinic class $H_f(p)$ of $f$ associated to $p$ is $C^1$-stably expansive if and only if $H_f(p)$ is hyperbolic, and $(iii)$ $C^1$-generically, the homoclinic class $H_f(p)$ is $C^1$-stably expansive if and only if $H_f(p)$ is $C^1$-persistently expansive.
2010, 27(3): 1147-1158
doi: 10.3934/dcds.2010.27.1147
+[Abstract](2177)
+[PDF](269.8KB)
Abstract:
Employing a homotopy argument and the Leray-Schauder degree theory, we show the existence of rotating modes for the Frenkel-Kontorova model with periodic interaction potential. The solutions describing rotating modes are periodic and called rotating oscillating solutions, in which the phase of a fixed rotator increases by $2\pi$ per period, while its neighbors oscillate with small amplitudes around their equilibrium positions. We also discuss a fundamental difference between the Frenkel-Kontorova model with periodic interaction potential and that with convex interaction potential by demonstrating the nonexistence of the rotating modes for the latter case.
Employing a homotopy argument and the Leray-Schauder degree theory, we show the existence of rotating modes for the Frenkel-Kontorova model with periodic interaction potential. The solutions describing rotating modes are periodic and called rotating oscillating solutions, in which the phase of a fixed rotator increases by $2\pi$ per period, while its neighbors oscillate with small amplitudes around their equilibrium positions. We also discuss a fundamental difference between the Frenkel-Kontorova model with periodic interaction potential and that with convex interaction potential by demonstrating the nonexistence of the rotating modes for the latter case.
2010, 27(3): 1159-1187
doi: 10.3934/dcds.2010.27.1159
+[Abstract](2579)
+[PDF](348.7KB)
Abstract:
We study the boundary stabilization of the two-dimensional Navier-Stokes equations about an unstable stationary solution by controls of finite dimension in feedback form. The main novelty is that the linear feedback control law is determined by solving an optimal control problem of finite dimension. More precisely, we show that, to stabilize locally the Navier-Stokes equations, it is sufficient to look for a boundary feedback control of finite dimension, able to stabilize the projection of the linearized equation onto the unstable subspace of the linearized Navier-Stokes operator. The feedback operator is obtained by solving an algebraic Riccati equation in a space of finite dimension, that is to say a matrix Riccati equation.
We study the boundary stabilization of the two-dimensional Navier-Stokes equations about an unstable stationary solution by controls of finite dimension in feedback form. The main novelty is that the linear feedback control law is determined by solving an optimal control problem of finite dimension. More precisely, we show that, to stabilize locally the Navier-Stokes equations, it is sufficient to look for a boundary feedback control of finite dimension, able to stabilize the projection of the linearized equation onto the unstable subspace of the linearized Navier-Stokes operator. The feedback operator is obtained by solving an algebraic Riccati equation in a space of finite dimension, that is to say a matrix Riccati equation.
2010, 27(3): 1189-1217
doi: 10.3934/dcds.2010.27.1189
+[Abstract](2223)
+[PDF](355.1KB)
Abstract:
The spatially uniform case of the problem of quasistatic evolution in small strain associative elastoplasticity with softening is studied. Through the introdution of a viscous approximation, the problem reduces to determine the limit behaviour of the solutions of a singularly perturbed system of ODE's in a finite dimensional Banach space. We see that the limit dynamics presents, for a generic choice of the initial data, the alternation of three possible regimes (elastic regime, slow dynamics, fast dynamics), which is determined by the sign of two scalar indicators, whose explicit expression is given.
The spatially uniform case of the problem of quasistatic evolution in small strain associative elastoplasticity with softening is studied. Through the introdution of a viscous approximation, the problem reduces to determine the limit behaviour of the solutions of a singularly perturbed system of ODE's in a finite dimensional Banach space. We see that the limit dynamics presents, for a generic choice of the initial data, the alternation of three possible regimes (elastic regime, slow dynamics, fast dynamics), which is determined by the sign of two scalar indicators, whose explicit expression is given.
2010, 27(3): 1219-1231
doi: 10.3934/dcds.2010.27.1219
+[Abstract](2380)
+[PDF](180.3KB)
Abstract:
In this paper we study a skew product map $F$ preserving an ergodic measure $\mu$ of positive entropy. We show that if on the fibers the map are $C^{1+\alpha}$ diffeomorphisms with nonzero Lyapunov exponents, then $F$ has ergodic measures of arbitrary intermediate entropies. To construct these measures we find a set on which the return map is a skew product with horseshoes along fibers. We can control the average return time and show the maximal entropy of these measures can be arbitrarily close to $h_\mu(F)$.
In this paper we study a skew product map $F$ preserving an ergodic measure $\mu$ of positive entropy. We show that if on the fibers the map are $C^{1+\alpha}$ diffeomorphisms with nonzero Lyapunov exponents, then $F$ has ergodic measures of arbitrary intermediate entropies. To construct these measures we find a set on which the return map is a skew product with horseshoes along fibers. We can control the average return time and show the maximal entropy of these measures can be arbitrarily close to $h_\mu(F)$.
2010, 27(3): 1233-1239
doi: 10.3934/dcds.2010.27.1233
+[Abstract](2039)
+[PDF](139.6KB)
Abstract:
We investigate the spatial asymptotics of decaying solutions of the Toda hierarchy and show that the asymptotic behaviour is preserved by the time evolution. In particular, we show that the leading asymptotic term is time independent. Moreover, we establish infinite propagation speed for the Toda lattice.
We investigate the spatial asymptotics of decaying solutions of the Toda hierarchy and show that the asymptotic behaviour is preserved by the time evolution. In particular, we show that the leading asymptotic term is time independent. Moreover, we establish infinite propagation speed for the Toda lattice.
Homoclinic orbits for superlinear Hamiltonian systems
without Ambrosetti-Rabinowitz growth condition
2010, 27(3): 1241-1257
doi: 10.3934/dcds.2010.27.1241
+[Abstract](1934)
+[PDF](248.8KB)
Abstract:
In this paper we prove the existence of homoclinic orbits for the first order non-autonomous Hamiltonian system
In this paper we prove the existence of homoclinic orbits for the first order non-autonomous Hamiltonian system
$\dot{z}=\mathcal {J}H_{z}(t,z),$
where $H(t,z)$ depends periodically on $t$. We establish some existence results of the homoclinic orbits for weak superlinear cases. To this purpose, we apply a new linking theorem to provide bounded Palais-Samle sequences.
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