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Discrete & Continuous Dynamical Systems - A
January 2001 , Volume 7 , Issue 1
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2001, 7(1): 1-33
doi: 10.3934/dcds.2001.7.1
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Abstract:
This is a systematic study of order-preserving (or monotone) random dynamical systems which are generated by cooperative random or stochastic differential equations. Our main results concern the long-term behavior of these systems, in particular the existence of equilibria and attractors and a limit set trichotomy theorem. Several applications (models of the control of the protein synthesis in a cell, of gonorrhea infection and of symbiotic interaction in a random environment) are treated in detail.
This is a systematic study of order-preserving (or monotone) random dynamical systems which are generated by cooperative random or stochastic differential equations. Our main results concern the long-term behavior of these systems, in particular the existence of equilibria and attractors and a limit set trichotomy theorem. Several applications (models of the control of the protein synthesis in a cell, of gonorrhea infection and of symbiotic interaction in a random environment) are treated in detail.
2001, 7(1): 35-50
doi: 10.3934/dcds.2001.7.35
+[Abstract](1791)
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Abstract:
Let $F_\mu(x_1,\cdots,x_k)=(x_2,\cdots,x_k,-x_1^2+\mu x_1)$. For any $G$ in a $C^2$ neighborhood $\mathcal{U}$ of the family $F_\mu$, the point at $\infty$ is an attractor (with basin denoted by $B_\infty$), and there exists a repelling fixed point in the boundary of $B_\infty$. This gives the initial step to the study of the whole boundary of $B_\infty$ and the changes it suffers: for perturbations of $F_\mu$ with $\mu$ small, the boundary of $B_\infty$ is an invariant codimension one manifold, while for large values of $\mu$ the basin $B_\infty$ is dense and its complementary set an expanding Cantor set. The techniques developed will be applied to delay endomorphisms.
Let $F_\mu(x_1,\cdots,x_k)=(x_2,\cdots,x_k,-x_1^2+\mu x_1)$. For any $G$ in a $C^2$ neighborhood $\mathcal{U}$ of the family $F_\mu$, the point at $\infty$ is an attractor (with basin denoted by $B_\infty$), and there exists a repelling fixed point in the boundary of $B_\infty$. This gives the initial step to the study of the whole boundary of $B_\infty$ and the changes it suffers: for perturbations of $F_\mu$ with $\mu$ small, the boundary of $B_\infty$ is an invariant codimension one manifold, while for large values of $\mu$ the basin $B_\infty$ is dense and its complementary set an expanding Cantor set. The techniques developed will be applied to delay endomorphisms.
2001, 7(1): 51-60
doi: 10.3934/dcds.2001.7.51
+[Abstract](2180)
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Abstract:
In this paper, we study almost periodic (a.p.) solutions of discrete dynamical systems. We first adapt some results on a.p. differential equations to a.p. difference equations, on the link between boundedness of solutions and existence of a.p. solutions. After, we obtain an existence result in the space of the Harmonic Synthesis for an equation $A_t (x_t,...,x_{t+p})=0$ when the dependance of $A$ on $t$ is a.p. and when $A_t$ and $D A_t$ are uniformly Lipschitz and satisfy another condition which is exactly the extension of a simple one for the basic linear system. The main tools for that are Nonlinear Functional Analysis and the Newton method.
In this paper, we study almost periodic (a.p.) solutions of discrete dynamical systems. We first adapt some results on a.p. differential equations to a.p. difference equations, on the link between boundedness of solutions and existence of a.p. solutions. After, we obtain an existence result in the space of the Harmonic Synthesis for an equation $A_t (x_t,...,x_{t+p})=0$ when the dependance of $A$ on $t$ is a.p. and when $A_t$ and $D A_t$ are uniformly Lipschitz and satisfy another condition which is exactly the extension of a simple one for the basic linear system. The main tools for that are Nonlinear Functional Analysis and the Newton method.
2001, 7(1): 61-84
doi: 10.3934/dcds.2001.7.61
+[Abstract](2010)
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Abstract:
Given an analytic integrable Hamiltonian with three or more degrees of freedom, we construct, arbitrarily close to it, an analytic perturbation with transition chains whose lengths only depend on the unperturbed Hamiltonian. Then we deduce that the perturbed system has Arnold diffusion. We provide the technical details of the tools we use.
Given an analytic integrable Hamiltonian with three or more degrees of freedom, we construct, arbitrarily close to it, an analytic perturbation with transition chains whose lengths only depend on the unperturbed Hamiltonian. Then we deduce that the perturbed system has Arnold diffusion. We provide the technical details of the tools we use.
2001, 7(1): 85-90
doi: 10.3934/dcds.2001.7.85
+[Abstract](2043)
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Abstract:
Recently A. Chenciner and R. Montgomery found a remarkable periodic orbit for a three-body problem by variational methods. On this orbit all masses chase each other along a figure-eight circuit without any collision, and the solution curve is indeed a minimizer of the action functional on a properly chosen path space. One technical difficulty, where numerical integration had been used in their proof, is to show that the minimizing orbit does not experience any collision. In this paper a short analytical proof will be presented.
Recently A. Chenciner and R. Montgomery found a remarkable periodic orbit for a three-body problem by variational methods. On this orbit all masses chase each other along a figure-eight circuit without any collision, and the solution curve is indeed a minimizer of the action functional on a properly chosen path space. One technical difficulty, where numerical integration had been used in their proof, is to show that the minimizing orbit does not experience any collision. In this paper a short analytical proof will be presented.
2001, 7(1): 91-113
doi: 10.3934/dcds.2001.7.91
+[Abstract](2421)
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A dynamical system equivariant with respect to a compact symmetry group induces a system on the orbit space. This (reduced) system inherits many important features of the given one, but the drifts along the group orbits disappear. Using invariant theory the orbit space along with the reduced system can be embedded into a real vector space. We consider the Lyapunov exponents of the reduced system, and prove formulas for these in terms of the Lyapunov exponents of the given system. These formulas enable us to make predictions about the latter using only the Lyapunov exponents of the reduced system.
A dynamical system equivariant with respect to a compact symmetry group induces a system on the orbit space. This (reduced) system inherits many important features of the given one, but the drifts along the group orbits disappear. Using invariant theory the orbit space along with the reduced system can be embedded into a real vector space. We consider the Lyapunov exponents of the reduced system, and prove formulas for these in terms of the Lyapunov exponents of the given system. These formulas enable us to make predictions about the latter using only the Lyapunov exponents of the reduced system.
2001, 7(1): 115-126
doi: 10.3934/dcds.2001.7.115
+[Abstract](1792)
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Abstract:
A transformation of binary sequences that is ergodic and mixing with respect to the equidistributed measure is constructed with the help of a cellular automaton. The transformation is the basic element for a pseudorandom number generator. The ratio of the number of seeds that generate equidistributed sequences to the number of all words goes to one as the length of words is increased. The evaluation of a hardware implementation of the generator confirms the statistical behavior of sequences as determined from the ergodic properties of the mathematical model of the generator. Unpredictability under random search attacks is attained by means of three coupled transformations.
A transformation of binary sequences that is ergodic and mixing with respect to the equidistributed measure is constructed with the help of a cellular automaton. The transformation is the basic element for a pseudorandom number generator. The ratio of the number of seeds that generate equidistributed sequences to the number of all words goes to one as the length of words is increased. The evaluation of a hardware implementation of the generator confirms the statistical behavior of sequences as determined from the ergodic properties of the mathematical model of the generator. Unpredictability under random search attacks is attained by means of three coupled transformations.
2001, 7(1): 127-146
doi: 10.3934/dcds.2001.7.127
+[Abstract](1973)
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Abstract:
The discrete Allen-Cahn and Cahn-Hilliard equations are (continuous time) lattice differential equations that are analogues of two well-studied parabolic partial differential equations of materials science, and that model the evolution of, respectively, a conserved, or nonconserved, quantity, which has two preferred homogeneous phases and which avoids spatial inhomogenieties. If the interaction length $\nu$ is small, equilibrium states are scattered throughout phase space, but as $\nu$ increases most of these equilibria disappear. As they do so, the system passes through a transitional stage known as dynamical metastability, during which true equilibrium solutions are difficult to distinguish from solutions that evolve slowly but eventually traverse long distances. This paper contains results on the grain sizes of genuine equilibria that help make this distinction possible. In particular, lower bounds on grain sizes and connections between the grain sizes of adjacent grains and next-nearest neighbors are established.
The discrete Allen-Cahn and Cahn-Hilliard equations are (continuous time) lattice differential equations that are analogues of two well-studied parabolic partial differential equations of materials science, and that model the evolution of, respectively, a conserved, or nonconserved, quantity, which has two preferred homogeneous phases and which avoids spatial inhomogenieties. If the interaction length $\nu$ is small, equilibrium states are scattered throughout phase space, but as $\nu$ increases most of these equilibria disappear. As they do so, the system passes through a transitional stage known as dynamical metastability, during which true equilibrium solutions are difficult to distinguish from solutions that evolve slowly but eventually traverse long distances. This paper contains results on the grain sizes of genuine equilibria that help make this distinction possible. In particular, lower bounds on grain sizes and connections between the grain sizes of adjacent grains and next-nearest neighbors are established.
2001, 7(1): 147-154
doi: 10.3934/dcds.2001.7.147
+[Abstract](1605)
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Abstract:
Let $(f,T^n,\mu)$ be a linear hyperbolic automorphism of the $n$-torus. We show that if $A\subset T^n$ has a boundary which is a finite union of $C^1$ submanifolds which have no tangents in the stable ($E^s$) or unstable $(E^u)$ direction then the induced map on $A$, $(f_A,A,\mu_A)$ is also Bernoulli. We show that Poincáre maps for uniformly transverse $C^1$ Poincáre sections in smooth Bernoulli Anosov flows preserving a volume measure are Bernoulli if they are also transverse to the strongly stable and strongly unstable foliation.
Let $(f,T^n,\mu)$ be a linear hyperbolic automorphism of the $n$-torus. We show that if $A\subset T^n$ has a boundary which is a finite union of $C^1$ submanifolds which have no tangents in the stable ($E^s$) or unstable $(E^u)$ direction then the induced map on $A$, $(f_A,A,\mu_A)$ is also Bernoulli. We show that Poincáre maps for uniformly transverse $C^1$ Poincáre sections in smooth Bernoulli Anosov flows preserving a volume measure are Bernoulli if they are also transverse to the strongly stable and strongly unstable foliation.
2001, 7(1): 155-176
doi: 10.3934/dcds.2001.7.155
+[Abstract](2261)
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Abstract:
A normal form theory for functional differential equations in Banach spaces of retarded type is addressed. The theory is based on a formal adjoint theory for the linearized equation at an equilibrium and on the existence of center manifolds for perturbed inhomogeneous equations, established in the first part of this work under weaker hypotheses than those that usually appear in the literature. Based on these results, an algorithm to compute normal forms on finite dimensional invariant manifolds of the origin is presented. Such normal forms are important in obtaining the ordinary differential equation giving the flow on center manifolds explicitly in terms of the original functional differential equation. Applications to Bogdanov-Takens and Hopf bifurcations are presented.
A normal form theory for functional differential equations in Banach spaces of retarded type is addressed. The theory is based on a formal adjoint theory for the linearized equation at an equilibrium and on the existence of center manifolds for perturbed inhomogeneous equations, established in the first part of this work under weaker hypotheses than those that usually appear in the literature. Based on these results, an algorithm to compute normal forms on finite dimensional invariant manifolds of the origin is presented. Such normal forms are important in obtaining the ordinary differential equation giving the flow on center manifolds explicitly in terms of the original functional differential equation. Applications to Bogdanov-Takens and Hopf bifurcations are presented.
2001, 7(1): 177-202
doi: 10.3934/dcds.2001.7.177
+[Abstract](2274)
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Abstract:
We study topological and ergodic properties of some almost hyperbolic diffeomorphisms on two dimensional manifolds. Under generic conditions, diffeomorphisms obtained from Anosov by an isotopy pushing together the stable and unstable manifolds to be tangent at a fixed point, are conjugate to Anosov. For a finite codimension subset at the boundary of Anosov there exist a SRB measure and an unique ergodic attractor.
We study topological and ergodic properties of some almost hyperbolic diffeomorphisms on two dimensional manifolds. Under generic conditions, diffeomorphisms obtained from Anosov by an isotopy pushing together the stable and unstable manifolds to be tangent at a fixed point, are conjugate to Anosov. For a finite codimension subset at the boundary of Anosov there exist a SRB measure and an unique ergodic attractor.
2001, 7(1): 203-218
doi: 10.3934/dcds.2001.7.203
+[Abstract](2241)
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Abstract:
We consider the singularly perturbed system $\dot\xi = f_0(\xi) + \varepsilon f_1(\xi,\eta,\varepsilon)$, $\dot\eta = \varepsilon g(\xi,\eta,\varepsilon )$ where $\xi\in\Omega\subset\mathbb R^n$, $\eta\in\mathbb R$ and $\varepsilon\in\mathbb R$ is a small real parameter. We assume that $\dot\xi = f_{0}(\xi)$ has a non degenerate heteroclinic solution $\g(t)$ and that the Melnikov function $\int_{-\infty}^{+\infty} \psi^{*}(t) f_{1}(\g(t),\alpha,0)\dt$ has a double zero at some point $\alpha_{0}$. Using a functional analytic approach we show that if a suitable second order Melnikov function is not zero, the above system has, in a neighborhood of $\{\gamma(t)\}\times\mathbb R$, two heteroclinic orbits for $\varepsilon$ on one side of $\varepsilon=0$ and none for $\varepsilon$ on the other side. We also study the transversality of the intersection of the center-stable and the center-unstable manifolds along these orbits.
We consider the singularly perturbed system $\dot\xi = f_0(\xi) + \varepsilon f_1(\xi,\eta,\varepsilon)$, $\dot\eta = \varepsilon g(\xi,\eta,\varepsilon )$ where $\xi\in\Omega\subset\mathbb R^n$, $\eta\in\mathbb R$ and $\varepsilon\in\mathbb R$ is a small real parameter. We assume that $\dot\xi = f_{0}(\xi)$ has a non degenerate heteroclinic solution $\g(t)$ and that the Melnikov function $\int_{-\infty}^{+\infty} \psi^{*}(t) f_{1}(\g(t),\alpha,0)\dt$ has a double zero at some point $\alpha_{0}$. Using a functional analytic approach we show that if a suitable second order Melnikov function is not zero, the above system has, in a neighborhood of $\{\gamma(t)\}\times\mathbb R$, two heteroclinic orbits for $\varepsilon$ on one side of $\varepsilon=0$ and none for $\varepsilon$ on the other side. We also study the transversality of the intersection of the center-stable and the center-unstable manifolds along these orbits.
2001, 7(1): 219-235
doi: 10.3934/dcds.2001.7.219
+[Abstract](2696)
+[PDF](268.3KB)
Abstract:
This paper investigates the exact rate of convergence in Ulam's method: a well-known discretization scheme for approximating the invariant density of an absolutely continuous invariant probability measure for piecewise expanding interval maps. It is shown by example that the rate is no better than $O(\frac{\log n}{n})$, where $n$ is the number of cells in the discretization. The result is in agreement with upper estimates previously established in a number of general settings, and shows that the conjectured rate of $O(\frac{1}{n})$ cannot be obtained, even for extremely regular maps.
This paper investigates the exact rate of convergence in Ulam's method: a well-known discretization scheme for approximating the invariant density of an absolutely continuous invariant probability measure for piecewise expanding interval maps. It is shown by example that the rate is no better than $O(\frac{\log n}{n})$, where $n$ is the number of cells in the discretization. The result is in agreement with upper estimates previously established in a number of general settings, and shows that the conjectured rate of $O(\frac{1}{n})$ cannot be obtained, even for extremely regular maps.
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