Results 1  10
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73
The Fredholm Alternative for Functional Differential Equations of Mixed Type
 J. Dynam. Differential Equations
, 1999
"... We prove a Fredholm alternative theorem for a class of asymptotically hyperbolic linear differential difference equations of mixed type. We also establish the cocycle property and the spectral flow property for such equations, providing an effective means of calculating the Fredholm index. Such syst ..."
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Cited by 67 (3 self)
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We prove a Fredholm alternative theorem for a class of asymptotically hyperbolic linear differential difference equations of mixed type. We also establish the cocycle property and the spectral flow property for such equations, providing an effective means of calculating the Fredholm index. Such systems can arise from equations which describe traveling waves in a spatial lattice.
Traveling wave solutions for systems of ODE's on a twodimensional spatial lattice
 SIAM J. APPL. MATH
, 1999
"... We consider infinite systems of ODE's on the twodimensional integer lattice, given by a bistable scalar ODE at each point, with a nearest neighbor coupling between lattice points. For a class of ideal nonlinearities, we obtain traveling wave solutions in each direction e i`, and we explore the ..."
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Cited by 61 (11 self)
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We consider infinite systems of ODE's on the twodimensional integer lattice, given by a bistable scalar ODE at each point, with a nearest neighbor coupling between lattice points. For a class of ideal nonlinearities, we obtain traveling wave solutions in each direction e i`, and we explore the relation between the wave speed c, the angle `, and the detuning parameter a of the nonlinearity. Of particular interest is the phenomenon of &quot;propagation failure, &quot; and we study how the critical value a = a
Traveling Waves in Lattice Dynamical Systems
"... In this paper, we study the existence and stability of traveling waves in lattice dynamical systems, in particular, in lattice ordinary differential equations (lattice ODE's) and in coupled map lattices (CML's). Instead of employing the moving coordinate approach as for partial differentia ..."
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Cited by 60 (2 self)
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In this paper, we study the existence and stability of traveling waves in lattice dynamical systems, in particular, in lattice ordinary differential equations (lattice ODE's) and in coupled map lattices (CML's). Instead of employing the moving coordinate approach as for partial differential equations, we construct a local coordinate system around a traveling wave solution of a lattice ODE, analogous to the local coordinate system around a periodic solution of an ODE. In this coordinate system the lattice ODE becomes a nonautonomous periodic differential equation, and the traveling wave corresponds to a periodic solution of this equation. We prove the asymptotic stability with asymptotic phase shift of the traveling wave solution under appropriate spectral conditions. We also show the existence of traveling waves in CML's which arise as timediscretizations of lattice ODE's. Finally, we show that these results apply to the discrete Nagumo equation. 1 Introduction This paper is concer...
Exponential dichotomies and WienerHopf factorizations for mixedtype functional differential equations
, 2001
"... We study linear systems of functional differential equations of mixed type, both autonomous and (asymptotically hyperbolic) nonautonomous. Such equations arise naturally in various contexts, for example, in lattice differential equations. We obtain a decomposition of the state space into stable and ..."
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Cited by 27 (0 self)
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We study linear systems of functional differential equations of mixed type, both autonomous and (asymptotically hyperbolic) nonautonomous. Such equations arise naturally in various contexts, for example, in lattice differential equations. We obtain a decomposition of the state space into stable and unstable subspaces with associated semigroups or evolutionary processes. In the autonomous case we additionally obtain representations of the semigroups in terms of retarded and advanced equations. We also obtain a factorization of the characteristic function, analogous to a WienerHopf factorization, with which we define an integer invariant for the system. Finally, we study the boundary value problem on intervals of long but finite length in the spirit of the finite section method.
Morse theory on spaces of braids and Lagrangian dynamics
 IN PREPARATION
, 2002
"... In the first half of the paper we construct a Morsetype theory on certain spaces of braid diagrams. We define a topological invariant of closed positive braids which is correlated with the existence of invariant sets of parabolic flows defined on discretized braid spaces. Parabolic flows, a type of ..."
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Cited by 22 (13 self)
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In the first half of the paper we construct a Morsetype theory on certain spaces of braid diagrams. We define a topological invariant of closed positive braids which is correlated with the existence of invariant sets of parabolic flows defined on discretized braid spaces. Parabolic flows, a type of onedimensional lattice dynamics, evolve singular braid diagrams in such a way as to decrease their topological complexity; algebraic lengths decrease monotonically. This topological invariant is derived from a MorseConley homotopy index. In the second half of the paper we apply this technology to second order Lagrangians via a discrete formulation of the variational problem. This culminates in a very general forcing theorem for the existence of infinitely many braid classes of closed orbits.
Analysis and Computation of Traveling Wave Solutions of Bistable DifferentialDifference Equations
 Nonlinearity
, 1999
"... We consider traveling wave solutions to a class of differentialdifference equations. Our interest is in understanding propagation failure, directional dependence due to the discrete Laplacian, and the relationship between traveling wave solutions of the spatially continuous and spatially discrete ..."
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Cited by 17 (7 self)
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We consider traveling wave solutions to a class of differentialdifference equations. Our interest is in understanding propagation failure, directional dependence due to the discrete Laplacian, and the relationship between traveling wave solutions of the spatially continuous and spatially discrete limits of this equation. The differentialdifference equations that we study include damped and undamped nonlinear wave and reactiondiffusion equations as well as their spatially discrete counterparts. Both analytical and numerical results are given.
Uniqueness and asymptotics of traveling waves of monostable dynamics on lattices
 SIAM J. Math. Anal
"... Abstract. Established here is the uniquenes of solutions for the traveling wave problem cU ′(x) = U(x+1)+U(x−1)−2U(x)+f(U(x)), x ∈ R, under the monostable nonlinearity: f ∈ C1([0, 1]), f(0) = f(1) = 0 < f(s) ∀ s ∈ (0, 1). Asymptotic expansions for U(x) as x → ±∞, accurate enough to capture the ..."
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Cited by 16 (3 self)
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Abstract. Established here is the uniquenes of solutions for the traveling wave problem cU ′(x) = U(x+1)+U(x−1)−2U(x)+f(U(x)), x ∈ R, under the monostable nonlinearity: f ∈ C1([0, 1]), f(0) = f(1) = 0 < f(s) ∀ s ∈ (0, 1). Asymptotic expansions for U(x) as x → ±∞, accurate enough to capture the translation differences, are also derived and rigorously verified. These results complement earlier existence and partial uniqueness/stability results in the literature. New tools are also developed to deal with the degenerate case f ′(0)f ′(1) = 0, about which is the main concern of this article.
Crystallographic Pinning: Direction Dependent Pinning in Lattice Differential Equations
, 2001
"... We study dynamical phenomena for a class of lattice differential equations, namely infinite systems of ordinary differential equations coordinatized by points on a spatial lattice. We examine in particular the dependence of traveling wave solutions on the direction of motion of the traveling wave. T ..."
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Cited by 10 (1 self)
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We study dynamical phenomena for a class of lattice differential equations, namely infinite systems of ordinary differential equations coordinatized by points on a spatial lattice. We examine in particular the dependence of traveling wave solutions on the direction of motion of the traveling wave. The phenomenon of crystallographic pinning occurs when there is a tendency for a wave to become pinned in selected directions. In previous work with J.W. Cahn and E.S. Van Vleck we demonstrated this phenomenon for a special class of systems with piecewise linear nonlinearities. In the present work we show how this phenomenon holds for a general class of systems with smooth nonlinearities and how it follows from general principles of dynamical systems.
Computation of mixed type functional differential bundary value problems
 SIADS
"... Abstract. We study boundary value dierentialdierence equations where the dierence terms may contain both advances and delays. Special attention is paid to connecting orbits, in particular to the modeling of the tails after truncation to a nite interval, and we reformulate these problems as function ..."
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Cited by 9 (1 self)
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Abstract. We study boundary value dierentialdierence equations where the dierence terms may contain both advances and delays. Special attention is paid to connecting orbits, in particular to the modeling of the tails after truncation to a nite interval, and we reformulate these problems as functional dierential equations over a bounded domain. Connecting orbits are computed for several such problems including discrete Nagumo equations, an Ising model and FrenkelKontorova type equations. We describe the collocation boundary value problem code used to compute these solutions, and the numerical analysis issues which arise, including linear algebra, boundary functions and conditions, and convergence theory for the collocation approximation on nite intervals.