# Showing papers in "Journal of Dynamics and Differential Equations in 1994"

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TL;DR: In this paper, the behavior of a continuous flow in the vicinity of a closed positively invariant subset in a metric space is investigated, and the main theorem in this part in some sense generalizes previous results concerning classification of the flow near a compact invariant set in a locally compact metric space.

Abstract: In this paper, the behavior of a continuous flow in the vicinity of a closed positively .invariant subset in a metric space is investigated. The main theorem in this part in some sense generalizes previous results concerning classification of the flow near a compact invariant set in a locally compact metric space which was described by Ura-Kimura (1960) and Bhatia (1969). By applying the obtained main theorem, we are able to prove two persistence theorems. In the first one, several equivalent statements are established, which unify and generalize earlier results based on Liapunov-like functions and those about the equiyalence of weak uniform persistence and uniform persistence. The second theorem generalizes the classical uniform persistence theorems based on analysis of the flow on the boundary by relaxing point dissipativity and invariance of the boundary. Several examples are given which show that our theorems will apply to a wider rarity of ecological models.

328 citations

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TL;DR: In this article, a criterion for the linear stability of relative equilibria of the Newtonian body problem is found in the case when n−1 of the masses are small and several stable periodic orbits of the problem are presented as examples.

Abstract: A criterion for the linear stability of relative equilibria of the Newtoniann-body problem is found in the case whenn−1 of the masses are small. Several stable periodic orbits of the problem are presented as examples.

86 citations

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TL;DR: In this paper, the existence of a bi-Hamiltonian structure for a completely integrable Hamiltonian system was shown to exist in a neighborhood of an invariant torus if, and only if, the graph of the Hamiltonian function is a hypersurface of translation relative to the affine structure determined by the action variables.

Abstract: We study the geometry of completely integrable bi-Hamiltonian systems and, in particular, the existence of a bi-Hamiltonian structure for a completely integrable Hamiltonian system. We show that under some natural hypothesis, such a structure exists in a neighborhood of an invariant torus if, and only if, the graph of the Hamiltonian function is a hypersurface of translation, relative to the affine structure determined by the action variables. This generalizes a result of Brouzet for dimension four.

52 citations

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TL;DR: In this article, it was shown that a stable isolated fixed point of an orientation preserving local homeomorphism on R2 has fixed point index 1, and that a number of existence methods for producing periodic solutions of differential equations in the plane always produce unstable solutions.

Abstract: In this paper, we prove that a stable isolated fixed point of an orientation preserving local homeomorphism onR2 has fixed point index 1. We also give a number of applications to differential equations. In particular, we deduce that a number of existence methods for producing periodic solutions of differential equations in the plane always produce unstable solutions.

51 citations

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TL;DR: In this paper, the stability of linear stochastic differential delay equations in the presence of additive or multiplicative white and colored noise was studied and sufficient conditions for mean square and stochastically stability were derived.

Abstract: We study the stability of linear stochastic differential delay equations in the presence of additive or multiplicative white and colored noise. Using a stochastic analog of the second Liapunov method, sufficient conditions for mean square and stochastic stability are derived.

39 citations

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TL;DR: In this article, a structured population model of a single population having two distinct life stages is considered and the model equations, consisting of a hyperbolic partial differential equation coupled to an ordinary differential equation, can be reduced to a single, scalar functional differential equation.

Abstract: A structured population model of a single population having two distinct life stages is considered. The model equations, consisting of a hyperbolic partial differential equation coupled to an ordinary differential equation, can be reduced to a single, scalar functional differential equation. This allows us to use the well-developed dynamical systems theory for functional differential equations in order to study the dynamical system generated by the more complicated coupled system. A precise relation is established between the dynamical systems generated by each system of equations and a correspondence between their respective global attractors is made. The two systems are topologically equivalent on their respective attractors. These relationships are used to determine sharp sufficient conditions for the uniform persistence of the population.

36 citations

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TL;DR: In this article, the existence and smoothness of global center, center-stable, and center-unstable manifolds for skew-product flows is studied. Smooth invariant foliations to the center stable and center unstable manifolds are also discussed.

Abstract: We study the existence and smoothness of global center, center-stable, and center-unstable manifolds for skew-product flows. Smooth invariant foliations to the center stable and center unstable manifolds are also discussed.

34 citations

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TL;DR: In this paper, the authors studied the stability of standing waves for a nonlinear Schrodinger equation derived from the generalized Davey-Stewartson system in the elliptic-elliptic case.

Abstract: We study the stability of standing waves for a nonlinear Schrodinger equation, which derives from the generalized Davey-Stewartson system in the elliptic-elliptic case. We prove the existence of stable standing waves under certain conditions.

28 citations

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TL;DR: In this article, the structural stability of the Cahn-Hilliard equation and the phase-field equations is investigated. And the authors show that the phase field equations are topologically conjugate to a decoupled system of a linear equation of infinite dimension and an ordinary differential equation which is the reduced equation on the inertial manifold.

Abstract: In this paper, we study the structural stability of the Cahn-Hilliard equation and the phase-field equations. We show that the Cahn-Hilliard equation and the phase-field equations are topologically conjugate to a decoupled system of a linear equation of infinite dimension and an ordinary differential equation which is the reduced equation on the inertial manifold; particularly, the flow nearby hyperbolic stationary solutions is structurally stable.

27 citations

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TL;DR: In this article, the stability of stationary solutions of nonlocal reaction-diffusion equations on a bounded intervalJ of the real line with homogeneous Dirichlet boundary conditions is studied and it is shown that it is possible to have stable stationary solutions which change sign once onJ in the case of constant diffusion when the reaction term does not depend explicitly on the space variable.

Abstract: The stability of stationary solutions of nonlocal reaction-diffusion equations on a bounded intervalJ of the real line with homogeneous Dirichlet boundary conditions is studied. It is shown that it is possible to have stable stationary solutions which change sign once onJ in the case of constant diffusion when the reaction term does not depend explicitly on the space variable. The problem of the possible types of stable solutions that may exist is considered. It is also shown that Matano's result on the lap-number is still true in the case of nonlocal problems.

26 citations

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TL;DR: In this article, the existence of dichotomies for strongly continuous evolution families (U(t,s)) on Hilbert and Banach spaces has been characterized under an exponential growth condition, and the concepts of hyperbolic evolution groups and exponentially dichotomic evolution families are equivalent.

Abstract: Recently, Ben-Artzi and Gohberg [2] used the concept ofC
0-semigroups in order to characterize the existence of dichotomies for nonautonomous differential equations on ℂn. A similar task was performed by Latushkin and Stepin [11] for dichotomies of linear skew-product flows. In this paper we will useC
o-semigroups to characterize existence of dichotomies for strongly continuous evolution families (U(t,s))
t.s∃ℝ on Hilbert and Banach spaces. Under an exponential growth condition we show that the concepts of hyperbolic evolution groups and exponentially dichotomic evolution families are equivalent.

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TL;DR: The Gidas-Ni-Nirenberg theorem for negatively invariant sets was proved in this paper for positive solutions of a Dirichlet problem for a strongly nonlinear parabolic partial differential equation in a symmetric domain.

Abstract: Symmetry properties of positive solutions of a Dirichlet problem for a strongly nonlinear parabolic partial differential equation in a symmetric domainD ⊂ R
n
are considered. It is assumed that the domainD and the equation are invariant with respect to a group {Q} of transformations ofD. In examples {Q} consists of reflections or rotations. The main result of the paper is the theorem which states that any compact inC(D) negatively invariant set which consists of positive functions consists ofQ-symmetric functions. Examples of negatively invariant sets are (in autonomous case) equilibrium points, omega-limit sets, alpha-limit sets, unstable sets of invariant sets, and global attractors. Application of the main theorem to equilibrium points gives the Gidas-Ni-Nirenberg theorem. Applying the theorem to omega-limit sets, we obtain the asymptotical symmetrization property. That means that a bounded solutionu(t) asr→∞ approaches subspace of symmetric functions. One more result concerns properties of eigenfunctions of linearizations of the equations at positive equilibrium points. It is proved that all unstable eigenfunctions are symmetric.

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TL;DR: In this article, Hopf bifurcation of an example reaction-diffusion system on a square domain with Robin boundary conditions; the Brusselator equations was investigated using a smooth homotopy of boundary conditions from Neumann to Dirichlet type.

Abstract: We investigate Hopf bifurcation of an example reaction-diffusion system on a square domain with Robin boundary conditions; the Brusselator equations. By performing a smooth homotopy of boundary conditions from Neumann to Dirichlet type, we observe the creation of branches of periodic solutions with submaximal symmetry in codimension two bifurcations, although we do not fully calculate the branching behaviour. We also note that mode interactions behave generically on varying the boundary conditions. The investigation is performed using a numerical Liapunov-Schmidt reduction technique of Ashwin, Bohmer, and Mei (1994) and an analysis of Swift (1988).

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TL;DR: In this paper, the singular perturbation problem for the periodic-parabolic logistic equation with indefinite weight functions subject to Dirichlet boundary conditions at the boundary is investigated, and it is shown that the positive periodic solution tends to the periodic solution of the purely kinetic model as the diffusion coefficient goes to zero.

Abstract: In this paper we investigate the singular perturbation problem for the periodic-parabolic logistic equation with indefinite weight functions subject to Dirichlet boundary conditions at the boundary. We show that the positive periodic solution of the diffusion model tends to the periodic solution of the purely kinetic model as the diffusion coefficient goes to zero, uniformly in time on compact subsets of the domain.

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TL;DR: In this paper, a bifurcation function was constructed such that the singular system has a heteroclinic orbit if and only ifG(α, ǫ) = 0 has a solutionα=α(ǫ).

Abstract: We consider singularly perturbed systems
$$\xi = f(\xi ,\eta ,\varepsilon ),\dot \eta = \varepsilon g(\xi ,\eta ,\varepsilon )$$
, such thatξ=f(ξ, αo, 0). αo∃ℝ
m
, has a heteroclinic orbitu(t). We construct a bifurcation functionG(α, ɛ) such that the singular system has a heteroclinic orbit if and only ifG(α, ɛ)=0 has a solutionα=α(ɛ). We also apply this result to recover some theorems that have been proved using different approaches.

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CSAV

^{1}TL;DR: In this paper, it was shown that the solution semigroup is generated by the evolutionary problem and possesses a global attractor in the energy space, which attracts bounded subsets exponentially with growing time.

Abstract: We prove that the solution semigroup
$$S_t \left[ {u_0 ,v_0 } \right] = \left[ {u(t),u_t (t)} \right]$$
generated by the evolutionary problem
$$\left\{ P \right\}\left\{ \begin{gathered} u_{tt} + g(u_t ) + Lu + f(u) = 0, t \geqslant 0 \hfill \\ u(0) = u_0 , u_t (0) = \upsilon _0 \hfill \\ \end{gathered} \right.$$
possesses a global attractorA in the energy spaceE
o=V×L
2(Ω). Moreover,A is contained in a finite-dimensional inertial setA attracting bounded subsets ofE
1=D(L)×V exponentially with growing time.

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TL;DR: In this article, a simple free boundary model associated with solid combustion and some phase transition processes is analyzed and it is shown that the basic uniform front propagation is asymptotically stable against perturbations for the bifurcation parameter above the instability threshold.

Abstract: In this paper, we analyze a simple free boundary model associated with solid combustion and some phase transition processes. There is strong evidence that this “one-phase” model captures all major features of dynamical behavior of more realistic (and complicated) combustion and phase transition models. The principal results concern the dynamical behavior of the model as a bifurcation parameter (which is related to the activation energy in the case of combustion) varies. We prove that the basic uniform front propagation is asymptotically stable against perturbations for the bifurcation parameter above the instability threshold and that a Hopf bifurcation takes place at the threshold value. Results of numerical simulations are presented which confirm that both supercritical and subcritical Hofp bifurcation may occur for physically reasonable nonlinear kinetic functions.

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TL;DR: In this paper, a new construction of exponential attractors based on the control of Lyapunov exponents over a compact, invariant set is presented, and the fractal dimension estimate of the exponential attractor thus obtained is of the same order as the one for global attractors estimated through Lyapuna exponents.

Abstract: In this paper we present a new construction of exponential attractors based on the control of Lyapunov exponents over a compact, invariant set. The fractal dimension estimate of the exponential attractor thus obtained is of the same order as the one for global attractors estimated through Lyapunov exponents. We discuss various applications to Navier-Stokes systems.

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TL;DR: In this paper, the role played by the Schwarzian derivative in controlling the convergence of Newton's map to a super stable fixed point was discussed, and it was shown that the role of Schwarzian derivatives in controlling Newton's convergence to a fixed point with orders of convergence greater than 2 was discussed.

Abstract: We discuss Newton's method with respect to obtaining convergence to a fixed point with orders of convergence greater than 2. We identify the role played by the Schwarzian derivative in controlling the convergence of Newton's map to a super stable fixed point.

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TL;DR: In this paper, the fundamental wave manifold for systems of two conservation laws with quadratic flux functions was constructed explicitly and the singular set of the shock foliation for this manifold was described.

Abstract: We construct explicitly the fundamental wave manifold for systems of two conservation laws with quadratic flux functions. We describe the shock foliation for this manifold, as well as the singular set of the foliation. We subdivide the manifold into regions where the shock curves form trivial foliations. Sonic surfaces are identified as well. We establish the stability of shock curves underC
3 perturbations of the flux functions in the Whitney topology.

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TL;DR: In this article, the generalized geometric entropy criterion for admissibility of shocks in systems which change type was validated by a nonlinear inviscid perturbation, where the real part of a complex characteristic replaces the characteristic speed itself.

Abstract: In this paper we validate the generalized geometric entropy criterion for admissibility of shocks in systems which change type. This condition states that a shock between a state in a hyperbolic region and one in a nonhyperbolic region is admissible if the Lax geometric entropy criterion, based on the number of characteristics entering the shock, holds, where now the real part of a complex characteristic replaces the characteristic speed itself. We test this criterion by a nonlinear inviscid perturbation. We prove that the perturbed Cauchy problem in the elliptic region has a solution for a uniform time if the data lie in a suitable class of analytic functions and show that under small perturbations of the data a perturbed shock and a perturbed solution in the hyperbolic region exist, also for a uniform time.

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TL;DR: In this article, it was shown that every time-periodic solution with a period smaller than ε is necessarily a stationary solution, and an explicit formula for ǫ is also provided.

Abstract: In the case of the 3D Navier-Stokes equations, it is proved that there exists a constantɛ>0 with the following property: Every time-periodic solution with a period smaller thanɛ is necessarily a stationary solution. An explicit formula for ɛ is also provided.

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TL;DR: The phase dynamics method has been used to understand in a heuristic way the stability of periodic patterns and the dynamics of slow relaxation to periodic patterns as discussed by the authors, and a rigorous mathematical foundation of the phase dynamic method through some simple model equations.

Abstract: The phase dynamics method has been used to understand in a heuristic way the stability of periodic patterns and the dynamics of slow relaxation to periodic patterns. We attempt to give a rigorous mathematical foundation of the phase dynamics method through some simple model equations.

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TL;DR: In this article, the Dirichlet boundary conditions for the reduced family of equations with finite-dimensional invariant manifolds of solutions were studied and the existence of transversal homoclinic orbits to periodic solutions and invariant sets with chaotic dynamics was proved.

Abstract: We consider the parametrized family of equations∂
tt
,u-∂
xx
u-au+∥u∥
2
2α
u=O,x∃(0,πL), with Dirichlet boundary conditions. This equation has finite-dimensional invariant manifolds of solutions. Studying the reduced equation to a four-dimensional manifold, we prove the existence of transversal homoclinic orbits to periodic solutions and of invariant sets with “chaotic” dynamics, provided that α=2, 3, 4,.... For α=1 we prove the existence of infinitely many first integrals pairwise in involution.

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TL;DR: In this article, it was shown that for certain parameter values b, the full attracting sets of the annulus maps described by Levi can be realized as inverse limits of circles, and that the map is topologically conjugate to the shift homeomorphism on the inverse limit space.

Abstract: In “Qualitative Analysis of the Periodically Forced Relaxation Oscillations,” Mark Levi (Mem. Am. Math. Soc. 32, No. 244, July 1981) gives a nice geometric description of annulus maps associated with the first return map for the forced van der Pol equation ex+Φ(x)x+ex=bp(t). In this paper, it is shown that for certain parameter valuesb, the full attracting sets of the annulus maps described by Levi can be realized as inverse limits of circles. Furthermore, we show that the annulus map is topologically conjugate to the shift homeomorphism on the inverse limit space.

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TL;DR: In this article, a squeezing property in L2(Ω) is established for orbits of the semigroup associated with the equations of motion of a nonlinear incompressible bipolar viscous fluid; it is assumed thatΩ=[0,L] n,n=2 or 3,L>0, and that the velocity vector satisfies a spatial periodicity condition.

Abstract: A squeezing property inL2(Ω) is established for orbits of the semigroup associated with the equations of motion of a nonlinear incompressible bipolar viscous fluid; it is assumed thatΩ=[0,L] n ,n=2 or 3,L>0, and that the velocity vector satisfies a spatial periodicity condition The proof depends, in an essential manner, on key estimates for both the nonlinear operator generated by the nonlinear viscosity term in the model and the time integral of theH3(Ω) norm of the velocity

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TL;DR: In this paper, the hyperbolicity of linear skew-product flows with infinite-dimensional fibers over a dynamical system on a compact metric space is described in terms of the existence and uniqueness of Green's function and the spectra for family of the semigroups of weighted composition operators acting in L 2(R;E) and parametrized by the points of the points.

Abstract: The hyperbolicity of linear skew-product flows with infinite-dimensional fibersE over a dynamical system on a compact metric spaceX is described in terms of the existence and uniqueness of Green's function and in terms of the spectra for family of the semigroups of weighted composition operators acting inL
2(R;E) and parametrized by the points ofX

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TL;DR: In this paper, the authors considered the Burgers equation with a nonhomogeneous drift term, in the limit of small dissipation, and derived a formal derivation of the dynamics on this manifold, in terms of a system of ordinary differential equations.

Abstract: We consider the Burgers equation with a nonhomogeneous drift term, in the limit of small dissipation. A finite-dimensional manifold of slowly varying shock-like solutions is described, and a formal derivation of the dynamics on this manifold, in terms of a system of ordinary differential equations, is given. We also discuss the interpretation of the stationary solutions to the Burgers equation imbedded on the slow manifold.

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TL;DR: In this paper, the authors consider a dynamical system on M(n) defined by the vector field V(X), X*, X, X], (X ∃ M(m)) and show that the global behavior of X(t) ast → ∞.

Abstract: LetM(n) be the algebra of alln×n complex matrices. We consider a dynamical system onM(n) defined by the vector fieldV(X)=[[X*,X],X], (X ∃ M(n)). It arises as the gradient flow for two kinds of variational problems onM(n). Given anyX0∃ M(n), letX(t) be the trajectory starting atX0. We study the global behavior ofX(t) ast → ∞. We show that, ifX0 is semisimple, thenX(t) converges exponentially to a normal matrix. IfX0 is not semisimple, then the behavior ofX(t) is completely different and difficult to analyze. We give some results also in this case. Furthermore, we discuss about a center manifold approach to our dynamical system.