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Global Pathfollowing of Homoclinic Orbits in TwoParameter Flows
 in Pitman Res. Notes Math
, 1996
"... The main goal of this paper is a global continuation theorem for homoclinic solutions of autonomous ordinary differential equations with two real parameters. In oneparameter flows, Hopf bifurcation serves as a starting point for global paths of periodic orbits. Bpoints, alias Arnol'dBogdanovTake ..."
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The main goal of this paper is a global continuation theorem for homoclinic solutions of autonomous ordinary differential equations with two real parameters. In oneparameter flows, Hopf bifurcation serves as a starting point for global paths of periodic orbits. Bpoints, alias Arnol'dBogdanovTakens points, play an analogous role for paths of homoclinic orbits in twoparameter flows. In fact, a path of homoclinic orbits emanating from a Bpoint can be continued in phase space until it terminates at another Bpoint, or becomes unbounded, or approaches a region with chaotic dynamics. This result is obtained via a new topological invariant for homoclinic orbits, based on approximation of the homoclinic orbit by nearby periodic orbits. Several local bifurcation results for homoclinic and heteroclinic orbits are reviewed, along the way, to illustrate scope, significance, and limitations of the global approach. The paper concludes with an extensive discussion, including "nongeneric" aspec...
Stability Of Pulses In Nonlinear Optical Fibers Using PhaseSensitive Amplifiers
 SIAM J. Appl. Math
, 1996
"... . We consider the stability of solitonlike pulses propagating in nonlinear optical fibers with periodicallyspaced phasesensitive amplifiers, a situation where the averaged pulse evolution is governed by a fourthorder nonlinear diffusion equation similar to the KuramotoSivashinsky or SwiftHohenb ..."
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. We consider the stability of solitonlike pulses propagating in nonlinear optical fibers with periodicallyspaced phasesensitive amplifiers, a situation where the averaged pulse evolution is governed by a fourthorder nonlinear diffusion equation similar to the KuramotoSivashinsky or SwiftHohenberg equations. A bifurcation and stability analysis of this averagedequation is carried out, and in the limit of small amplifier spacing, a steadystate pulse solution is shown to be asymptotically stable. Furthermore, both a saddlenode bifurcation and a subcritical bifurcation from the zero solution are found. Analytical results are confirmed using the bifurcation software package auto. The analysis provides evidence for the existence of stable pulse solutions for a wide range of parameter values, including those corresponding to physically realizable soliton communications systems. Key words. solitons, nonlinear optical pulse propagation, optical fibers, bifurcation theory AMS subject c...
Collocation Methods for Boundary Value Problems with an Essential Singularity
 IN LARGESCALE SCIENTI COMPUTING
, 2004
"... We investigate collocation methods for the efficient solution of singular boundary value problems with an essential singularity. We give numerical evidence that this approach indeed yields high order solutions. Moreover, ..."
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Cited by 5 (5 self)
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We investigate collocation methods for the efficient solution of singular boundary value problems with an essential singularity. We give numerical evidence that this approach indeed yields high order solutions. Moreover,
A Numerical Bifurcation Function For Homoclinic Orbits
 SIAM J. Numer. Anal
, 1998
"... . We present a numerical method to locate periodic orbits near homoclinic orbits. Using a method of X.B. Lin and solutions of the adjoint variational equation, we get a bifurcation function for periodic orbits whose period is asymptotic to infinity on approaching a homoclinic orbit. As a bonus, a l ..."
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. We present a numerical method to locate periodic orbits near homoclinic orbits. Using a method of X.B. Lin and solutions of the adjoint variational equation, we get a bifurcation function for periodic orbits whose period is asymptotic to infinity on approaching a homoclinic orbit. As a bonus, a linear predictor for continuation of the homoclinic orbit is easily available. Numerical approximations of the homoclinic orbit and the solutions of the adjoint variational equation are discussed. In particular, we consider a context in which the effects of continuous symmetries of equations can be incorporated. Applying the method to an ordinary differential equation on R 3 studied by Freire et al. we show the bifurcation function can give good agreement with pathfollowed solutions even down to low period. As an example of an application to a parabolic partial differential equation, we examine the bifurcation function for a homoclinic orbit in the KuramotoSivashinsky equation. Key wor...
Difference schemes for nonlinear BVPs on the halfaxis
"... The scalar boundary value problem (BVP) d2u dx2 − m2u = −f (x, u) , x ∈ (0, ∞), u (0) = µ1, lim u (x) = 0, x→∞ for a second order differential equation on the infinite interval [0, ∞) is considered. Under some natural assumptions it is shown that on an arbitrary finite grid there exists a unique t ..."
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The scalar boundary value problem (BVP) d2u dx2 − m2u = −f (x, u) , x ∈ (0, ∞), u (0) = µ1, lim u (x) = 0, x→∞ for a second order differential equation on the infinite interval [0, ∞) is considered. Under some natural assumptions it is shown that on an arbitrary finite grid there exists a unique threepoint exact difference scheme (EDS), i.e., a difference scheme of which the solution coincides with the projection of the exact solution of the given differential equation onto the underlying grid. A constructive method is proposed to derive from the EDS a socalled truncated difference scheme (nTDS) of rank n, where n is a freely selectable natural number and [·] denotes the entire part of the expression in brackets. The nTDS has the order of accuracy ¯n = 2[(n+1)/2], i.e., the global error is of the form O(h  ¯n), where h  is the maximum step size. The nTDS is represented by a system of nonlinear algebraic equations for the approximate values of the exact solution on the grid. Iterative methods for its numerical solution are discussed. The theoretical and practical results are used to develop a new algorithm which has all the advantages known from the modern IVPsolvers. Numerical examples are given which illustrate the theorems presented in the paper and demonstrate the reliability of the
THREEPOINT DIFFERENCE SCHEMES OF VARIABLE ORDER FOR NONLINEAR BVPS ON THE HALFAXIS
"... Abstract. The scalar BVP d 2 u dx 2 − m2 u = −f (x, u) , x ∈ (0, ∞), u (0) = µ1, lim u (x) = 0, x→∞ on the infinite interval [0, ∞) is considered. Under some natural assumptions it is shown that on an arbitrary finite grid there exists a unique threepoint exact difference scheme (EDS), i.e., a di ..."
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Abstract. The scalar BVP d 2 u dx 2 − m2 u = −f (x, u) , x ∈ (0, ∞), u (0) = µ1, lim u (x) = 0, x→∞ on the infinite interval [0, ∞) is considered. Under some natural assumptions it is shown that on an arbitrary finite grid there exists a unique threepoint exact difference scheme (EDS), i.e., a difference scheme of which the solution coincides with the projection onto the grid of the exact solution of the corresponding differential equation. A constructive algorithm is proposed to derive from the EDS a socalled truncated difference scheme (TDS) of a given rank ¯n = 2[(n + 1)/2], provided that the righthand side possesses n continuous derivatives between a finite number of discontinuity points. Here [·] denotes the entire part of the expression in brackets. The ¯nTDS possesses the order of accuracy O(h  ¯n) w.r.t. the maximal step size h. The ¯nTDS is represented by a system of nonlinear algebraic equations for the approximate values of the exact solution on the grid. Iterative methods for its numerical solution are discussed. Numerical examples are given which illustrate the theorems proved. 1.
New methods for nonlinear BVPs on the halfaxis using RungeKutta IVPsolvers I.P. GAVRILYUK ∗
"... The scalar BVP d 2 u dx2 − m2u = −f (x, u) , x ∈ (0, ∞), u (0) = µ1, lim u (x) = 0, x→∞ on the infinite interval [0, ∞) is considered. Under some natural assumptions it is shown that on an arbitrary finite grid there exists a unique threepoint exact difference scheme (EDS), i.e., a difference sch ..."
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The scalar BVP d 2 u dx2 − m2u = −f (x, u) , x ∈ (0, ∞), u (0) = µ1, lim u (x) = 0, x→∞ on the infinite interval [0, ∞) is considered. Under some natural assumptions it is shown that on an arbitrary finite grid there exists a unique threepoint exact difference scheme (EDS), i.e., a difference scheme of which the solution coincides with the projection onto the grid of the exact solution of the corresponding differential equation. A constructive algorithm is proposed to derive from the EDS a socalled truncated difference scheme (TDS) of a given rank ¯n = 2[(n + 1)/2], provided that the righthand side possesses n continuous derivatives between a finite number of discontinuity points. Here [·] denotes the entire part of the expression in brackets. The ¯nTDS possesses the order of accuracy O(h  ¯n) w.r.t. the maximal step size h. The ¯nTDS is represented by a system of nonlinear algebraic equations for the approximate values of the exact solution on the grid. Iterative methods for its numerical solution are discussed. Numerical examples are given which illustrate the theorems proved.
POSITIVE SOLUTIONS FOR FIRST ORDER NONLINEAR FUNCTIONAL BOUNDARY VALUE PROBLEMS ON INFINITE INTERVALS
"... In this paper we study a boundary value problem for a first order functional differential equation on an infinite interval. Using fixed point theorems on appropriate cones in Banach spaces, we derive multiple positive solutions for our boundary value problem. 1. ..."
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In this paper we study a boundary value problem for a first order functional differential equation on an infinite interval. Using fixed point theorems on appropriate cones in Banach spaces, we derive multiple positive solutions for our boundary value problem. 1.