Bifurcation of a reversible Hamiltonian system from a fixed point with fourfold eigenvalue zero

Wagenknecht, Thomas

doi:10.1080/14689360110089831

Cited by 5 publications

(17 citation statements)

References 19 publications

(33 reference statements)

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“…and (14), (15) into the equation for w in system (12), and equating coefficients as above, yields the following results:…”

Section: A Inner Zone Analysismentioning

confidence: 99%

“…It has been studied in detail by a number of authors, see the reviews [11,12]. The case we are interested in, with ε > 0, is generally classified as the Consequently, former studies of normal forms near equilibria with fourfold eigenvalue zero in [14,15] do not discuss the accumulation of ESs observed in Eqs. (12).…”

Section: The Two-wave System As a Normal Formmentioning

confidence: 99%

“…(15). This can be done using the known connection formula [16] of the WKB approximation, with the result…”

Section: A Inner Zone Analysismentioning

confidence: 99%

“…Moreover, w 0 (x) = 1 leads to a divergence in Eq. (15), which has to be compensated by setting sin(φ(x)) = 0. Obviously, there are two values of x for which w 0 (x) − 1 vanishes.…”

Section: A Inner Zone Analysismentioning

confidence: 99%

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Accumulation of embedded solitons in systems with quadratic nonlinearity

Malomed

Wagenknecht

Champneys

et al. 2005

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Previous numerical studies have revealed the existence of embedded solitons (ESs) in a class of multi-wave systems with quadratic nonlinearity, families of which seem to emerge from a critical point in the parameter space, where the zero solution has a fourfold zero eigenvalue. In this paper, the existence of such solutions is studied in a three-wave model.An appropriate rescaling casts the system in a normal form, which is universal for models supporting ESs through quadratic nonlinearities. The normal-form system contains a single irreducible parameter ε, and is tantamount to the basic model of type-I second-harmonic generation. An analytical approximation of WKB type yields an asymptotic formula for the distribution of discrete values of ε at which the ESs exist. Comparison with numerical results shows that the asymptotic formula yields an exact value of the scaling index, −6/5, and a fairly good approximation for the numerical factor in front of the scaling term. In this work, we resolve these issues, by developing an asymptotic analysis and verifying the predictions against numerical results. As a result, we derive a universal asymptotic approximation (normal form) for models with quadratic nonlinearity that support embedded solitons. The normal form amounts to a system of two second-order equations, which is well known as a basic model for ordinary solitons in second-harmonic-generating systems. There was a rather common belief that all soliton solutions had 3 Accumulation of embedded solitons been found in this much-studied system. Nevertheless, in this work we are able to predict the existence of an infinite series of previously unknown embedded solitons in the model. This is done in an analytical form, by realizing that the embedded solitons feature a broad inner zone, the solution in which must be matched to exponentially decaying ones in outer zones. Further, a method (Bohr-Sommerfeld quantization rule) borrowed from quantum mechanics is applied to the solution in the inner zone. The most essential part of the eventual analytical result is an asymptotic distribution law for values of the single control parameter of the normal-form system at which the embedded solitons exist. Comparison with numerical results clearly shows that the analysis has produced an exact value of the respective scaling index.

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“…and (14), (15) into the equation for w in system (12), and equating coefficients as above, yields the following results:…”

Section: A Inner Zone Analysismentioning

confidence: 99%

Section: The Two-wave System As a Normal Formmentioning

confidence: 99%

“…(15). This can be done using the known connection formula [16] of the WKB approximation, with the result…”

Section: A Inner Zone Analysismentioning

confidence: 99%

“…Moreover, w 0 (x) = 1 leads to a divergence in Eq. (15), which has to be compensated by setting sin(φ(x)) = 0. Obviously, there are two values of x for which w 0 (x) − 1 vanishes.…”

Section: A Inner Zone Analysismentioning

confidence: 99%

See 2 more Smart Citations

Accumulation of embedded solitons in systems with quadratic nonlinearity

Malomed

Wagenknecht

Champneys

et al. 2005

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…In the survey [9], P. Rabinowitz, who has given fundamental contributions to this field, presents the main results obtained in the last twenty years, describes some methods and lists some open problems. Among the previous studies of heteroclinic orbits are those of [2][3][4]6,8,10,12,13]. Homoclinic solutions are considered for example in [1,5,7,11,13].…”

Section: Introductionmentioning

confidence: 99%

Heteroclinic solutions for a class of the second order Hamiltonian systems

Izydorek

Janczewska

2007

Journal of Differential Equations

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We shall be concerned with the existence of heteroclinic orbits for the second order Hamiltonian system q + V q (t, q) = 0, where q ∈ R n and V ∈ C 1 (R × R n , R), V 0. We will assume that V and a certain subset M ⊂ R n satisfy the following conditions. M is a set of isolated points and #M 2. For every sufficiently small ε > 0 there exists δ > 0 such that for all (t, z) ∈ R×R n , if d(z, M) ε then −V (t, z) δ. The integrals ∞ −∞ −V (t, z) dt, z ∈ M, are equi-bounded and −V (t, z) → ∞, as |t| → ∞, uniformly on compact subsets of R n \ M. Our result states that each point in M is joined to another point in M by a solution of our system.

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About a homoclinic pitchfork bifurcation in reversible systems with additional ₂-symmetry

Wagenknecht

2002

Nonlinearity

Self Cite

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This paper studies bifurcations from a homoclinic orbit to a degenerate fixed point. We consider reversible Z 2 -symmetric systems of ordinary differential equations (ODEs) and assume the existence of a symmetric homoclinic orbit to a fixed point which itself undergoes a pitchfork bifurcation. We are interested in bifurcations from the primary homoclinic orbit in an unfolding of the degenerate situation.The studies are motivated by numerical investigations on a model-system of second-order ODEs. There one finds a similar behaviour in the local and the global bifurcation. While locally two new fixed points are created, numerical computations show that at the same time two homoclinic orbits to these fixed points bifurcate from the primary orbit. We call the global scenario a reversible homoclinic pitchfork bifurcation.An analysis of this homoclinic bifurcation is performed in a general frame. Depending on the sign of a higher order coefficient in the normal form we distinguish two cases of the local pitchfork bifurcation: the eye case (which is the one encountered in the model-system) and the figure-eight case. Adopting Lin's method to the non-hyperbolic situation the bifurcation of one-homoclinic orbits to the local centre manifold of the fixed point is investigated. Rigorous existence results for homoclinic orbits to fixed points and periodic orbits are derived. The global bifurcation picture is found to depend crucially on the local bifurcation.

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Bifurcation of a reversible Hamiltonian system from a fixed point with fourfold eigenvalue zero

Cited by 5 publications

References 19 publications

Accumulation of embedded solitons in systems with quadratic nonlinearity

Accumulation of embedded solitons in systems with quadratic nonlinearity

Heteroclinic solutions for a class of the second order Hamiltonian systems

About a homoclinic pitchfork bifurcation in reversible systems with additional ₂-symmetry

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Bifurcation of a reversible Hamiltonian system from a fixed point with fourfold eigenvalue zero

Cited by 5 publications

References 19 publications

Accumulation of embedded solitons in systems with quadratic nonlinearity

Accumulation of embedded solitons in systems with quadratic nonlinearity

Heteroclinic solutions for a class of the second order Hamiltonian systems

About a homoclinic pitchfork bifurcation in reversible systems with additional 2-symmetry

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Product

Resources

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About a homoclinic pitchfork bifurcation in reversible systems with additional ₂-symmetry