Homoclinic time reversible system

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Author: Dr. Thomas Wagenknecht, Department of Applied Mathematics, University of Leeds, UK
Author: Dr. Alan R. Champneys, Department of Engineering Mathematics, University of Bristol, UK

Dr. Thomas Wagenknecht accepted the invitation on 22 April 2008 (self-imposed deadline: 22 September 2008).

Dr. Thomas Wagenknecht, Department of Applied Mathematics, University of Leeds, UK, was invited on 11 April 2008.

Dr. Alan R. Champneys accepted the invitation on 4 April 2008 (self-imposed deadline: 4 June 2008).

Homoclinic orbits of time reversible systems refers to homoclinic bifurcations of continuous time dynamical systems that are invariant under reversal of time. Typically such systems of ordinary-differential equations (ODEs) arise as steady state or travelling-wave reductions of spatially even-order partial differential equations (PDEs). Canonical examples include the soliton solutions of the KdV and NLS equations. Given higher-order spatial terms though far more complex things can occur such as characteristic snaking bifurcation diagrams or so-called embedded solitons. The theory often allows one to study what other coherent structures (either periodic or localised) exist in a neighbourhood of the homoclinic trajectory.

1 Formal definitions
- 1.1 Robust existence of symmetric homoclinics
- 1.2 An example
2 Dynamics near symmetric homoclinic orbits
3 Continuation of symmetric homoclinic orbits
4 Other topics
5 References

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Formal definitions

Consider a $2n$ -dimensional system of ODEs $\dot{x}=f(x)$ , $x \in \Bbb{R}^{2n}$ , and assume that $0$ is an equilibrium solution, i.e. $f(0)=0$ . A (non-constant) solution $x(t)$ of the equation with orbit $X$ is called homoclinic to $0$ , if $\lim_{t\to \pm \infty} x(t)=0$ .

The equation is called reversible if there exists a (linear) involution $R: \Bbb{R}^{2n} \to \Bbb{R}^{2n}$ , $R^2=id$ , such that $f(Rx)=-Rf(x)$ . If $x(t)$ is a solution of a reversible ODE, then so is $Rx(-t)$ . A solution is called symmetric, if $x(t)=Rx(-t)$ . Note that a solution is symmetric if and only if the orbit $X$ intersects the symmetry section $\mbox{Fix}\, (R):=\{x \in \Bbb{R}^{2n}: Rx=x \}$ .

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Robust existence of symmetric homoclinics

In the typical situation, we have $\dim (\mbox{Fix}\, (R))=n$ , and the equilibrium at $0$ is hyperbolic. Because of the reversibility the spectrum of the linearised vector field is symmetric with respect to the imaginary axis, such that $0$ has an $n$ -dimensional stable and $n$ -dimensional unstable manifold, respectively. A symmetric homoclinic orbit to $0$ exists if the stable manifold $W^s(0)$ intersects the symmetry section $\mbox{Fix}\, (R)$ . Hence, such orbits can exist robustly in reversible systems, i.e. they are codimension zero objects.

In this aspect (and some others), symmetric homoclinic orbits behave similar to homoclinic orbits in Hamiltonian systems.

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An example

The Korteweg de Vries (KdV) equation $\frac{\partial u}{\partial t} + \frac{\partial^3 u}{\partial x^3} + 6u \frac{\partial u}{\partial x} = 0$ describes shallow water waves in a channel. We look for travelling wave solutions $u(x,t)=U(x-ct)=U(\xi)$ with wave speed $c$ . Inserting this ansatz into the PDE we obtain the ODE $U'''+6UU'-c U'=0,$ where the prime denotes differentiation with respect to $\xi$ . This equation can be integrated once to yield the second order ODE $U''+ 3U^2-cU = 0.$ (Note that the integration constant has been set to $0$ .)

The phase portrait of this equation for $c>0$ in the $(U,U')$ is shown in panel a) of the Figure to the right. The orbits are symmetric with respect to the $U$ -axis, because of the reversibility of the equation with respect to the transformation $R:(U,U') \mapsto (U, -U')$ . In particular we find a homoclinic orbit connecting the origin to itself. The corresponding solution profile $U(\xi)$ is shown in panel (b) of the Figure. Such a solution exists for all $c>0$ .

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Dynamics near symmetric homoclinic orbits

The phase portrait above shows a one-parameter family of periodic orbits existing in a neighbourhood of the homoclinic orbit. The period of the orbits goes to infinity as the homoclinic orbit is approached, (Vanderbauwhede and Fiedler, 1992). This blue sky catastrophe is a generic feature of the dynamics near symmetric homoclinic orbits. Moreover, if $0$ is a saddle with real leading eigenvalues, then, typically, this is the whole recurrent dynamics near the homoclinic orbit.

On the other hand, if $0$ is a saddle focus equilibrium, i.e. if its leading eigenvalues are complex, then besides the periodic orbits more complex dynamics can be found near a homoclinic orbit. More precisely, infinitely many $N$ -homoclinic and $N$ -periodic orbits exist near the orbit for each $N>1$ . (Here, $N$ -homoclinic or periodic orbits refer to orbits intersecting a cross section to the primary orbit $N$ times.) These orbits can be characterised by the times they spent in their passage past the equilbrium.

[maybe include some picture of an $N$-pulse here]

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Continuation of symmetric homoclinic orbits

Since symmetric homoclinic orbits are structurally stable in the class of reversible systems, their bifurcations can be studied in one-parameter families of equations $\dot{x}=f(x,\lambda)$ , with a parameter $\lambda \in \Bbb{R}$ . Several scenarios have been discussed.

Non-transversal homoclinics

The intersection of $W^s(0)$ and $\mbox{Fix}\, (R)$ can become non-transverse at $\lambda=\lambda_0$ . Typically, a quadratic tangency will occur, leading to a fold bifurcation of symmetric homoclinic orbits (Knobloch, 1997)

Homoclinic orbits and local bifurcations

Homoclinic orbits are typically destroyed or created in local bifurcations of symmetric equilbrium solutions. Well known examples include the saddle-node or transcritical bifurcation and the reversible 1:1 resonance (Hamiltonian Hopf bifurcation). In the latter case, the equilibrium changes its type from a center into a saddle focus, and - depending on the sign of a nonlinear term in the local normal form - two symmetric homoclinic orbits emerge (Iooss and Peroueme, 1993).

Orbit flip homoclinics

There are codimension-one mechanisms that can create a complicated dynamics near real saddle homoclinic orbit. These mechanisms usually involve the global geometry of stable and unstable manifolds of the equilibrium. For example, in an orbit-flip bifurcation, where the homoclinic orbit approaches $0$ in the strong (un)stable manifold, $N$ -homoclinic orbits can be found in perturbations of the primary orbit. In contrast to the case of a saddle focus homoclinic orbit, there exists a unique $N$ -homoclinic orbit for each $N$ .

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References

Beck, M., Knobloch, J., Lloyd, D.J.B., Sandstede, B. and Wagenknecht, T. (2009). Snakes, ladders, and isolas of localised patterns. SIAM Journal on Mathematical Analysis 41, 936-972.

Champneys, A.R. (1998), Homoclinic orbits in reversible systems and their applications in mechanics, fluids and optics. Physica D 112, 158-186.

Champneys, A.R. (1999), Homoclinic orbits in reversible systems II : multi-bumps and saddle-centres. CWI Quarterly 12(3-4), 185–212.

Champneys, A. R. and Härterich, J. (2000), Cascades of homoclinic orbits to a saddle-centre for reversible and perturbed Hamiltonian systems, Dyn. Stab. Syst. 15(3), 231–252.

Devaney, R. (1976), Reversible diffeomorphisms and flows, Trans. Amer. Math. Soc. 218, 89–113.

Iooss, G. and Peroueme, M.C. (1993), Perturbed Homoclinic Solutions in Reversible 1-1 Resonance Vector Fields, J. Dif. Eq. 102, 62-88.

Knobloch, J. (1997), Bifurcation of degenerate homoclinics in reversible and conservative systems, J. Dyn. Diff. Eq. 9(3), 427–444.

Lamb, J.S.W. and Roberts, J.A.G. (1998), Time-reversal symmetry in dynamical systems: A survey, Physica D 112, 1–39.

Sandstede, B. (1997). Instability of localized buckling modes in a one-dimensional strut model. Phil. Trans. of the Royal Society of London A 355, 2083-2097.

Sandstede, B., Jones, C.K.R.T. and Alexander, J.C. (1997), Existence and stability of N-pulses on optical fibers with phase-sensitive amplifiers. Physica D 106, 167-206.

Vanderbauwhede, A. and Fiedler, B. (1992), Homoclinic period blow-up in reversible and conservative systems, ZAMP 43, 292–318.

Wagenknecht, T. and Champneys, A.R. (2003), When gap solitons become embedded solitons: a generic unfolding, Physica D 177, 50–70.

Invited by:

Dr. Eugene M. Izhikevich, Editor-in-Chief of Scholarpedia, the peer-reviewed open-access encyclopedia

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Categories: Dynamical Systems

Homoclinic time reversible system

From Scholarpedia

Contents

Formal definitions

Robust existence of symmetric homoclinics

An example

Dynamics near symmetric homoclinic orbits

Continuation of symmetric homoclinic orbits

Other topics

References

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