User:Anonymous/Proposed/Homoclinic orbits of time-reversible systems
Homoclinic orbits of time reversible systems refers to homoclinic bifurcations of continuous time dynamical systems that are invariant under reversal of time. Often 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.
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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 the corresponding orbit \(X\) is mapped onto itself by \(R\ .\) Note that a solution is symmetric if and only if there exists a time \(\tau\ ,\) such that \(x(\tau)=Rx(\tau)\ ,\) that is, if and only if the orbit \(X\) intersects the symmetry section \(\mbox{Fix}\, (R):=\{x \in \Bbb{R}^{2n}: Rx=x \}\) (Vanderbauwhede and Fiedler 1992; Lamb and Roberts 1998; Champneys 1998).
Robust existence of symmetric homoclinic orbits
The majority of studies of homoclinic orbits in reversible systems concerns orbits homoclinic to a hyperbolic equilibrium. In this case, the symmetry section necessarily has dimension \(\dim (\mbox{Fix}\, (R))=n\ .\) Moreover, because of the reversibility, the spectrum of the linearized vector field is symmetric with respect to the imaginary axis. This means that \(0\) has an \(n\)-dimensional stable manifold \(W^s(0)\) and an \(n\)-dimensional unstable manifold \(W^u(0)\ ,\) respectively, and we have \(W^s(0)=RW^u(0)\ .\) A symmetric homoclinic orbit to \(0\) exists if the stable manifold \(W^s(0)\) intersects the symmetry section \(\mbox{Fix}\, (R)\ .\) This intersection can be transverse, and 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.
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 \(c>0\ .\)
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 period blow-up 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 equilibrium (Devaney 1976; Härterich 1998).
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 equilibrium 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 \(N\) (Sandstede et. al. 1997).
Other topics
- Saddle center homoclinic orbits
Homoclinic orbits to saddle centers have attracted interest since they can describe embedded soliton solutions of PDEs (Champneys et.al. 2001). In 4-dimensional phase space a saddle center has a pair of imaginary and a pair of real eigenvalues, and a homoclinic orbit exists if the one-dimensional stable manifold intersects the symmetry section \(\mbox{Fix}\, (R)\ .\) Consequently, such orbits are of codimension one.
The dynamics near saddle center homoclinic orbits can be rather complex and the complete story is not fully understood. It is known that such orbits are embedded into a family of homoclinics to periodic orbits with a small amplitude. Furthermore, if a saddle center homoclinic orbit exists at a parameter value \(\lambda_0\ ,\) then there is a sequence of parameter values accumulating at \(\lambda_0\ ,\) for which symmetric 2-homoclinic orbits exist (Champneys and Härterich 2000).
As codimension one objects, saddle center homoclinic orbits exist along curves in the parameter space of two-parameter families of differential equations. These curves typically end at points along which the equilibrium undergoes a bifurcation, changing its type to a center or a real saddle. Corresponding studies can be found in (Lombardi 2000, Wagenknecht and Champneys 2003).
- Homoclinic snaking
Recently, symmetric homoclinic orbits near heteroclinic cycles have been studied widely, since they give rise to homoclinic snaking. This term refers to the existence of a snaking curve of homoclinic orbits, along which infinitely many fold bifurcations of homoclinic orbits occur. The corresponding solutions spread out and develop additional oscillations about their middle part. The Figure shows the snaking bifurcation diagram for homoclinic solutions in the Swift-Hohenberg equation with forcing parameter \(\mu\ .\) Along the double-helix structure symmetric solutions exist, as illustrated in the panels on the left (Beck et.al. 2009).
Homoclinic snaking can explain the emergence of infinitely many localized patterns in the Swift-Hohenberg equation and has also been found to occur in problems in nonlinear optics, nonlinear elasticity and fluid mechanics.
- Non-symmetric homoclinic orbits
If \(X_1\) is a non-symmetric homoclinic orbit of a reversible system, then so is its \(R\)-image \(X_2=RX_1\ .\) Hence, such orbits exist as pairs. The behaviour of and the dynamics near the single orbits is the same as for homoclinic orbits in generic systems see (Homoclinic Orbit), but studies in (Homburg and Knobloch, 2006) show that the interplay of the 2 orbits can lead to a highly complicated dynamics. The authors consider the case where the 2 orbits approach a saddle equilibrium from the same side, forming a so-called homoclinic bellows, and they prove the existence of a multitude of periodic, homoclinic and heteroclinic orbits in an unfolding of this configuration.
- Homoclinic orbits in odd-dimensional systems
There has been relatively little work on homoclinic orbits in odd-dimensional reversible systems, with the main reason being that symmetric equilibrium points in such systems are necessarily non-hyperbolic, and therefore symmetric homoclinic orbits are of higher codimension. Similar to the case of even-dimensional systems, however, homoclinic and heteroclinic orbits can be created in local bifurcations of such equilibrium points, and they can give rise to a very interesting and complicated dynamics, see (Lamb et. al. 2005) for an investigation of this in a Hopf-zero bifurcation.
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.
- Champneys, A. R.,Malomed, B. A., Yang, J. and Kaup, D. J. (2001), Embedded solitons: solitary waves in resonance with the linear spectrum, Physica D 152-153, 340-354.
- Devaney, R. (1976), Reversible diffeomorphisms and flows, Trans. Amer. Math. Soc. 218, 89–113.
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- Homburg, A. J. and Knobloch, J. (2006), Multiple homoclinic orbits in conservative and reversible systems, Trans. Am. Math. Soc. 358(4), 1715-1740.
- 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.
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- Lamb, J. S. W., Teixeira, M.-A. and Webster, K. N. (2005), Heteroclinic bifurcations near Hopf-zero bifurcation in reversible vector fields in \(\Bbb{R}^3\ ,\) J. Dif. Eq. 219, 78-115.
- Lombardi, E. (2000), Oscillatory integrals and phenomena beyond all algebraic orders: with applications to homoclinic orbits in reversible systems, Springer.
- 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.
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- Wagenknecht, T. and Champneys, A. R. (2003), When gap solitons become embedded solitons: a generic unfolding, Physica D 177, 50–70.
