We will show that the ODE in Equation (5.4) can be formulated as a reversible, four- dimensional, Hamiltonian dynamical system. We refer the reader to the work of Champ- neys [12] for a review of the theory and application of homoclinic orbits to equilibria in reversible dynamical systems. Champneys concentrates on even-order reversible systems in four or more dimensions where the homoclinic orbit and the equilibrium are both reversible.
Reversible structure
Consider an even-dimensional system ˙
x=f(x), x∈R2n. (5.5)
All derivatives are with respect to time, that is, ˙x= dx/ dt. We will study systems with linear reversing symmetries.
Definition 5.2.1 Alinear involution ofR2n is a linear map R:R2n→R2n whose square
is the identity, ie. R2 = Id.
Definition 5.2.2 A systemx˙ =f(x) is invariant under time reversal if
Rf(x) =−f(Rx).
Definition 5.2.3 Let fix(R) be the set of all points in R2n that are fixed under the
reversibility R. A symmetric section S is a linear subspace of R2n such that S= fix(R).
Consider an involution R that fixes half the variables in the system and under which the system is invariant after time reversal. Then the system has a setS= fix(R) where S is the symmetric section of the reversibilityR.
Hamiltonian structure
Consider the system x˙ = f(x) where x = (q(t),p(t)) ∈ R2n. Then the system is a Hamiltonian system if it can be written in the form
dqi dt = ∂H ∂pi , dpi dt =− ∂H ∂qi
for aHamiltonian functionwherei= 1, . . . , n. This is often simply called theHamiltonian, H(p,q, t), and comes from a generalised form of Newton’s laws of motion where p ∈Rn is the generalised momentum and q ∈ Rn is the generalised coordinate of a mechanical system of degree of freedom n. Very frequently the Hamiltonian is of the form
H= n X
i=1
p2i +V(q)
where p = (p1, p2, . . . , pn) and V:Rn → R. Such a Hamiltonian is called “separable” if it is additively separable in each coordinate. If the Hamiltonian system is also reversible
with the reversibility R: (q, p) →(q,−p), then this form of the Hamiltonian is separable underR. The reversibilityRusually reverses all the momentum variables. A Hamiltonian system preserves volume in 2n-dimensional phase space.
Definition 5.2.4 A symmetric fixed point is a fixed point of a reversible system that is invariant under the linear involution, R, used to define reversibility.
Consider a 2n-dimensional reversible system with a symmetric fixed point that lies in S. Take the case where the unstable manifold, Wu, of the fixed point has dimension n. This occurs when symmetric homoclinic orbits are of codimension-zero and asymmetric homoclinic orbits are of codimension-one.
Definition 5.2.5 An orbit is said to be symmetric if it is invariant under the reversibility
R. Therefore a symmetric orbit is mapped into itself under the reversibility R.
Definition 5.2.6 An orbit is said to be asymmetric if it is not invariant under the reversibility R. Asymmetric orbits are mapped into the time reversal of themselves under the reversibilityR.
Thus asymmetric homoclinic orbits come in pairs where each orbit is the time reversal of the other underR. See Figure 5.3 for a schematic of homoclinic orbits in phase space. The left plot shows a homoclinic orbit that is symmetric under time reversal. That is, the orbit is symmetric under reflection in the x-axis. The right plot shows a pair of asymmetric homoclinic orbits. Let Υ1,Υ2 be the two asymmetric orbits. The orbits are not symmetrical under time reversal. Each orbit is the time reversal of the other, so R(Υ1) = Υ2 and R(Υ2) = Υ1.
Our system
We now show that our system in Equation (5.4) has both reversible structure and Hamilto- nian structure, although the Hamiltonian is not separable and the involution Ris not the standard one. The independent variable in our system is space, not time, so all derivatives are with respect to space. That is,u0 =du/dx. First, we write Equation (5.4) in the form
u(iv)+au00+cu+g(u) = 0 (5.6)
where a = 2(1−b2), c = (b2+ 1)2 and g(u) = −4b(b2 + 1)f(u). The threshold is set at θ= 1.5 in this section. Following the approach of [34], we choose position variablesuand
Figure 5.3: Schematics of homoclinic orbits in (x, x0) phase space wherex0 = dx/ dt. Left: The homoclinic orbit is symmetric under reflection in the x-axis and is invariant under time reversal. Right: A pair of asymmetric homoclinic orbits. Each orbit is the time reversal of the other.
v and conjugate momentapu and pv so that
u0 = v v0 = pv
p0u = cu+g(u) p0v = −pu−av.
(5.7)
The Hamiltonian is defined as
H(u, v, pu, pv) =puv+ p2v 2 + av2 2 − cu2 2 −G(u) (5.8) where G(u) =− 8b (b2+ 1) Z u 0 e−r/(s−θ)2Θ(s−θ)ds. (5.9)
Note thatG0(u) =g(u). The system given by Equations (5.6)–(5.7) has the reversibility
R: (x, u, v, pu, pv)7→(−x, u,−v,−pu, pv). (5.10)
Classical Hamiltonian dynamical systems are invariant under time reversal and the reversal of all momentum variables. R is a non-standard reversibility as it does not reverse both the momentum variables. H is a conserved quantity
H0 = ∂H ∂uu 0+ ∂H ∂pu p0u+∂H ∂vv 0+ ∂H ∂pv p0v = 0 (5.11)
and is non-separable. Also, u0 = ∂H ∂pu , v0 = ∂H ∂pv , p0u =−∂H ∂u, (5.12) p0v =−∂H ∂v.
There is another way to determine the Hamiltonian, or energy function [70, 98], where the ODE in Equation (5.6) is multiplied by u0 and integrated to obtain
u0u000−1 2(u 00)2+a 2(u 0)2+ c 2u 2+G(u) = 0. (5.13)
G(u) is as in Equation (5.9). A small amount of algebra confirms the lefthand side of Equation (5.13) is the same as the Hamiltonian function in Equation (5.8).
We define level sets of the energy in Equation (5.8) by setting
H(u) =e, e∈R. (5.14)
Energy is conserved on the level sets H(u) =e and hence for any homoclinic orbits that lie upon them. The system in Equations (5.7)–(5.8) is a canonical Hamiltonian system. The system can therefore be written in the standard form
y0 =J∇H(y) (5.15)
wherey= [u, v, pu, pv]T and Jis the skew–symmetric matrix in R4 of
J= 0 Id −Id 0 . (5.16)
In matrix J,Idis the 2×2 identity matrix and0 is the 2×2 zero matrix.
Let Ws be the stable manifold and Wu the unstable manifold of a fixed point of the system. Then homoclinic orbits lie in Ws∩Wu. In a reversible, non-Hamiltonian system, symmetric homoclinic orbits are codimension-zero, therefore they persist under a generic perturbation that preserves reversibility. However, asymmetric homoclinic orbits are codimension-one and are destroyed by a generic perturbation that breaks the conserved quantity but still preserves reversibility. In a non-reversible Hamiltonian system, both symmetric and asymmetric homoclinic orbits are codimension-zero. This also holds for a Hamiltonian-reversible system.
When a fixed point of a reversible system is a saddle-focus, there are infinitely many N-pulses for eachN >1 as well as the primary transverse symmetric homoclinic orbit [12]. In Chapter 3 we found that Equation (5.4) has a saddle-focus fixed point at the origin with eigenvalues of ±(b±i). Numerical results presented in Chapter 3 showed multiple bump solutions (homoclinic orbits to the fixed point at the origin) as seen in the solution curve (see Figure 5.1). This is the manifestation of the multipleN-pulse solutions in [12].
The stable and unstable manifolds of a fixed point intersect transversally within a level set of the conserved integral for a conservative system. Homoclinic orbits lie on energy surfaces, therefore the topology of the level sets {u : H(u) = e} can change only where the level set contains a critical point. That is, where ∇H(u) = 0. These critical points are the fixed points of the system. For a conservative and reversible system, there are two codimension-one ways to lose transversality of the manifolds Wu and Ws within a level set without a local bifurcation, given a symmetric homoclinic orbit. This can happen either through coalescence or bifurcation. In Chapter 3, breaks were found in the solution curves when the firing rate function was sufficiently steep (see Figure 5.2). Homoclinic orbits disappeared as the parameter r was varied. We have only considered homoclinic orbits to the fixed point at the origin. However, there are other nonzero fixed points of our system for certain parameter ranges for a step firing rate function, a piecewise linear firing rate function and smooth firing rate function. To find a possible bifurcation causing the disappearance of solutions, we study the other fixed points of Equation (5.4) in later sections of this chapter.