Bifurcations in Hamiltonian Systems with a Reflecting Symmetry

A reflecting symmetry $${q \mapsto -q}$$ of a Hamiltonian system does not leave the symplectic structure $${{\rm d}q \wedge {\rm d}p}$$ invariant and is therefore usually associated with a reversible Hamiltonian system. However, if $${q \mapsto -q}$$ leads to $${H \mapsto -H}$$ , then the equations of motion are invariant under the reflection. Such a symmetry imposes strong restrictions on equilibria with q = 0. We study the possible bifurcations triggered by a zero eigenvalue and describe the simplest bifurcation triggered by non-zero eigenvalues on the imaginary axis.


Introduction
Classical mechanical systems H = T + V , with a potential V = V (q) depending only on the position q and a kinetic energy T quadratic in the momenta p, are reversible with respect to the involution (q, p) → (q, −p) , cf. [12,6] and references therein. Indeed, the Hamiltonian H is invariant under (1) while the symplectic form ω = dq ∧ dp turns under (1) into −ω, whence the Hamiltonian vector field X H defined by ω(X H , ..) = dH similarly turns into −X H . On the other hand, the π-rotation (q, p) → (−q, −p) leaves the symplectic form ω fixed but yields reversibility X H → −X H for Hamiltonians satisfying H(−q, −p) = −H(q, p), see [7,11]. Combination of these two mechanisms leads to an involution (q, p) → (−q, p) that transforms both ω → −ω and H → −H but preserves the equations of motioṅ defined by X H . The following theorem from [14] shows that co-ordinates (q, p) as above can be chosen for any involution that transforms ω → −ω and H → −H. of the eigenvalues of A "made Hamiltonian" (ensuring that for every eigenvalue λ also −λ,λ and −λ belong to the spectrum). On the linear level the dissipative subsystemṗ = Ap has its direct counterpartq = −A T . Eigenvalues λ that pass through the imaginary axis exchange with −λ -there is no nilpotent part that enforces e.g. a Krein collision. In particular, the linear part at a bifurcating equilibrium (q, p) = (0, 0) in one degree of freedom vanishes. This paper is organized as follows. In the next section we detail the possible bifurcations in one degree of freedom. The stable families of reflectionally anti-symmetric Hamiltonians without moduli turn out to constitute three families, according to their 3and 4-jets. In section 3 we consider the double Hopf bifurcation that can occur in two degrees of freedom. In the concluding section 4 we discuss how the necessary parameters can be accounted for by phase space variables.

One degree of freedom
Up to scaling time the only structurally stable equilibrium on the symmetry manifold {q = 0} of (2) is the saddle with quadratic part H 2 (q, p) = pq of the Hamiltonian. Under variation of parameters one can also encounter equilibria with vanishing quadratic part H 2 . There are two local bifurcations of co-dimension 1, three local bifurcations of co-dimension 2, six local bifurcations of co-dimension 3 and four local bifurcations of codimension 4. From co-dimension 5 on moduli may occur, increasing the number of terms in a versal unfolding.
Two Hamiltonians H and K on R 2 are called right equivalent if there is a diffeomorphism ψ with K = ±H•ψ and singularity theory shows how this can be used to remove higher order terms, see [3,18,1] and references therein. In case ψ is a symplectomorphism it furthermore constitutes a conjugacy between the vector fields X K and ±X H . As shown in [4,10] this can also be achieved if ψ does not respect the symplectic structure by allowing for a time re-parametrisation (making ψ an equivalence between the vector fields).
A singularity of the Hamiltonian H at (0, 0) is non-degenerate (having finite multiplicity) if and only if it is finitely determined, see [1]. A sufficiently high Taylor polynomial should then contain a monomial of the form p l or p l q and a monomial of the form pq j or q j , otherwise one could factor q 2 or p 2 . The reflecting symmetry furthermore excludes p l and requires j to be odd. As shown in [17] a versal unfolding (within the reflectionally symmetric universe) can be obtained from the universal unfolding (within the universe of all singularities) by restricting to monomials p l q j with odd j.

The reflecting umbilic bifurcation
At a bifurcation the quadratic part H 2 vanishes and the cubic part H 3 becomes important. The zero set H −1 3 (0) = {(q, p) ∈ R 2 | H 3 (q, p) = 0} of the homogeneous polynomial H 3 consists of 3 lines, counted with multiplicity (unless H 3 vanishes completely). In the simplest situation these are the p-axis {q = 0} and the two lines {3ap 2 + bq 2 = 0}. Here the sign of ab distinguishes between the elliptic case D − 4 where the additional two lines are real (mapped to each other by the reflector τ ) and the hyperbolic case D + 4 with two complex lines. The ensuing umbilical bifurcations have co-dimension 3 for Hamiltonian systems without additional symmetry, see [3,18] for the umbilical catastrophes and [4] for the resulting bifurcation diagrams. Under the reversible symmetry (1) the co-dimension shrinks to 2 with bifurcation diagrams given in [8]. The present reflecting symmetry (2) brings the codimension down to 1 and a versal unfolding is provided by λq. This yields the equations of motionq = apq  Note that elliptic equilibria always come in τ -related pairs as centres cannot exist on the p-axis.

Higher order umbilics
The versal unfolding of the singularities D ± 4 is the first member of the series where we again assume a > 0. All equilibria are on the axes; for λ 1 < 0 there are two saddles (q, p) = (0, ± − for equilibria (q, p) = (q 0 , 0) on the q-axis is reminiscent of the generalized Hopf bifurcation treated in [19] and results in the same local bifurcation set, see figure 3 for the case k = 4 of the singularities D ± 8 . The bifurcation diagrams of D ± 6 are derived in [2], see figures 4 and 5.
When passing through the λ 2 -axis (outside of the origin), the q 5 -term becomes a higher order term in the versal unfolding of D ± 6 and a reflecting umbilic bifurcation takes place -elliptic for λ 2 < 0 and hyperbolic for λ 2 > 0. Moreover, there are subordinate centre-saddle bifurcations (cf. e.g. [6]) on the q-axis; because of the reflecting symmetry these always come in τ -related pairs. The line of centre-saddle bifurcations has the parametrisation and only exists for negative The reflecting D − 6 furthermore has a subordinate global bifurcation. Indeed, for the centres born in the hyperbolic reflecting umbilic bifurcation to engage in a centre-saddle bifurcation with the saddles still present from the elliptic reflecting umbilic bifurcation the latter have to let their separatrices form a homoclinic loop around the former. This happens when crossing the line which lies between (4) and the positive λ 2 -axis. From k ≥ 4 on a second type of global bifurcation occurs subordinate to D ± 2k , where two saddles off the p-axis become connected by heteroclinic orbits (as do their τ -related counterparts). Also interactions of local and global bifurcations become possible, e.g. parabolic equilibria getting connected during their centre-saddle bifurcation.

The third bifurcation of co-dimension 2
For the higher order umbilics D ± 2k , k ≥ 3 the zero set H −1 3 (0) of the cubic part consists of the p-axis and the q-axis, the latter with multiplicity 2. A (single) line with multiplicity 3 necessarily is the p-axis and then the cubic part is a multiple of q 3 . The simplest situation is the singularity . The reflecting symmetry (2) decreases the co-dimension from 6 to 2 and a versal unfolding is provided by λ 1 q + λ 2 pq, see [17,10,2]. This yields the equations of motionq where we scaled both a and b to 1. The bifurcation diagram of E 7 has been derived in [2], see figure 6.
To check the τ -related pair of subordinate centre-saddle bifurcations at the negative λ 1 -axis we expand H around (q, p) = (q 0 , 0) with q 0 = ± −2λ 1 and obtain where we omitted the constant terms. The first two terms in (5b) are of higher order and can be removed by a right equivalence, see [3,1]. Then the shear transformation removes the third term in (5b) and the translation Figure 6: Bifurcation diagram of the reflecting E 7 .
takes care of the resulting P 2 -term. Omitting once more constant terms and replacing λ 2 by the remaining polynomial (5a) displays a centre-saddle bifurcation as λ passes through 0. For the subordinate reflecting umbilic bifurcations we expand H around (q, p) = (0, p 0 ) satisfying and obtain where we again omitted the constant terms. After a right equivalence removing the higher order term in (6b) the translation takes care of the second term in (6b). Omitting the new constant term and replacing the coefficient of Q by the remaining polynomial (6a) displays a reflecting umbilic bifurcation as λ passes through 0 -elliptic 1 if p 0 < 0 and hyperbolic if p 0 > 0. Thus, the cusp parametrises elliptic reflecting umbilic bifurcations subordinate to E 7 for λ 1 < 0 and hyperbolic reflecting umbilic bifurcations for λ 1 > 0.

Non-vanishing 3-jet
The versal unfolding of the singularity E 7 is the first member of the series of unfolded singularities. In case of odd k = 2 + 1 the singularity has the label E 6 +1 and when k = 2 + 2 is even the label is J ± 2 ,0 with ± = sgn(ab). For k = 4 the equations of motion areq and we assume a > 0 to fix thoughts. Looking for the singularity E 7 adjacent to J ± 2,0 we expand H around (q, p) = (0, p 0 ) with and, omitting constant terms, obtain Note that during the bifurcation there are heteroclinic connections between the monkey saddle and the second (hyperbolic) saddle -for topological reasons these are unavoidable. Also, it is necessarily the middle saddle that inherits these connecting orbits. where In particular, the curve of singularities E 7 can be parametrised by p 0 = 0, see figure 7. Similarly, the surface of reflecting umbilic bifurcations is parametrised by p 0 = 0 and λ 3 = − a 2 connecting the two arcs of the E 7 -curve yields the elliptic case with the remaining part yielding the hyperbolic case, while it is the other way around for negative b.
The second co-dimension 2 bifurcation is a pair of τ -related cusp bifurcations, unfolded by λ 2 and λ 3 , along the negative λ 1 -axis for b > 0 and along the positive λ 1 -axis for b < 0 ; in the latter case a pair of dual cusp bifurcations. Expanding H around (q, p) = (q 0 , 0) with removing higher order terms, applying the shear transformation to remove the P Q-term and omitting constant terms, we indeed obtain with μ 1 and μ 2 given by (7). Where the third and fourth term vanish simultaneously the Hamiltonian displays a D ± 4 -singularity. The expressions (7a) and (7b) are polynomials in p 0 , with μ 2 (p 0 ) = μ 1 (p 0 ). In the three-dimensional λ-space this yields the swallowtail λ 1 λ 2 Figure 9: Bifurcation diagram of the reflecting J − 2,0 sliced at fixed λ 3 < 0. The lines intersecting on the λ 1 -axis parametrise reflecting elliptic umbilic bifurcations, the curve connecting the two cusps is parametrising a reflecting hyperbolic umbilic bifurcation and the two vertical lines emanating from the cusps each parametrise a τ -related pair of centresaddle bifurcations. The part of the λ 1 -axis above the intersection point parametrises a τ -related pair of connection bifurcations.
surface. In particular, the self-intersection occurs where μ 1 is even in p 0 , leading to λ 2 = 0, then ap 2 0 = −6λ 3 from (7b) and finally (9) from (7a). The coefficient of P 2 Q is (half of) the second derivative of μ 1 (p 0 ) and, as discussed above, distinguishes together with the sign of b between the two cases of the reflecting umbilic bifurcation, with the singularities E 7 (where this coefficient vanishes) parametrised by the curve (8).
For b > 0 the pair of τ -related co-dimension two bifurcations at the negative λ 1 -axis are cusps unfolding A + 4 -in the symmetric sub-scenario that λ 3 decreases through zero within the plane λ 2 = 0 the bifurcating equilibria turn from centres into saddles and both split off two centres, which are encircled by the figures eight formed by the separatrices of the saddles. This does not lead to a global bifurcation upon crossing the half plane λ 2 = 0, λ 3 < 0.
For b < 0 the pair of τ -related co-dimension two bifurcations at the positive λ 1 -axis are dual cusps unfolding A − 4 -in the symmetric sub-scenario that λ 3 decreases through zero within the plane λ 2 = 0 the bifurcating equilibria turn from saddles into centres and each split off two saddles, which are conneced by their separatrices. When approaching (9), the τ -related pairs of saddles come closer and meet in the two simultaneous elliptic reflecting umbilic bifurcations. Thus, when crossing the half plane λ 2 = 0, λ 3 < 0 a global bifurcation takes place at points (λ 1 , 0, λ 3 ) with 2aλ 1 > 2λ 2 3 . In particular, the line (9) is a true line of co-dimension two bifurcations instead of merely the intersection of two co-dimension one surfaces as in the case b > 0. Figure 10: The Z 2 × Z 2 -symmetric bifurcations of co-dimension 1 subordinate to X ± 9 , increasing λ 2 above for X − 9 with a = 1 and b = −1 and below for X + 9 with a = b = 1.
In both cases b > 0 and b < 0 all open regions of the bifurcation diagram extend to negative λ 3 , whence it suffices to give the slices in figures 8 and 9.

Vanishing 3-jet
If next to the quadratic part H 2 also the cubic part H 3 of the equilibrium vanishes, then the first non-zero terms are of (at least) fourth order. Counted with multiplicity, the zero set H −1 4 (0) of the homogeneous polynomial H 4 = 0 consists of 4 lines. Where these are in general position, the cross ratio leads to a modulus. Under the present reflecting symmetry (2) two of the lines have to coincide with the p-and q-axes and no modulus occurs. Put differently, the terms that we cannot transform away already have coefficient 0. The simplest situation where all 4 lines have multiplicity 1 is unfolded as q-axis, including the two saddles at This leads to the global bifurcation of co-dimension 2 connecting all five saddles when crossing the half plane {λ 1 = 0, λ 2 < 0}. See figure 11 for the resulting partial bifurcation diagram.
Other singularities with vanishing 3-jet have either the p-axis or the q-axis as a triple line of H −1 4 (0). The versal unfolding of X ± 9 is the first member of the series of unfolded reflecting simple singularities. In case of even k = 2 the singularity has the label Z 6 and when k = 2 + 1 is odd the label is Z ± −1,0 -with the exception of k = 3 for which the label is X ± 9 and not Z ± 0,0 .

Moduli
As beared witness by [13], one has to stop somewhere when studying families of dynamical systems. In the previous subsections we encountered three series of reflecting simple singularities, see figure 12 for the adjacency diagram. These contain all local bifurcations on the p-axis of co-dimensions 1, 2, 3 and 4. A 5-parameter family of Hamiltonian systems with reflecting symmetry may also encounter the singularity with label N m,± 16 . Here a and b are coefficients that can be scaled to 1 and ±1 = sgn(ab), while m is a modulus. This singularity is adjacent to D ± 6 and Z 12 . As exemplified in [9], such a modulus does not have dramatic consequences, merely leading to a discrete set of exceptional values separating topologically inequivalent behaviour from each other. The difference to varying an unfolding parameter is that varying a modulus leads to topological equivalences that are not differentiable. For more details see [9,10] and references therein.
In the same way that D ± 4 can be included in the E&J series (under its alias J ± 1,0 ), the singularity X ± 9 is a prequel to the series of singularities with vanishing 3-jet for which H −1 4 (0) has the p-axis as a triple line. Already the first of these, the singularity Z 15 given by has a modulus. Next to X ± 9 this singularity is adjacent to D ± 8 and the co-dimension is 6.

Two degrees of freedom
An equilibrium on the symmetry manifold {q = 0} of (2) with a zero eigenvalue has at least two corresponding linearly independent eigenvectors. Unless unfolded by parameters, the remaining eigenvalues are expected to be hyperbolic. Then a centre manifold brings us back to the previous section. In two degrees of freedom the resulting hyperbolic equilibria on the symmetry manifold are of saddle-saddle type. The second type of structurally stable equilibria on the symmetry manifold in two degrees of freedom are hyperbolic of focus-focus type. Indeed, elliptic equilibria bifurcate away immediately, leading to focus-focus equilibria during a double Hopf bifurcation. Here two pairs of complex conjugate eigenvalues exchange their stability when meeting on their way through the imaginary axis. This phenomenon is of co-dimension 1 and the quadratic part of H can be brought into the form H 2 (q, p) = ω(q 1 p 2 − q 2 p 1 ) + λ(q 1 p 1 + q 2 p 2 ) =: ωS + λP .
At λ = 0 the linear system DX H (0) = X S generates an S 1 -symmetry and a normalization procedure pushes this symmetry through the Taylor series of H. The truncated normal form can then be written as a function of the Hilbert basis of the S 1 -action formed by S, P , Compared to the Hamiltonian Hopf bifurcation there is no nilpotent part to further normalize H as in [16,15], but the reflecting symmetry (2) excludes many terms from the outset and the fourth order normal form reads as show that indeed two Hopf bifurcations take place in the q-and p-plane, respectively. It is instructive to reduce the S 1 -symmetry, fixing S = σ and re-writing the Hamiltonian as H σ (N, M, P ) = ασN + βσM + (λ + aN + bM)P (omitting the constant term ωσ). The reduced phase space is the same space as for the Hamiltonian Hopf bifurcation; for more details see [10] and references therein.
This equilibrium has the whole N-axis as stable manifold and it inherits the intersection of the plane λaN + bM = 0 with U σ as unstable manifold. For λ < 0 the rôles of the Mand N-axes are interchanged, the regular equilibrium at having the whole M-axis as stable manifold. See also figure 13. For σ = 0 the regular equilibrium persists as a saddle, while the saddle at the singular point does not lead to a second regular equilibrium. All orbits with value of H different from that at the equilibrium are unbounded both for t → −∞ and for t → +∞. Upon reconstructing the flow of X H in two degrees of freedom the singular point gives rise to the equilibrium at (q, p) = (0, 0) while the regular equilibria give rise to periodic orbits, with (10) in the q-plane and (11) in the p-plane. Within these two planes, two supercritical Hopf bifurcations take place simultaneously. As λ increases through zero, in the q-plane the origin loses stability and a stable periodic orbit branches off. In the p-plane this happens as λ decreases through zero, i.e. for increasing λ a stable periodic orbit shrinks to the unstable origin and bequeaths its normal linear behaviour. As a result, the origin is always of focus-focus type (except at λ = 0 where a semi-simple 1:−1 resonance occurs). The periodic orbits are hyperbolic (normally hyperbolic within the energy shells).
When b is negative, i.e. has the same sign as a, the plane λ+aN +bM = 0 intersects the cone only for λ ≥ 0. Thus, for λ ≤ 0 the equilibrium at the singular point (N, M, P ) = 0 has both the N-axis as stable manifold and the M-axis as unstable manifold and no further bounded motions occur. For λ > 0, these rôles are interchanged, the stable direction pointing along the M-axis and the unstable direction pointing along the N-axis. The resulting (un)stable manifolds extend to the equilibria (11) and (10), respectively, which are again saddles. The intersection of the plane λ + aN + bM = 0 with U 0 connects these two saddles, providing both the unstable manifold of (10) and the stable manifold of (11). Encircled by the two triangles of separatrices are two centres at Between these equilibria and the two triangles extend two families of periodic orbits, see also figure 14. For σ = 0 the regular equilibria persist, while the saddle at the singular point again disappears. Each of the two families of periodic orbits persists as well, extending between a centre and two of the three separatrices connecting the two saddles. All other orbits are again unbounded. Upon reconstructing the flow of X H in two degrees of freedom the singular point gives rise to the equilibrium at (q, p) = (0, 0) while the regular equilibria give rise to periodic orbits, with (10) in the q-plane and (11) in the p-plane. As λ increases through zero, in the q-plane the origin loses stability and a stable periodic orbit branches off in a supercritical Hopf bifurcation. In the p-plane a subcritical Hopf bifurcation takes place as λ increases through zero, with the origin gaining stability and an unstable periodic orbit branching off. As a result, the origin is always of focus-focus type (except at λ = 0 where a semi-simple 1:−1 resonance occurs). The periodic orbits within the q-and p-planes are hyperbolic, while the periodic orbits reconstructed from (12) are elliptic. Furthermore, the two simultaneous Hopf bifurcations interact in this case of positive sgn(ab) and lead to two families of (Lagrangean) 2-tori branching off from the bifurcating equilibrium.
If the system depends on more than one parameter, one may encounter the generalized Hopf bifurcations of [19]. Note that the degeneracies can build up independently in the q-and p-planes. A 2-parameter family of Hamiltonian systems in two degrees of freedom with reflecting symmetry may furthermore encounter a double Bogdanov-Takens bifurcation.

Conclusions
For the bifurcation theory of the previous sections to apply one needs the proper dependence on parameters. Where quantities enter the equations of motion that can be influenced or have to be measured, these can play the rôle of external parameters. The bifurcation diagram then serves as an inventory for the various open regions of typical behaviour (if on a boundary: measure again).
In Hamiltonian systems also phase space variables can be parameters, such internal parameters are distinguished with respect to external parameters. While equilibria typically are isolated, periodic orbits form 1-parameter families, mostly parametrised by the energy. As shown in [13], periodic orbits do bifurcate under variation of the energy.
In the present situation of a reflecting symmetry, the normal dynamics of the family of periodic orbits is expected to behave according to the scenarios derived in the previous sections. In this way one should encounter periodic reflecting umbilic bifurcations and periodic double Hopf (or double Neȋmark-Sacker) bifurcations. Note that reflecting bifurcations always concern (partially) hyperbolic periodic orbits, of saddle-... and of focus-focus-... type.
Invariant n-tori T n form n-parameter families in integrable Hamiltonian systems, parametrised by the actions y conjugate to the toral angles x. In this way bifurcations of co-dimension up to n − 1 may occur -unlike periodic orbits the T n × {y} persist a non-integrable perturbation only as a Cantor family, whence isolated bifurcations of codimension n may disappear in a resonance gap. These quasi-periodic bifurcations persist under non-integrable perturbations as the reflecting symmetry can be dragged through the proofs in [5,10]. The relevant part H(x, y, q, p) = ω | y + H(q, p) turns under the anti-symplectic involution τ : (x, y, q, p) → (x, −y, −q, p) (13) into −H. If we want y to account for the parameters λ in H(q, p), then these parameters inherit the behaviour λ → −λ under the reflecting symmetry (13). Consequently, the unfolding monomials p l q j should be invariant under (2) instead of turning into −p l q j . This dramatically increases the number of unfolding parameters. At the same time we also want y to control the frequency vector ω ∈ R n . The Kolmogorov condition det D 2 y H = 0 requires y → ω(y) to be a diffeomorphism. This not only precludes the use of y as unfolding parameters, but is furthermore made impossible by the behaviour H → −H under y → −y. Indeed, pure terms in y may enter H only with odd order, whence D 2 y H ≡ 0. The Rüssmann condition that the partial derivatives up to order L ∈ N of the frequency mapping span the whole frequency space R n can still be satisfied, though, leaving y free to account for the unfolding parameters λ. Alternatively, one can search for bifurcating n-tori that are invariant under q → −q, but not under y → −y. To these tori the bifurcation theory developed in the previous sections does apply and furthermore the y-directions not needed to unfold the bifurcation can be used for linear control of the frequency vector ω (with additional non-linear control according to (14)). Such tori are situated at y 0 = 0 and (13) yields a counterpart at −y 0 . In this way the quasi-periodic reflecting bifurcations come in τ -related pairs.