Bifurcations in VolumePreserving Systems
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Abstract
We give a survey on local and semilocal bifurcations of divergencefree vector fields. These differ for low dimensions from ‘generic’ bifurcations of structureless ‘dissipative’ vector fields, up to a dimensionthreshold that grows with the codimension of the bifurcation.
Keywords
KAM theory Divergencefree vector field Volumepreserving Hopf bifurcation Double Hopf bifurcation Quasiperiodic stability1 Introduction
Local bifurcations are bifurcations of equilibria of vector fields and bifurcations of fixed points for mappings. The latter can always be interpreted as Poincaré mappings, see [23], with the fixed points giving rise to periodic orbits, which in principle can be of nonlocal influence. However, the term semilocal bifurcation is usually reserved for bifurcations of invariant tori in dynamical systems—whether given by the flow of a vector field or by iterating an invertible mapping. All these invariant sets can be attractors for dynamical systems that are dissipative; in particular for generic dynamical systems that preserve no structure whatsoever. But also e.g. reversible systems do admit attractors (together with a repelling counterpart obtained by applying the reversing symmetry to the attractor).
Indeed, resonant tori are expected to be destroyed by the perturbation, while (strongly!) nonresonant tori can be shown to persist if the perturbation is sufficiently small—which means very small as there are no further conditions on the perturbation except for preserving the structure at hand. In frequency space both resonant and nonresonant frequency vectors \(\omega \in \mathbb {R}^{n}\) form dense subsets and it is here that the dependence \(\mu \mapsto \omega ( \mu )\) on the parameter \(\mu \in \mathbb {R}^{s}\) becomes important. If the frequency mapping \(\omega : \mathbb {R}^{s} \longrightarrow \mathbb {R}^{n}\) is a submersion—which requires \(s \geq n\)—then the set of strongly nonresonant tori is of full measure, see below, and kam Theory allows to conclude persistence of most quasiperiodic tori. On the other hand, the perturbation generically—within the universe of admissible perturbations, preserving the structure at hand—opens up the dense resonances to an open (albeit of small masure) complement of the union of persistent tori. We colloquially say that the corresponding nonpersistent tori disappear in a resonance gap.
The topological ‘size’ of this measuretheoretically large set is ‘small’ as the complement is open and dense. Locally in the frequency space \(\mathbb {R}^{n+m}\) this set has a product structure: half lines times a Cantor set. Indeed, when \((\omega , \alpha )\) satisfies (5) then also \((\varsigma \omega , \varsigma \alpha )\) satisfies (5) for all \(\varsigma \geq 1\). The intersection of the set of all Diophantine frequency vectors with a sphere of radius \(R > 0\) is closed and totally disconnected; by the theorem of Cantor–Bendixson [25] it is the union of a countable and a perfect set, the latter necessarily being a Cantor set.
The simplest bifurcation for dissipative vector fields is the quasiperiodic saddlenode bifurcation [4, 11, 40] with \(m=1\) and (6b) becoming \(\dot{z} = \mu  z^{2}\), i.e.\(\sigma (\mu ) \equiv \mu \) and \(\varOmega (\mu ) \equiv 0\). This example shows that it is in general not possible to achieve \(\sigma (\mu ) \equiv 0\), while \(\sigma (\mu _{0}) = 0\) can always be achieved. The other two quasiperiodic bifurcations of codimension 1 that structureless dissipative vector fields may undergo are the quasiperiodic Hopf bifurcation [3, 4, 11] and the frequencyhalving or quasiperiodic flip bifurcation [4, 9, 11]. For both bifurcations \(\sigma (\mu ) \equiv 0\) can be achieved. While the former needs \(m \geq 2\) normal dimensions, the latter can be put into the form \((6)\) only after the passage to a \(2{:}1\) covering space.
Remarks 1

Research of bifurcations in volumepreserving systems started with Broer et al. [5, 6, 13, 16]. This is an interesting class of dynamical systems, where the standard techniques of transversality, normal forms, stability and instability could be applied, combined with kam Theory. One of the results concerns the occurrence of infinitely many moduli of stability and another that of infinite (c.q. exponential) flatness.
 The Shilnikov bifurcation concerns the 2dimensional stable manifold of a 3dimensional saddle point with a complex conjugate pair of eigenvalues \(\beta \pm \mathrm {i}\alpha \), \(\alpha \neq 0\), \(\beta < 0\) containing the 1dimensional unstable manifold as a homoclinic orbit spiralling back to the saddle. This bifurcation was first studied by Shil’nikov and Gavrilov in [22, 37] for dissipative systems but it also occurs in volumepreserving systems, where the stable eigenvalue is given by \(2 \beta > 0\) as dictated by (2). The Shilnikov bifurcation occurs subordinately in the Hopf–Saddle Node bifurcation, both in the dissipative and volumepreserving context. In the present paper the latter case is termed the “volumepreserving Hopf bifurcation” as this is the counterpart of the (dissipative) Hopf bifurcation for volumepreserving systems. As can be inferred from Fig. 1 below this involves a spherical structure consisting of 2dimensional invariant manifolds, enclosing a Cantor foliation of invariant 2tori shrinking down to an elliptic periodic orbit. René Thom [private communication] here spoke of “smoke rings”.

The persistence of the quasiperiodic invariant tori inside the spherical structure as described above was first proven by Broer and Braaksma in [2, 7], using kam techniques.

Another local study on volumepreserving vector fields is given by Gavrilov [21]. As in [5, 6, 13, 16] the classification is modulo topological equivalence. In dimension 2 the analysis essentially reduces to catastrophe theory on corresponding Hamiltonian functions. Again the focus is on the dimensions 3 or 4.

Dullin and Meiss [19] study the dynamics of a family of volumepreserving diffeomorphisms on \(\mathbb {R}^{3}\). This family unfolds a bifurcation of codimension 2, triggered by a fixed point with a triple Floquet multiplier 1. As in the above case of vector fields a spherical structure emerges, termed “vortex bubble” in [19]. In this spherical structure a Shilnikov–like situation occurs, where the inclusion of the 1dimensional unstable manifold in the 2dimensional stable manifold is replaced by sequences of generic tangencies. This setting involves both invariant circles and invariant 2tori for the diffeomorphism. In particular, a string of pearls occurs that creates multiple copies of the original spherical structure for an iterate of the mapping.

Lomelí and RamírezRos [30] concentrate on the splitting of separatrices as the stable and unstable manifold forming the above spherical structure (for which they use the terms spheromak and Hill’s spherical vortices) cease to coincide due to a volumepreserving perturbation.

Meiss et al. [33] also consider 3dimensional diffeomorphisms, where chaotic orbits are studied near the spherical structure described above. In particular it is found that trapping times exhibit an algebraic decay.
For general \((n, m)\) it turns out that the system, up to smooth dynamical equivalence, preserves a certain volume form that explodes at the boundary of \(S_{n} \times S_{m}\). This property was discovered by Akin, see [20]. This means that the dynamics can be imagined as the motion of a particle in an incompressible fluid.
This paper is organized as follows. In the next three sections we focus on the normal dynamics defined by (6b) with \(x\)independent higher order terms \(\mathcal {O}(z^{2})\). This covers the case \(n=0\) of equilibria, of which we describe the linear theory in Sect. 2 and the nonlinear theory in Sects. 3 and 4. The case \(n=1\) of periodic orbits then is addressed in Sect. 5. In Sect. 6 we come back to quasiperiodic bifurcations in volumepreserving dynamical systems and Sect. 7 concludes the paper.
2 Linear Systems
In this section we treat linear systems \(\dot{z} = \varOmega z\) on \(\mathbb {R}^{m}\) that preserve the standard volume, i.e. satisfy \(\operatorname {trace}\varOmega = 0\). That the last eigenvalue equals minus the sum of all other eigenvalues, cf. (2), puts a severe restriction on the spectrum of \(\varOmega \) if the dimension \(m\) is low, but for high \(m\) this merely results in \(m1\) or \(m2\) eigenvalues in general position that completely determine the last eigenvalue or the real part of the last pair of eigenvalues.
2.1 \(m=1\)
Here the restriction \(\operatorname {trace}\varOmega = 0\) immediately results in \(\varOmega = 0\). However, this same restriction applies to volumepreserving perturbations of \(\varOmega = 0\) as well whence this (motionless) dynamics is in fact structurally stable.
2.2 \(m=2\)
In this case one can speak of areapreservation instead of volumepreservation and the (local) dynamics is in fact Hamiltonian. The dimensional restriction \(m = 2 < 4\) prevents eigenvalues outside the real and imaginary axes. For elliptic eigenvalues \(\pm \mathrm {i}\alpha \) the restriction \(\operatorname {trace}\varOmega = 0\) is automatically satisfied, while hyperbolic eigenvalues must take the form \(\pm \beta \). An eigenvalue 0 has necessarily algebraic multiplicity 2 whence we distinguish the case \(\varOmega = 0\) of geometric multiplicity 2 from the parabolic case where the Jordan normal form is \(\varOmega = \bigl({}^{0}_{0} {}^{1}_{0} \bigr)\).
2.3 \(m=3\)
Here the restriction \(\operatorname {trace}\varOmega = 0\) still gives complete freedom for the choice of the first eigenvalue. If that eigenvalue does not lie on the real axis then its complex conjugate is also an eigenvalue (and we order them according to \(\alpha _{1} = \alpha > 0\), \(\alpha _{2} =  \alpha < 0\)) and the third eigenvalue is real with \(\beta _{3} =  2 \beta \), \(\beta = \beta _{1} = \beta _{2}\). The remaining possibility satisfying (2) is that there are 3 real eigenvalues; we notice that also then the eigenvalue that is ‘alone’ on its side of the imaginary axis yields a stronger attraction/repulsion than each of the other 2 eigenvalues because of \(\beta _{1} + \beta _{2} + \beta _{3} = 0\).
 both real:

the eigenvalue 0 triggers a normally hyperbolic bifurcation and the only difference to a dissipative normally hyperbolic bifurcation with 2 real eigenvalues of opposite sign is that at the bifurcation these 2 eigenvalues have the same absolute value. We colloquially say that the main characteristic is dissipative.
 both imaginary:
 under parameter variation the pair \(\beta (\mu ) \pm \mathrm {i}\alpha (\mu )\), \(\beta (0) = 0\) passes through the imaginary axis, forcing the third eigenvalue \(2 \beta (\mu )\) to pass through zero—in the opposite direction. The genericity condition \(\beta ^{\prime }(0) \neq 0\) yields, after reparametrisation, the normal formThis 1parameter unfolding of the (linear) foldHopf singularity has a decidedly volumepreserving character, we therefore speak of the volumepreserving Hopf bifurcation.$$ \varOmega \;\; = \;\; \left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c} \mu &  \alpha & 0 \\ \alpha & \mu & 0 \\ 0 & 0 &  2 \mu \end{array}\displaystyle \right ) \enspace . $$(8)
 both zero:
 then all 3 eigenvalues vanish and we have three subcases, compare with [1], according to whether \(\varOmega ^{2} \neq 0\) with codimension 2, \(\varOmega = 0\) with codimension 8 and, in between, \(\varOmega \neq 0\) but \(\varOmega ^{2} = 0\) with codimension 4 and Jordan normal form(the Jordan normal form of the case \(\varOmega ^{2} \neq 0\) has a single Jordan chain of length 3).$$\varOmega \;\; = \;\; \left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\displaystyle \right ) $$
2.4 \(m=4\)
 2 complex conjugate pairs\(\beta _{j} \pm \mathrm {i}\alpha _{j}\):

here the volumepreserving character is already weak, but still strongest compared to the following cases. Volume preservation enforces \(\beta _{2} =  \beta _{1}\) whence attraction and repulsion of the 2 foci balance each other out, while the rotational velocities \(\alpha _{1}\) and \(\alpha _{2}\) are independent of each other.
 1 complex conjugate pair and 2 real eigenvalues:

there are two subcases. Either the 2 real eigenvalues are on the opposite side with respect to the imaginary axis of the complex conjugate pair, yielding 2 attracting and 2 repelling eigenvalues, or one of the real eigenvalues is on the same side (yielding 3 eigenvalues on one side of the imaginary axis) and the other is ‘alone’ on the opposite side—with a strong attraction/repulsion.
 4 real eigenvalues:

to satisfy (2) these cannot all be on the same side with respect to the imaginary axis, so again there are either 2 attracting and 2 repelling eigenvalues or a single eigenvalue on one side of the imaginary axis balances out 3 eigenvalues on the other side.
 2 complex pairs\(\beta _{j}(\mu ) \pm \mathrm {i}\alpha _{j}(\mu )\),\(\beta _{1}(0) =  \beta _{2}(0) = 0\):

this 1parameter unfolding of the Hopf–Hopf singularity has a decidedly volumepreserving character and we speak of the volumepreserving double Hopf bifurcation. If furthermore \(\alpha _{1}(0) = \alpha _{2}(0)\) the codimension becomes 2 in the nonsemisimple case and 4 in the semisimple case, again compare with [1]. This may be interpreted as a \(1{:}1\) resonance and also other low order resonances can come into play, see §3.4.2 below.
 1 purely imaginary pair:

then the 2 remaining eigenvalues are real with opposite signs and coinciding absolute value, leading to a normally hyperbolic bifurcation of dissipative character. In case the 2 real eigenvalues vanish as well the codimension becomes 2—in the unfolding we expect an interaction of a normally hyperbolic (dissipative) Hopf bifurcation with a normally elliptic Hamiltonian bifurcation and containing the volumepreserving double Hopf bifurcation in a subordinate way.
 a single zero eigenvalue:

the 3 remaining eigenvalues are in one of the two generic configurations detailed at the beginning of §2.3, leading to a normally hyperbolic bifurcation of dissipative character.
 a double zero eigenvalue:

unless the 2 other eigenvalues form a purely imaginary pair—a possibility we already discussed—the 2 other eigenvalues are real and lead to a normally hyperbolic Bogdanov–Takens bifurcation, with the second unfolding parameter redistributing hyperbolicity to the bifurcation.
 all 4 eigenvalues vanish:

here we have the subcases \(\varOmega ^{3} \neq 0\) with codimension 3 of a single Jordan chain, \(\varOmega ^{3} = 0\) but \(\varOmega ^{2} \neq 0\) of codimension 5 of a Jordan chain of length 3, the cases \(\varOmega ^{2} = 0\) but \(\varOmega \neq 0\) of 2 or 1 Jordan chain(s) of length 2 of codimensions 7 and 9, respectively, and the most degenerate subcase \(\varOmega = 0\) of codimension 15, see [1].
2.5 \(m=5\)
 codimension 2:
 all eigenvalues are simple and on the imaginary axis, whence the spectrum is \(\{ \pm \mathrm {i}\alpha _{1}, \pm \mathrm {i}\alpha _{2}, 0 \}\). A linear versal unfolding is given byand contains in a subordinate way both the volumepreserving Hopf bifurcation and the volumepreserving double Hopf bifurcation.$$ \varOmega \;\; = \;\; \left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} \mu _{1} &  \alpha _{1} & 0 & 0 & 0 \\ \alpha _{1} & \mu _{1} & 0 & 0 & 0 \\ 0 & 0 & \mu _{2} &  \alpha _{2} & 0 \\ 0 & 0 & \alpha _{2} & \mu _{2} & 0 \\ 0 & 0 & 0 & 0 &  (\mu _{1} + \mu _{2}) \end{array}\displaystyle \right ) $$(9)
 codimension 3:

the 5 eigenvalues on the imaginary axis are no longer all simple, so either 0 is a simple eigenvalue and \(\pm \mathrm {i}\alpha \) is a double pair of purely imaginary eigenvalues—with the codimension rising to 5 in the semisimple case—or \(\pm \mathrm {i}\alpha \) is a simple pair of purely imaginary eigenvalues and 0 is a triple eigenvalue—with rising codimensions for shorter Jordan chains, compare with §2.3. The codimension of (9) also rises to 3 where \(\alpha _{1}\) and \(\alpha _{2}\) are in \(1{:}2\) or \(1{:}3\) resonance, see 3.4.2.
 codimension 4:

if 0 is an eigenvalue of algebraic multiplicity 5, then the codimension is determined by the Jordan chains—see [1]—starting with a single maximal Jordan chain of codimension 4.
2.6 \(m \geq 6\)
These cases form two series according to whether \(m\) is odd, where the situation resembles that of §2.5, or \(m\) is even. For low codimension \(c\) compared to the dimension \(m\), the theory is equivalent to the dissipative one. All such bifurcations are normally hyperbolic versions of dissipative bifurcations—see [11, 23, 29] for a classification of the ones of codimension 2—and the only remainder of volume preservation is that at the bifurcation the sum of the hyperbolic eigenvalues vanishes. To be precise, the theory is dissipative for odd \(m\) whenever \(2 c \le m  3\) and for even \(m\) whenever \(2 c \le m  4\). For instance, in dimension \(m = 6\) the codimension \(c\) must be at least 2 for the bifurcation to have volume preserving characterics.
Again, bifurcations have volumepreserving characteristics of dimension \(m\) only if all eigenvalues are on the imaginary axis. Then the lowest codimension occurs if all eigenvalues are simple. For \(m=6\) this means that the eigenvalues form \(3 = \frac{1}{2}m\) pairs \(\pm \mathrm {i}\alpha _{1}\), \(\pm \mathrm {i}\alpha _{2}\), \(\pm \mathrm {i}\alpha _{3}\); for odd \(m\) compare with §2.5. Multiple eigenvalues on the imaginary axis again lead to higher codimensions, as do other low order resonances.
3 Normal Forms
The standard approach to local bifurcations triggered by nonhyperbolic eigenvalues—which we follow here as well—is twofold, compare with [23, 29]. The first step is to reduce to a centre manifold. For volumepreserving dynamical systems the restriction \(\operatorname {trace}\varOmega = 0\) enforces the sum of hyperbolic eigenvalues to be 0—indeed, the pairs of eigenvalues \(\pm \mathrm {i}\alpha \) on the imaginary axis add up to 0 as well. Correspondingly, the centre manifold is always truly hyperbolic—neither attracting nor repelling—but there are no further restrictions for the flow on the centre manifold. In particular, it is not true that the system on the centre manifold has to be again volumepreserving; this gives more flexibility to the bifurcation unfolding on the centre manifold, which therefore typically has a dissipative character. Thus, in the sequel we may assume that at the bifurcation all \(m\) eigenvalues of the bifurcating equilibrium are on the imaginary axis, i.e. no hyperbolic directions have to be split off through a centre manifold reduction.
The second step in the standard approach followed here is to compute a suitable normal form. Indeed, every pair of purely imaginary eigenvalues \(\pm \mathrm {i}\alpha \) generates an \(\mathbb {S}^{1}\)action on \(\mathbb {R}^{m}\). If all eigenvalues are simple, on the imaginary axis and share no resonances, then they yield a \(\mathbb {T}^{\ell }\)action on \(\mathbb {R}^{m}\) with \(\ell = \lfloor \frac{1}{2}m \rfloor \). Normal form theory allows to push this symmetry through the Taylor series and it is the order up to which this normalization is performed that decides which resonances are of low order and which are of higher order. Low order resonances result in additional ‘resonant terms’ in the normal form and lower the dimension of the resulting symmetry group \(\mathbb {T}^{\ell }\) to some \(\mathbb {T}^{\ell ^{\prime }}\), \(\ell ^{\prime } < \ell \). High order resonances have no influence on the normal form up to the chosen order. Truncating the not normalized higher order terms then yields a \(\mathbb {T}^{\ell }\)equivariant (or a \(\mathbb {T}^{\ell ^{\prime }}\)equivariant) approximation of the original vector field.
At this point, the standard approach is first to study the symmetric normal form dynamics and then to show which features survive the perturbation back to the original system. In this section we concentrate on the dynamics defined by the truncated normal form and we treat the perturbation problem in Sect. 4.
3.1 \(m=1\)
3.2 \(m=2\)
3.3 \(m=3\)
As we have seen in §2.3, a bifurcating equilibrium necessarily has an eigenvalue 0. In case the other 2 eigenvalues are \(\pm \beta \) we can reduce to a centre manifold where, under variation of a single parameter, generically a saddlenode bifurcation takes place; for details see [15, 23, 29]. At the bifurcation two equilibria on the centre manifold meet, one attracting, one repelling. The corresponding eigenvalue is the difference (in absolute values) of the 2 hyperbolic eigenvalues which no longer cancel each other out. This is the only remaining influence of the vector field being volumepreserving, also in case of degeneracies of higher order terms that lead to bifurcations on the centre manifold of higher codimension.
3.3.1 The VolumePreserving Hopf Bifurcation
3.3.2 Bifurcations of Codimension 2
In the unfolding (16) we required \(a(0) b(0) \neq 0\) and later even scaled to \(a=1\), \(b = \pm 1\). This makes \(a=0\) or \(b=0\) a degenerate situation, triggering a bifurcation of codimension 2 that includes both the hyperbolic and elliptic volumepreserving Hopf bifurcation in a subordinate way. In the corresponding normal form the zero coefficient gets replaced by the—second—unfolding parameter.
3.4 \(m=4\)
A single eigenvalue 0 triggers a normally hyperbolic saddlenode bifurcation and a pair of purely imaginary eigenvalues \(\pm \mathrm {i}\alpha \), \(\alpha > 0\) triggers a normally hyperbolic (dissipative) Hopf bifurcation, see [15, 23, 29] for details. Next to these two bifurcations of dissipative character there is a third bifurcation of codimension 1, see §3.4.1 below. Degeneracies in the higher order terms lead to a normally hyperbolic cusp bifurcation and to a normally hyperbolic degenerate Hopf bifurcation, respectively. A third bifurcation of codimension 2 is the normally hyperbolic Bogdanov–Takens bifurcation, unfolding a double eigenvalue 0. For the other bifurcations of codimension 2 see §3.4.2 below. A triple eigenvalue 0 is necessarily a fourfold eigenvalue 0 and the codimension is determined by the length(s) of the Jordan chain(s), see §2.4.
3.4.1 The VolumePreserving Double Hopf Bifurcation
For the elliptic volumepreserving double Hopf bifurcation also the ‘ordinary’ Hopf bifurcation within the \((z_{3}, z_{4})\)plane \(\{ \tau _{1} = 0 \}\) is subcritical—the repelling equilibrium at the origin becomes attracting as a within \(\{ \tau _{1} = 0 \}\) attracting periodic orbit shrinks down; all attracting and repelling characterisations within the plane are again balanced by repelling and attracting behaviour normal to the plane because of volume preservation. The heteroclinic connection outside the \(\tau _{i}\)axes in the reduced system reconstructs to a toroidal cylinder \(\mathbb {T}^{2} \times \mathopen ] \lambda \sqrt{2}, 0 \mathclose [\) consisting of heteroclinic orbits from the hyperbolic periodic orbit in the \((z_{3}, z_{4})\)plane \(\{ \tau _{1} = 0 \}\) to the hyperbolic periodic orbit in the \((z_{1}, z_{2})\)plane \(\{ \tau _{2} = 0 \}\) and the union of these is the 3sphere \(\{ \tau _{1} + \tau _{2} =  \lambda \} \subseteq \mathbb {R}^{4}\). Finally, the equilibria with both \(\tau _{i} \neq 0\) that exist for \(\lambda < 0\) lead to normally elliptic invariant 2tori surrounded by invariant 3tori. For \(\lambda > 0\) the only critical element after the elliptic volumepreserving double Hopf bifurcation is the hyperbolic equilibrium at the origin.
3.4.2 Bifurcations of Codimension 2
As in §3.3.2 the nonlinear degeneracies \(c_{1} = 0\) and \(c_{2} = 0\) lead to degenerate volumepreserving double Hopf bifurcations and for the necessary higher order normalization more resonances (20) have to be excluded. The codimension also increases to 2 where the 2 normal frequencies satisfy a low order resonance; scaling \(\alpha _{1} = 1\) this happens for the \(1{:}1\) resonance \(\alpha _{2} = 1\), the \(1{:}2\) resonance \(\alpha _{2} = 2\) and for the \(1{:}3\) resonance \(\alpha _{2} = 3\). We remark that there are no ‘indefinite’ volumepreserving resonances. The remaining bifurcation of codimension 2—triggered by a pair \(\pm \mathrm {i}\alpha _{1} = \pm \mathrm {i}\) of purely imaginary eigenvalues and a double zero eigenvalue \(\pm \mathrm {i}\alpha _{2} = 0\)—may also be termed a \(1{:}0\) resonance.
3.5 \(m \geq 5\)
There are no more truly volumepreserving bifurcations of codimension 1, but for \(m=5\) and \(m=6\) it is of codimension 2 that the spectrum consists of simple eigenvalues on the imaginary axis. A versal unfolding has 1 parameter for each real part to pass through 0, except for the last eigenvalue or the real part of the last pair of eigenvalues which because of (2) is determined by the sum of the other eigenvalues, compare with (9). To normalize with respect to the \(\mathbb {T}^{\ell }\)action, \(\ell = \lfloor \frac{1}{2} m \rfloor \) generated by the \(\ell \) rotations in the \((z_{2j1}, z_{2j})\)planes (\(j = 1, \ldots , \ell \)) we again exclude low order resonances \(k_{1} \alpha _{1} + \cdots + k_{\ell } \alpha _{\ell } = 0\), \(0 \neq k \leq 4\). The resulting normal forms generalize (22) for \(m\) even and generalize a combination of (16) and (22) for \(m\) odd. The codimension increases where coefficients in these normal forms vanish or where normal frequencies are in low order resonances, including the resonances of multiple eigenvalues 0.
4 Nonlinear Bifurcations
Truncated normal forms provide standard models for bifurcations and an important question is whether the dynamical properties of the approximating truncation persist when perturbing back to the original family. This is certainly the case where—after a suitable reparametrisation—the flows of the two systems are conjugate. To avoid that the periods of occurring periodic orbits act as moduli we weaken the notion of conjugacy to that of an equivalence of the two systems, i.e. allow for time reparametrisation along the orbits. For the same reason the equivalences need to be only homeomorphisms and the parameter changes continuous, i.e. not necessarily smooth. Note that we do not require the dependence of the equivalences on the parameter to be continuous. This still leaves e.g. the rotation numbers of invariant 2tori as possible moduli and we shall see what can be said in such more involved situations.
For \(m=1\) all flows with an equilibrium are equivalent because they are all equal—being volumepreserving enforces all other points to be equilibria as well. For \(m=2\) the smooth right equivalences between simple singularities provide equivalences between the local flows and the moduli of high codimension can be dealt with by passing to continuous right or leftright equivalences, see [24, 38] and references therein. For \(m=3\) we have the volumepreserving Hopf bifurcation detailed in §4.1 below and the normally hyperbolic saddlenode bifurcation. For the latter the flow is locally topologically conjugate to the flow on the centre manifold superposed with the linear flow \(\dot{z} = \bigl({}_{1}^{0} {}_{0}^{1} \bigr) z\) and the flow on the centre manifold is locally topologically equivalent with the flow of the standard saddlenode bifurcation, see [15, 23, 29] and references therein.
For \(m=4\) we have next to the normally hyperbolic saddlenode bifurcation also a normally hyperbolic (dissipative) Hopf bifurcation—locally topologically equivalent to the standard Hopf bifurcation superposed with \(\dot{z} = \bigl({}_{1}^{0} {}_{0}^{1} \bigr) z\), for details see [15, 23, 29] and references therein—and the volumepreserving double Hopf bifurcation detailed in §4.2 below. There are thus four bifurcations of codimension 1 when \(m \geq 3\): two truly volumepreserving ones in dimensions \(m=3\) and \(m=4\), respectively and two normally hyperbolic ones of dissipative character which take place on a centre manifold of dimension \(m=1\) or \(m=2\), respectively. While it is of course possible to have e.g. a normally hyperbolic Hopf–Hopf bifurcation in dimension \(m \geq 6\), this bifurcation then acquires a dissipative character and in particular has codimension 2. For results on truly volumepreserving bifurcations of codimension 2 see [21].
4.1 The VolumePreserving Hopf Bifurcation
We have seen in §3.3.1 that there are two different cases distinguished by the sign of the product \(a(0) b(0)\) in (16). As proposed after (18) we scale to \(a=1\) and \(b = \pm 1\), the sign of \(a(0) b(0)\), and speak of the hyperbolic volumepreserving Hopf bifurcation if \(b=1\) and of the elliptic volumepreserving Hopf bifurcation if \(b = 1\). The simpler of the two families is the hyperbolic one and this family also allows for the stronger result.
Theorem 1
(Hyperbolic case in \(\mathbb {R}^{3}\))
Generic 1parameter families of volumepreserving vector fields on \(\mathbb {R}^{3}\)with normalized 2jet (16), \(a(0) b(0) > 0\)are locally structurally stable.
For the proof see [5, 6]. Next to \(\beta ^{\prime }(0) \neq 0\) and \(\lambda ^{\prime }(0) \neq 0\) allowing to achieve (8) and (18) the genericity condition concerns the saddle connection along the vertical axis in the dynamics of (16); this connection needs to be broken up by the perturbation from the normal form (16) to the original vector fields for all parameter values for the latter family to satisfy the genericity condition. Note that this means that the \(\mathbb {S}^{1}\)symmetry is broken, in particular it is not possible to read off from the coefficients of any normal form whether the genericity condition is satisfied. As proven in [13], the family of equivalences can be chosen continuous for \(\lambda \leq 0\), but not for \(\lambda > 0\).
The dynamics of the elliptic volumepreserving Hopf bifurcation is more involved—when \(\lambda > 0\), see Fig. 1. When \(\lambda < 0\) there are no equilibria near the origin and we have local structural stability by the flow box theorem, merely using the height as a Lyapunov function. The full complexity of the volumepreserving Hopf bifurcation occurs for \(b = 1\), \(\lambda > 0\). Indeed, for the normal form (16) the two saddles are not only connected by a heteroclinic orbit along the vertical axis, but also by a whole 2sphere of spiralling heteroclinic orbits. Furthermore, there is a family of conditionally periodic 2tori around the elliptic periodic orbits filling up the inside of this sphere; in [19] this is termed a vortex bubble. Therefore the situation under perturbation from the normal form (16) back to the original family of vector fields is less clear.
Remarks 2

It is generic that this perturbation breaks the \(\mathbb {S}^{1}\)symmetry. Also the 1dimensional saddle connection generically breaks as the proof for \(b = 1\) applies here as well. This situation is described in [16]; the phenomena are infinitely flat and for analytic vector fields probably exponentially small.

The 2spheres of coinciding stable and unstable manifolds generically do break up as the stable and unstable manifolds do not coincide anymore. For a generic volumepreserving flow these manifolds meet transversely along spiralling heteroclinic orbits and within a generic family the set of parameter values for which the intersection is not transverse is at most countable, again see [16].

There are infinitely many horseshoes related to subordinate Shilnikovhomoclinic bifurcations invoked by the breakup of both the 1 and the 2dimensional stable and unstable manifolds; these bifurcations have codimension 1 and occur for a discrete set of parameter values accumulating on 0. See [16] for more details. Since the horseshoes are connected the corresponding symbolic dynamics needs an infinite alphabet.

The family of invariant 2tori persists as a Cantor family with inside the gaps at least one periodic orbit corresponding to the rational frequency ratio opening that gap. The Cantor family of quasiperiodic tori extends all the way to the broken 2sphere and the broken line. The infinite (c.q. exponential) flatness makes many things possible, see also [12].
In particular we have the following result proven in [5, 7], weaker than Theorem 4.1. Where \(\varOmega \)stability is structural stability of the restriction of the system to the nonwandering set, quasiperiodic stability is structural stability after a further restriction to a measuretheoretically large union of quasiperiodic tori.
Theorem 2
(Elliptic case in \(\mathbb {R}^{3}\))
Generic 1parameter families of volumepreserving vector fields on \(\mathbb {R}^{3}\)with normalized 2jet (16), \(a(0) b(0) < 0\)are locally quasiperiodically stable.
4.2 The VolumePreserving Double Hopf Bifurcation
As we have seen in §3.4.1 there are two different cases distinguished by the sign of \(c_{2} = \pm 1\) in (22). Here the hyperbolic case is the one with the lower signs \(c_{2} =  1\), while the upper signs \(c_{2} = + 1\) yield the elliptic volumepreserving double Hopf bifurcation. This choice is made for the periodic orbits in the reduced system to again occur in the elliptic case. For the hyperbolic volumepreserving double Hopf bifurcation the only critical elements are periodic orbits in the \((z_{1}, z_{2})\) and \((z_{3}, z_{4})\)planes, the equilibria at the origin and the heteroclinic orbits between the latter and the former.
Theorem 3
(Hyperbolic case in \(\mathbb {R}^{4}\))
Generic 1parameter families of volumepreserving vector fields on \(\mathbb {R}^{4}\)with normalized 3jet\(M(\tau ; \lambda ) z\)given by (22), lower signs are locally structurally stable.
The proof runs along the same lines as the proof of Theorem 4.1 in [5, 6]. In the elliptic case there is structural stability for \(\lambda \geq 0\) by the flow box theorem as there are no critical elements other than the equilibria at the origin. The invariant 2 and 3tori at \(\lambda < 0\) prevent such a result to hold true for the whole family.
Theorem 4
(Elliptic case in \(\mathbb {R}^{4}\))
Generic 1parameter families of volumepreserving vector fields on \(\mathbb {R}^{4}\)with normalized 3jet\(M(\tau ; \lambda ) z\)given by (22), upper signs are locally quasiperiodically stable.
For the proof see [2]. Regarding the various heteroclinic phenomena not much has been explicitly written down as compared to §4.1, but the infinite (c.q. exponential) flatness [12, 14, 16] is expected to be similar. It is generic for stable and unstable manifolds to no longer coincide. Mere counting of the dimensions—2 for both the stable and unstable manifold of the equilibrium at the origin which in the unperturbed case coincide with the unstable resp. stable manifold of the periodic orbit resulting from the bifurcation—shows that generically these manifolds cease to even intersect.
As the 3sphere consisting of heteroclinic orbits between the periodic orbits breaks up, volume preservation enforces that the 3dimensional stable and unstable manifolds still intersect after perturbation. Generically this intersection is transverse, so similar to the 2sphere in the elliptic volumepreserving Hopf bifurcation one would expect the set of parameter values for which this is not the case to be an at most countable subset of \(\{ \lambda < 0 \}\). Again this breakup of stable and unstable manifolds invokes subordinate Shilnikovlike homoclinic bifurcations, which are further complicated by the additional circular dimension, compare with [30].
5 Bifurcations of Periodic Orbits
Floquet’s theorem yields near a periodic orbit the reducibility of the equations of motion to Floquet form \((6)\) on \(\mathbb {T}\times \mathbb {R}^{m}\) with parameter \(\mu \in \mathbb {R}^{s}\) and \(\sigma (0) = 0\), making \(\mathbb {T}\times \{ 0 \}\) the periodic orbit for \(\mu = 0\). To avoid repetitious reductions to a centre manifold we assume that all \(m\) Floquet multipliers are on the unit circle. Then the condition of Floquet’s theorem is that if −1 is a Floquet multiplier, then it is of even multiplicity and the associated Jordan blocks come in equal pairs. In particular, the Floquet multipliers and the Floquet exponents are in \(1{:}1\) correspondence, the exponential mapping turning the latter into the former.
The second step after reduction to a centre manifold is to compute a suitable normal form. In the periodic case a truncated normal form acquires a \(\mathbb {T}^{\ell + 1}\)symmetry, coming from \(\ell \) pairs of purely imaginary eigenvalues \(\pm \mathrm {i}\alpha (0) \neq 0\) of \(\varOmega (0)\) and invariance under translation along the first factor \(\mathbb {T}\) of the phase space \(\mathbb {T}\times \mathbb {R}^{m}\). Additional nonresonance conditions between the internal frequency \(\omega (0)\) and the normal frequencies \(\alpha _{1}(0)\), …, \(\alpha _{\ell }(0)\) are needed to avoid new resonance terms in the normal form.
To preserve the oriented volume the Floquet multiplier −1 has to be of even algebraic multiplicity. Recall that the condition of Floquet’s theorem furthermore requires that also the geometric multiplicity is even as the Jordan blocks have to come in equal pairs. In case the condition is not satisfied this can be easily remedied by passing to a double cover \(\mathbb {T}\times \mathbb {R}^{m}\) of the phase space, with the deck group \(\mathbb {Z}_{2}\) as additional symmetry group. Correspondingly, there is a third type of bifurcation for periodic orbits that does not exist for equilibria on manifolds—the flip or period doubling bifurcation. Under the assumption that \(\omega (0) \neq 0\) in (6a) we can take \(\{ x_{0} \} \times \mathbb {R}^{m}\), \(x_{0} \in \mathbb {T}\) as a Poincaré section and study the resulting volumepreserving Poincarémapping. Since the normal form is independent of \(x\) we may perform a partial symmetry reduction to \(\mathbb {R}^{m}\)—the time–1mapping of this reduced flow then is the Poincarémapping of the normal form dynamics. One also speaks of an integrable Poincarémapping, and while the Poincarémapping of the ‘original’ volumepreserving dynamical system is in general not integrable, the approximation by the normal form shows that it is close to an integrable one.
5.1 \(m=1\)
Normalizing around the periodic orbit \(\mathbb {T}\times \{ 0 \}\) and reducing the resulting \(\mathbb {T}\)symmetry leads to a volumepreserving flow on ℝ with equilibrium \(z=0\), whence all \(z_{0} \in \mathbb {R}\) are equilibria. This reconstructs to a flow on the cylinder \(\mathbb {T}\times \mathbb {R}\) where all \(\mathbb {T}\times \{ z _{0} \}\), \(z_{0} \in \mathbb {R}\) are periodic orbits. The Poincarémapping on \(\{ x_{0} \} \times \mathbb {R}\), \(x_{0} \in \mathbb {T}\) is the identity mapping.
A small perturbation of the identity mapping on ℝ is monotonous. This allows to interpolate the mapping by a flow on ℝ—and to preserve volume, this flow must be a constant translation (10), see §3.1. Since the perturbation is by higher order terms in the normalizing coordinates, the point \(z=0\) remains an equilibrium whence the translation remains the identity mapping—all volumepreserving flows on \(\mathbb {T}\times \mathbb {R}\) with a periodic orbit \(\mathbb {T}\times \{ z_{0} \}\) are periodic flows.
We remark that the cylinder \(\mathbb {T}\times \mathbb {R}\) cannot be the double cover of a phase space with a flow preserving an oriented volume. Indeed, dividing out a deck group \(\mathbb {Z}_{2}\) turns the cylinder into the Möbius band which is not orientable and hence cannot carry a volume, or area form.
5.2 \(m=2\)
The Poincarémapping on \(\{ x_{0} \} \times \mathbb {R}^{2}\), \(x_{0} \in \mathbb {T}\) is an areapreserving mapping. In addition to the periodic centresaddle bifurcation inherited from §3.2, triggered by a (double) eigenvalue 1 of the Poincarémapping, there is the perioddoubling bifurcation triggered by a (double) eigenvalue −1 of the Poincarémapping. While Hamiltonian dynamical systems do preserve volume, it would be out of proportion to discuss this vast theory in the context of volumepreserving dynamical systems. We therefore refer to [31] for further details on bifurcations of areapreserving mappings.
5.3 \(m=3\)
Normalizing around the periodic orbit \(\mathbb {T}\times \{ 0 \} \subseteq \mathbb {T}\times \mathbb {R}^{3}\) with Floquet multipliers \(\mathrm {e}^{\pm \mathrm {i}\alpha }\) and 1 and reducing the resulting \(\mathbb {T}^{2}\)action leads to the same family of Hamiltonian systems as in §3.3.1, with additional nonresonance conditions on the internal frequency \(\omega (0)\) and the normal frequency \(\alpha (0)\). Reconstructing the reduced dynamics back to \(\mathbb {T}\times \mathbb {R}^{3}\) amounts to superposing that Hamiltonian flow with a conditionally periodic motion on \(\mathbb {T}^{2}\), or to superpose the flow of (16) with the periodic motion of (6a), where furthermore the \(\mathcal {O}(z)\)term is \(x\)independent. In this way the equilibria of (16) on the vertical axis become periodic orbits, the periodic orbits around the vertical axis become invariant 2tori and the invariant 2tori shrinking down to elliptic periodic orbits become invariant 3tori shrinking down to normally elliptic invariant 2tori. Furthermore, the heteroclinic connections along the vertical axis become cylinders of heteroclinic orbits spiralling between periodic orbits \(\mathbb {T}\times \{ (0, 0, z_{3}^{j}) \}\), \(j = 1, 2\) and the 2sphere \(\mathbb {S}^{2}\) of heteroclinic orbits turns into the product \(\mathbb {T}\times \mathbb {S}^{2}\) consisting of heteroclinic orbits, compare with [30].
Theorem 1
(Periodic elliptic case in \(\mathbb {R}^{3}\))
Note that we may restrict to \(X\) being the truncated normal form \((24)\).
Outline of proof
First construct an equivalence between the two periodic orbits in \(U \cap \{ \beta = 0 \}\) and \(V \cap \{ \beta = \beta _{0} \}\) by mapping the point \((0, 0; 0)\) on the former to a point \((x_{0}, z_{0}; \beta _{0})\) on the latter and extending to all of \(\mathbb {T}\times \{ 0 \} \times \{ 0 \}\) using the flow and rescaling time to account for the possibly different periods of the unperturbed and perturbed periodic orbits. The flow box theorem then provides for an extension to all of \(U\), taking \(V = \varPhi (U)\), with the desired properties. □
The elliptic case \(b = 1\) of the volumepreserving Hopf bifurcation has for \(\beta > 0\) invariant 2tori already when \(n=0\), see §3.3.1, and in the periodic case \(n=1\) these turn into invariant 3tori; moreover the elliptic periodic orbits turn into normally elliptic invariant 2tori. In the hyperbolic case \(b=1\) of the periodic volumepreserving Hopf bifurcation there are normally hyperbolic invariant 2tori for \(\beta < 0\), while for \(\beta \geq 0\) the only critical elements are the periodic orbits \(\mathbb {T}\times \{ (0, 0, z_{3}) \}\)—coming from the equilibria \((0, 0, z_{3}) \in \mathbb {R}^{3}\) of (16)—and their stable and unstable manifolds.
Theorem 2
(Periodic hyperbolic case in \(\mathbb {R}^{3}\))
Again we may restrict to \(X\) being the truncated normal form \((24)\). Note that here we do not claim the family \(\phi _{\beta }\) of equivalences to depend continuously on the parameter \(\beta \), in particular \(\varPhi (x, z; \beta ) := ( \phi _{\beta }(x, z); \varphi (\beta ) )\) does not necessarily define a homeomorphism from \(U\) to \(V\). The obstruction to such a homeomorphism is formed by infinitely many moduli [13]; these are provided by the winding around each other of the broken heteroclinic connections.
Outline of proof
Fixing \(\beta _{1} \geq 0\), first construct an equivalence between the periodic orbits in \(U \cap \{ \beta = \beta _{1} \}\) and \(V \cap \{ \beta = \varphi (\beta _{1}) \}\) as in the proof of Theorem 5.1. The flow box theorem then provides for an extension to all of \(U\), taking \(V = \varPhi (U)\), with the desired properties. □
The two proofs show that \(\varOmega \)stability can lead to structural stability. However, where the critical elements are conditionally periodic tori the occurring resonances make \(\varOmega \)stability too strong a notion to achieve—compare e.g. with the case \(b = 1\), \(\lambda > 0\) in §4.1 above. We shall therefore have to weaken our statements to quasiperiodic stability.
5.4 \(m=4\)
Additional nonresonance conditions on the internal frequency \(\omega (0)\) and the normal frequencies \(\alpha _{1}(0)\) and \(\alpha _{2}(0)\) allow to normalize around the periodic orbit \(\mathbb {T}\times \{ 0 \} \subseteq \mathbb {T}\times \mathbb {R}^{4}\) with Floquet multipliers \(\mathrm {e}^{\pm \mathrm {i}\alpha _{1}}\) and \(\mathrm {e}^{\pm \mathrm {i}\alpha _{2}}\). Reducing the resulting \(\mathbb {T}^{3}\)action leads to the same family of Hamiltonian systems as in §3.4.1 and reconstructing the reduced dynamics back to \(\mathbb {T}\times \mathbb {R}^{4}\) amounts to superposing the flow defined by (22) with the periodic motion of (6a), where furthermore the \(\mathcal {O}(z)\)term is \(x\)independent. In this way the equilibria of (22) at the origin become periodic orbits, the periodic orbits within the planes \(\{ \tau _{i} = 0 \}\) become invariant 2tori and the invariant 3tori shrinking down to normally elliptic 2tori become invariant 4tori shrinking down to normally elliptic 3tori. Furthermore, the heteroclinic connections within the planes \(\{ \tau _{i} = 0 \}\) become toroidal cylinders of heteroclinic orbits spiralling between periodic orbits \(\mathbb {T}\times \{ 0 \}\) and normally hyperbolic 2tori while the 3sphere \(\mathbb {S}^{3}\) consisting of heteroclinic orbits turns into the product \(\mathbb {T}\times \mathbb {S}^{3}\).
Theorem 3
(Periodic elliptic case in \(\mathbb {R}^{4}\))
We may restrict to \(X\) being the truncated normal form (30).
Outline of proof
By the Implicit Mapping Theorem the vector field \(Y_{\lambda }\) has for all \(\lambda \) a periodic orbit close to \(\mathbb {T}\times \{ 0 \}\). Denote by \(\lambda _{0}\) the unique parameter value where the pairs of Floquet multipliers of the periodic orbit of \(Y_{\lambda }\) both meet the unit circle and define \(\varphi (\lambda ) := \lambda _{0} + \lambda \). Then first construct an equivalence between the 2 periodic orbits in \(U \cap \{ \lambda = \lambda _{1} \}\) and \(V \cap \{ \lambda = \varphi (\lambda _{1}) \}\) by mapping the point \((0, 0; \lambda _{1})\) on the former to a point \((x_{0}, z_{0}; \varphi (\lambda _{1}))\) on the latter and extending to all of \(\mathbb {T}\times \{ 0 \} \times \{ \lambda _{1} \}\) using the flow and rescaling time to account for the possibly different periods of the unperturbed and perturbed periodic orbits. The flow box theorem then provides for an extension to all of \(U\), taking \(V = \varPhi (U)\), with the desired properties. □
The invariant tori for \(\lambda < 0\) in the elliptic case and for all \(\lambda \neq 0\) in the hyperbolic case again prevent such a strong result to hold true. However, using the same approach as in §5.3 we do get quasiperiodic stability for both cases of the periodic volumepreserving double Hopf bifurcation.
6 Quasiperiodic Bifurcations
In the previous sections we studied (semi)local bifurcations on \(\mathbb {T}^{n} \times \mathbb {R}^{m}\) around \(\mathbb {T}^{n} \times \{ 0 \}\) for \(n=0\) and \(n=1\). While for \(n=1\) a ‘resonant periodic orbit’ corresponds to a \(\mathbb {T}^{1}\) consisting of equilibria and is easily discarded, the conditionally periodic flow \(x \mapsto x + t \omega \) on \(\mathbb {T}^{n}\), \(n \geq 2\) has a dense set of resonant frequency vectors \(\omega \) inside the set of all frequency vectors \({\omega \in \mathbb {R}^{n}}\). Thus, for \(n \geq 2\) even the critical elements undergoing a volumepreserving bifurcation of codimension 1 may under perturbation of the whole unfolding family disappear in a resonance gap. Hence, at least \(s \geq n\) parameters are needed to control both the bifurcation scenario and the frequency ratio of the bifurcating \(n\)tori. This high number of parameters can be brought down by using Rüssmann’s nondegeneracy condition.
6.1 \(m=1\)
6.2 \(m=2\)
6.3 \(m \geq 3\)
We concentrate on the two quasiperiodic bifurcations of codimension 1—the quasiperiodic volumepreserving Hopf bifurcation and the quasiperiodic volumepreserving double Hopf bifurcation with normal dimensions \(m=3\) and \(m=4\), respectively. The phase space remains \(\mathbb {T}^{n} \times \mathbb {R}^{m}\), \(n \geq 2\) and for simplicity we first assume that \(s \geq n + 2\) for the dimension of the parameter space \(\mathbb {R}^{s}\). For the quasiperiodic volumepreserving Hopf bifurcation, triggered for \(m=3\) by 3 simple Floquet exponents on the imaginary axis, this allows to generalize (25) to achieve \(\mu = (\beta , \omega , \alpha , \hat{\mu })\) and to drop the mute parameter \(\hat{\mu } \in \mathbb {R}^{sn2}\), effectively restricting to \(s=n+2\).
Then the hyperbolic quasiperiodic volumepreserving Hopf bifurcation—with the upper sign \(b = + 1\) in \((24)\) with \(x \in \mathbb {T}^{n}\)—is quasiperiodically stable, see [11] for more details. The same holds true for the elliptic quasiperiodic volumepreserving Hopf bifurcation where again \(\eta \) is used as a local parameter to control the frequency vector of the invariant \(({n+2})\)tori and control is restricted to the ratio \([\omega : \alpha : \eta ] \in \mathbb {R}\mathbb {P}^{n+2}\) for the normally elliptic invariant \((n+1)\)tori.
Using a single parameter \(\beta \in \mathbb {R}\) is still possible, but requires explicit Diophantine conditions on \((\omega (0), \alpha (0)) \in \mathbb {R}^{n+1}\) to avoid that the complete bifurcation scenario disappears in a resonance gap. Indeed, the considerations in [8, 10, 17] concerning the (dissipative) quasiperiodic Hopf bifurcation apply here as well. In the hyperbolic case it is simpler to describe the bifurcation for \(\beta \) decreasing through 0: two normally hyperbolic \(n\)tori meet and acquire an additional frequency, resulting in one normally hyperbolic \((n+1)\)torus after bifurcation. The gaps between the \(\beta \)values with quasiperiodic tori are filled by \(\beta \)values for which there are still normally hyperbolic invariant tori, but the flow on these may be asymptotically periodic or chaotic.
Also in the elliptic case \(b =  1\) it is simpler to describe the bifurcation for \(\beta \) decreasing: two families of normally hyperbolic \(n\)tori meet a family of normally elliptic \((n+1)\)tori surrounded by invariant \((n+2)\)tori and all tori vanish at the bifurcation. Again, all resonance gaps of the \(n\)tori are filled by normal hyperbolicity, but the higherdimensional tori are parametrised by a Cantor subset of the parameter space.
For the quasiperiodic volumepreserving double Hopf bifurcation on \(\mathbb {T}^{n} \times \mathbb {R}^{4}\), \(n \geq 2\) we first assume that \(s = n + 3\) for the dimension of the parameter space \(\mathbb {R}^{s}\) with \(\mu = (\lambda , \omega , \alpha _{1}, \alpha _{2})\). For \(s > n + 3\) we drop mute parameters and at the end we see how the single parameter \(\lambda \in \mathbb {R}\) suffices.
The hyperbolic quasiperiodic volumepreserving double Hopf bifurcation—with the lower sign \(c_{2} =  1\) in (30) with \(x \in \mathbb {T}^{n}\)—is quasiperiodically stable, see [11] for more details. Restricting to a single parameter \(\lambda \in \mathbb {R}\) again requires explicit Diophantine conditions on \((\omega (0), \alpha (0)) \in \mathbb {R}^{n+2}\). Using normal hyperbolicity, the gaps between the \(\lambda \)values with quasiperiodic tori are filled by \(\lambda \)values of invariant tori with phaselocked flow. The elliptic quasiperiodic volumepreserving double Hopf bifurcation—with the upper sign \(c_{2} = 1\) in (30) with \(x \in \mathbb {T}^{n}\)—is quasiperiodically stable as well, see [11]. While the gaps between quasiperiodic normally hyperbolic tori are filled by phaselocked tori, the normally elliptic tori form a Cantor family, as do the ‘maximal’ invariant tori \(\mathbb {T}^{n+3}\).
7 Conclusions
Bifurcations of codimension 1 in volumepreserving systems that truly have a volumepreserving character occur in low (normal) dimension \(m\). These are the centresaddle bifurcation—a bifurcation of Hamiltonian systems—the volumepreserving Hopf bifurcation and the volumepreserving double Hopf bifurcation, respectively. For the latter two the normal form approximations can be reduced to a Hamiltonian system in 1 degree of freedom. In case of higher (normal) dimensions \(m \geq 5\) there are no bifurcations of codimension 1 that truly have a volumepreserving character. The case \(m=1\) of a single normal dimension is special as volume preservation puts severe bounds on the dynamics in that normal direction and rather gives it the character of an—internal—parameter.
For higher codimensions \(c \geq 2\) there is a dimension threshold \(m \leq 2(c+1)\) up to which the bifurcation may truly have a volumepreserving character. The codimension increases by putting more eigenvalues/Floquet exponents on the imaginary axis—as a volumepreserving character requires that all eigenvalues/Floquet exponents are on the imaginary axis, this necessarily increases \(m\). The codimension also increases in case of multiple eigenvalues/Floquet exponents on the imaginary axis and for low order resonances. Finally, failing to fulfil the nondegeneracy conditions of bifurcations of lower codimension—e.g. that certain coefficients in the normal form be nonzero—increases the codimension as well, see [24, 38] for the Hamiltonian case \(m=2\). Typically such coefficients then become parameters in the normal form.
For mappings, next to Floquet multipliers \(\mathrm {e}^{\pm \mathrm {i}\alpha }\), \(\alpha \notin \pi \mathbb {Z}\) and \(+1\) also Floquet multipliers −1 trigger bifurcations. For instance, when \(m=2\) the Hamiltonian flip bifurcation—with linearization \(\bigl({}_{\phantom{}0}^{1} \; {}_{1}^{\phantom{}1}\bigr)\) at the bifurcating fixed point—may occur, leading to a perioddoubling bifurcation of the corresponding volumepreserving flow on \(\mathbb {T}\times \mathbb {R}^{2}\), see [24, 31] for details. When \(m=4\) then a double Floquet multiplier −1 can be accompanied by a pair \(\mathrm {e}^{\pm \mathrm {i}\alpha }\) on the unit circle, triggering a volumepreserving combination of perioddoubling and Hopf bifurcations of codimension 2.
When \(m=3\) a set of 3 Floquet multipliers \(\mathrm {e}^{\pm \mathrm {i}\alpha }\) and −1 stems from the linearization of a Poincaré mapping that does not preserve the volume form \(\mathrm {d}z_{1} \wedge \mathrm {d}z_{2} \wedge \mathrm {d}z_{3}\). While passing to a covering space with deck group \(\mathbb {Z}_{2}\) should lead to an unfolding for which the normal form is a \(\mathbb {Z}_{2}\)equivariant version of (16) that does preserve the volume form and under dividing out the symmetry \(\mathbb {T}^{2} \times \mathbb {Z}_{2}\) reduces to (21) with \(\tau _{1} = \tau \) and \(\tau _{2} = \frac{1}{2} \zeta ^{2}\), dividing out only \(\mathbb {Z}_{2}\) would then again yield a dynamical system on a nonorientable manifold. Such a system on \((\mathbb {T}\times \mathbb {R}^{3})/\mathbb {Z}_{2}\) is volumepreserving in the sense that the orientation cover carries a volume form that is preserved by the lift of the flow to \(\mathbb {T}\times \mathbb {R}^{3}\). Future work should also be concerned with the volumepreserving codimension 2 bifurcation triggered by a single Floquet multiplier \(+1\) and a double Floquet multiplier −1; a counterpart to the triple Floquet multiplier \(+1\) studied in [19].
All bifurcations of periodic orbits have quasiperiodic counterparts and for invariant tori that are reducible to Floquet form no additional bifurcations emerge. To prove persistence of such bifurcation scenarios under small perturbations amounts to replacing the Implicit Mapping Theorem by kam Theory—in particular the dense set of resonances among the frequencies weakens the possible results to quasiperiodic stability; compare with [11] and references therein. For volumepreserving bifurcations we have seen in Theorems 4.2 and 4.4 that restricting to quasiperiodic stability already becomes necessary for the elliptic cases of volumepreserving Hopf and double Hopf bifurcations of equilibria in 1parameter families.
Notes
Acknowledgement
We thank Florian Wagener for his help during the preparation of this paper. Furthermore we thank the referees for their comments and remarks.
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