Generic twistless bifurcations (original) (raw)

(Vanishing) Twist in the Saddle-Centre and Period-Doubling Bifurcation

Physica D, 2003

The lowest order resonant bifurcations of a periodic orbit of a Hamiltonian system with two degrees of freedom have frequency ratio 1:1 (saddle-centre) and 1:2 (period-doubling). The twist, which is the derivative of the rotation number with respect to the action, is studied near these bifurcations. When the twist vanishes the nondegeneracy condition of the (isoenergetic) KAM theorem is not satisfied, with interesting consequences for the dynamics. We show that near the saddle-centre bifurcation the twist always vanishes. At this bifurcation a ``twistless'' torus is created, when the resonance is passed. The twistless torus replaces the colliding periodic orbits in phase space. We explicitly derive the position of the twistless torus depending on the resonance parameter, and show that the shape of this curve is universal. For the period doubling bifurcation the situation is different. Here we show that the twist does not vanish in a neighborhood of the bifurcation.

Another look at the saddle-centre bifurcation: Vanishing twist

PHYSICA D-NONLINEAR PHENOMENA, 2005

The lowest order resonant bifurcations of a periodic orbit of a Hamiltonian system with two degrees of freedom have frequency ratio 1 : 1 (saddle-centre) and 1 : 2 (period-doubling). The twist, which is the derivative of the rotation number with respect to the action, is studied near these bifurcations. When the twist vanishes the nondegeneracy condition of the (isoenergetic) KAM theorem is not satisfied, with interesting consequences for the dynamics. We show that near the saddle-centre bifurcation the twist always vanishes. At this bifurcation a "twistless" torus is created, when the resonance is passed. The twistless torus replaces the colliding periodic orbits in phase space. We explicitly derive the position of the twistless torus depending on the resonance parameter, and show that the shape of this curve is universal. For the period doubling bifurcation the situation is different. Here we show that the twist does not vanish in a neighborhood of the bifurcation.

Periodic orbits for dissipative twist maps

Ergodic Theory and Dynamical Systems, 1987

We develop simple topological criteria for the existence of periodic orbits in maps of the annulus. These are applied to one-parameter families of dissipative twist maps of the annulus and their attractors. It follows that many of the motions found by variational methods in area preserving twist maps also occur in the dissipative case.

On the stability of some periodic orbits of a new type for twist maps

Nonlinearity, 2002

We study a two-parameter family of twist maps defined on the torus. This family essentially determines the dynamics near saddle-centre loops of four-dimensional real analytic Hamiltonian systems. A saddle-centre loop is an orbit homoclinic to a saddle-centre equilibrium (related to pairs of pure real, ±ν, and pure imaginary, ±ωi, eigenvalues). We prove that given any period n we can find an open set of parameter values such that this family has an attracting n-periodic orbit of a special type. This has interesting consequences on the original Hamiltonian dynamics.

A torus bifurcation theorem with symmetry

Journal of Dynamics and Differential Equations, 1990

A general theory for the study of degenerate Hopf bifurcation in the presence of symmetry has been carried out only in situations where the normal form equations decoupte into phase/amplitude equations. In this paper we prove a theorem showing that in general we expect such degeneracies to lead to secondary torus bifurcations. We then apply this theorem to the case of degenerate Hopf bifurcation with triangular (D3) symmetry, proving that in codimension two there exist regions of parameter space where two branches of asymptotically stable 2-tori coexist but where no stable periodic solutions are present. Although this study does not lead to a theory for degenerate Hopf bifurcations in the presence of symmetry, it does present examples that would have to be accounted for by any such general theory.

Vanishing twist in the Hamiltonian Hopf bifurcation

Physica D: Nonlinear Phenomena, 2005

The Hamiltonian Hopf bifurcation has an integrable normal form that describes the passage of the eigenvalues of an equilibrium through the 1 : −1 resonance. At the bifurcation the pure imaginary eigenvalues of the elliptic equilibrium turn into a complex quadruplet of eigenvalues and the equilibrium becomes a linearly unstable focus-focus point. We explicitly calculate the frequency map of the integrable normal form, in particular we obtain the rotation number as a function on the image of the energy-momentum map in the case where the fibres are compact. We prove that the isoenergetic non-degeneracy condition of the KAM theorem is violated on a curve passing through the focus-focus point in the image of the energy-momentum map. This is equivalent to the vanishing of twist in a Poincaré map for each energy near that of the focus-focus point. In addition we show that in a family of periodic orbits (the nonlinear normal modes) the twist also vanishes. These results imply the existence of all the unusual dynamical phenomena associated to non-twist maps near the Hamiltonian Hopf bifurcation.

Invariance of bifurcation equations for high degeneracy bifurcations of non-autonomous periodic maps

arXiv (Cornell University), 2014

Bifurcations of the Aµ class in Arnold's classification, in nonautonomous p-periodic difference equations generated by parameter depending families with p maps, are studied. It is proved that the conditions of degeneracy, non-degeneracy and unfolding are invariant relative to cyclic order of compositions for any natural number µ. The main tool for the proofs is local topological conjugacy. Invariance results are essential to the proper definition of the bifurcations of class Aµ, and lower codimension bifurcations associated, using all the possible cyclic compositions of the fiber families of maps of the p-periodic difference equation. Finally, we present two actual examples of class A 3 or swallowtail bifurcation occurring in period two difference equations for which the bifurcation conditions are invariant.

Local Bifurcations of a Quasiperiodic Orbit

International Journal of Bifurcation and Chaos, 2012

We consider the local bifurcations that can occur in a quasiperiodic orbit in a three-dimensional map: (a) a torus doubling resulting in two disjoint loops, (b) a torus doubling resulting in a single closed curve with two loops, (c) the appearance of a third frequency, and (d) the birth of a stable torus and an unstable torus. We analyze these bifurcations in terms of the stability of the point at which the closed invariant curve intersects a "second Poincaré section". We show that these bifurcations can be classified depending on where the eigenvalues of this fixed point cross the unit circle.

On dynamics and bifurcations of area-preserving maps with homoclinic tangencies

Nonlinearity, 2015

We study bifurcations of area-preserving maps, both orientable (symplectic) and nonorientable, with quadratic homoclinic tangencies. We consider one and two parameter general unfoldings and establish results related to the appearance of elliptic periodic orbits. In particular, we find conditions for such maps to have infinitely many generic (KAM-stable) elliptic periodic orbits of all successive periods starting at some number.

On the bifurcation of periodic orbits

arXiv: Dynamical Systems, 2001

This article is a survey on recent contributions to an effective version of Bautin's theory about the bifurcation of periodic orbits (limit cycles). The analysis of Hopf bifurcations of higher order is possible by use of the return mapping. Explicit estimates of the size of the domain on which the number of limit cycles is controlled are important in many applications of bifurcation theory (for instance in Biology and Physiology).