Jeroen S. W. Lamb

Reversing symmetries in dynamical systems

Dynamical systems may possess, in addition to symmetries that leave the equations of motion invariant, reversing symmetries that invert the equations of motion. Such dynamical systems are called (weakly) reversible. Some consequences of the existence of reversing symmetries for dynamical systems with discrete time (mappings) are discussed. A reversing symmetry group is introduced and it is shown that every (weakly) reversible mapping L can be decomposed into two mappings K0 and K1 of the same order 2l (limit l to infinity included) such that K02 K12=I. Some applications are discussed briefly.

Reversing k -symmetries in dynamical systems

We generalize the concept of (reversing) symmetries of a dynamical system, i.e. we study dynamical systems that possess symmetry properties only if considered on a proper time scale. In particular (considering dynamical systems with discrete time), the kth iterate of a map may possess more (reversing) symmetries than the map itself. In this way the concepts of (reversing) symmetries and (reversing) symmetry groups are generalized to (reversing) k-symmetries and (reversing) k-symmetry groups. Furthermore, a method is studied for finding orbits that are (k-) symmetric with respect to reversing (k-)symmetries. Firstly an existing method for finding orbits that are symmetric with respect to one reversing symmetry is extended to the case of more than one reversing symmetry and secondly a generalization of this method to the case of reversing k-symmetries is introduced. Some physically relevant examples of dynamical systems possessing reversing k-symmetries are discussed briefly.

Equivariant Hopf bifurcation for neutral functional differential equations

In this paper we employ an equivariant Lyapunov-Schmidt procedure to give a clearer understanding of the one-to-one correspondence of the periodic solutions of a system of neutral functional differential equations with the zeros of the reduced bifurcation map, and then set up equivariant Hopf bifurcation theory. In the process we derive criteria for the existence and direction of branches of bifurcating periodic solutions in terms of the original system, avoiding the process of center manifold reduction.

Local bifurcations in k-symmetric dynamical systems

A map is called a (reversing) k-symmetry of the dynamical system represented by the map if k is the smallest positive integer for which U is a (reversing) symmetry of the kth iterate of L. We study generic local bifurcations of fixed points that are invariant under the action of a compact Lie group that is a reversing k-symmetry group of the map L, on the basis of a normal form approach. We derive normal forms relating the local bifurcations of k-symmetric maps to local steady-state bifurcations of symmetric flows of vector fields. Alternatively, we also discuss the derivation of normal forms entirely within the framework of Taylor expansions of maps. We illustrate our results with some examples.

Bifurcation from relative periodic solutions

Relative periodic solutions are ubiquitous in dynamical systems with continuous symmetry. Recently, Sandstede, Scheel and Wulff derived a center bundle theorem, reducing local bifurcation from relative periodic solutions to a finite-dimensional problem. Independently, Lamb and Melbourne showed how to systematically study local bifurcation from isolated periodic solutions with discrete spatiotemporal symmetries. In this paper, we show how the center bundle theorem, when combined with certain group theoretic results, reduces bifurcation from relative periodic solutions to bifurcation from isolated periodic solutions. In this way, we obtain a systematic approach to the study of local bifurcation from relative periodic solutions.

Local bifurcations on the plane with reversing point group symmetry

Abstract Local bifurcations of fixed points, being symmetric with respect to a reversing point group consisting of mirrors, two- and fourfold rotations, are studied for dynamical systems in R 2 . These bifurcations are of interest in connection to symmetry breaking phenomena, in particular with respect to the occurrence of dissipative features in (weakly) reversible systems.

Newhouse regions for reversible systems with infinitely many stable, unstable and elliptic periodic orbits

For reversible two-dimensional diffeomorphisms we establish a new type of Newhouse regions (regions of structural instability density). We prove that in these regions there exists a dense set of diffeomorphisms having, simultaneously, infinitely many stable, infinitely many unstable, and infinitely many elliptic type periodic orbits.

Bifurcation from periodic solutions with spatiotemporal symmetry, including resonances and mode interactions

Abstract We study local bifurcation in equivariant dynamical systems from periodic solutions with a mixture of spatial and spatiotemporal symmetries. In previous work, we focused primarily on codimension one bifurcations. In this paper, we show that the techniques used in the codimension one analysis can be extended to understand also higher codimension bifurcations, including resonant bifurcations and mode interactions. In particular, we present a general reduction scheme by which we relate bifurcations from periodic solutions to bifurcations from fixed points of twisted equivariant diffeomorphisms, which in turn are linked via normal form theory to bifurcations from equilibria of equivariant vector fields. We also obtain a general theory for bifurcation from relative periodic solutions and we show how to incorporate time-reversal symmetries into our framework.

k-symmetry and return maps of spacetime symmetric flows

A diffeomorphism is called a (reversing) k-symmetry of a dynamical system in represented by the diffeomorphism if k is the smallest positive integer for which U is a (reversing) symmetry of (the k-times iterate of f), i.e. . In this paper we show how k-symmetry naturally arises in the context of return maps of flows with spacetime symmetries. We discuss the connection between periodic orbits of k-symmetric maps and symmetric periodic orbits of the flows they represent, and illustrate the application of our results in local bifurcation theory. We also provide a geometric interpretation of formal (Birkhoff) normal form symmetries for diffeomorphisms as a time-shift symmetry of a locally constructed spacetime symmetric suspension flow. Finally, we explain the occurrence of dual (representations of) reversing k-symmetry groups in k-symmetric maps in relation to different choices for the position of the surface of section for return maps of spacetime symmetric flows.

A natural class of generalized Fibonacci chains

In this paper we propose a class of substitution rules that generate quasiperiodic chains sharing their typid properties with the quasiperiodic Fibonacci chain. For a subclass we explicitly con&vct the atomic surface. Moreover, scaling properties of the energy specmm are discussed in relation to the dynamics of trace maps.