Henk Broer

Geometrical aspects of stability theory for Hill's equations

where a and b are real parameters. Apparently, the first stability diagram was drawn in the classical paper by B. vaN DER POL & M. J. O. STI~UTT [-15]. Since then many papers and textbooks have appeared on the subject; we mention J. J. STOKER [-13], J. MEIXNER & F. W. SCH~FKE [12], D. M. LEVY 8~ J. B. KELLER [-11], M. I. WEINSTEIN & J. B. KELLER [,16, 17], J. POSCHEL & E. TRUBOWlTZ [-9] and V. I. ARNOLD [-3, 5, 6]. This literature contains extensive estimates on the order of tangency of resonance tongues and the behavior of stability boundaries at infinity. A full understanding of the stability diagrams is still lacking. For instance, the appearance of instability pockets in some cases (see Figure 1; cf. [15]) calls for a geometric explanation. It should be noted that this phenomenon does not occur in the classical Mathieu case; cf. [12]. For a preliminary study near resonances using singularity theory, we refer to AFSHARNEJAD [, 1]. Related bifurcational aspects of nonlinear perturbations of Mathieus equation near resonances were studied by BROER & VEGTER [7]. The global geometry of the symplectic group was used for a stability proof of a Hill equation in LEVI [10]. We introduce the main object of study of this paper.

Bifurcations and strange attractors in the Lorenz-84 climate model with seasonal forcing

A low-dimensional model of general circulation of the atmosphere is investigated. The differential equations are subject to periodic forcing, where the period is one year. A three-dimensional Poincare mapping depends on three control parameters F, G, and , the latter being the relative amplitude of the oscillating part of the forcing. This paper provides a coherent inventory of the phenomenology of F,G,. For small, a Hopf-saddle-node bifurcation of fixed points and quasi-periodic Hopf bifurcations of invariant circles occur, persisting from the autonomous case = 0. For = 0.5, the above bifurcations have disappeared. Different types of strange attractors are found in four regions (chaotic ranges) in {F,G} and the related routes to chaos are discussed.

Hill's equation with quasi-periodic forcing: resonance tongues, instability pockets and global phenomena

AbstractA simple example is considered of Hills equation

Bifurcations of normally parabolic tori in Hamiltonian systems

\ddot x + (a^2 + bp(t))x = 0

On the Destruction of Resonant Lagrangean Tori in Hamiltonian Systems

, where the forcing termp, instead of periodic, is quasi-periodic with two frequencies. A geometric exploration is carried out of certain resonance tongues, containing instability pockets. This phenomenon in the perturbative case of small |b|, can be explained by averaging. Next a numerical exploration is given for the global case of arbitraryb, where some interesting phenomena occur. Regarding these, a detailed numerical investigation and tentative explanations are presented.

Chaos in periodically driven systems

We present Kolmogorov–Arnold–Moser (or KAM) Theory regarding typicality of quasi-periodic invariant tori, partly from a historical and partly from a pedagogical point of view. At the same time we aim at a unified approach of the theory in various dynamical settings: the ‘classical’ Hamiltonian setting of Lagrangean tori, the Hamiltonian lower dimensional isotropic tori, the dissipative case of quasi-periodic attractors, etc. Also we sketch the theory of quasi-periodic bifurcations, where resonances cause Cantorization and fraying of the bifurcation sets known from the cases of equilibrium points and periodic orbits. Here the concept of Whitney differentiability plays a central role, which locally organizes the nowhere dense union of persistent quasi-periodic invariant tori, of positive measure. At the level of torus bundles this Cantorization is observed as well, where the geometry of the torus bundles turns out to be persistent. In the meantime we briefly deal with the natural affine structure of quasi-periodic tori, with uniqueness of most of the KAM tori, and with the mechanisms of the destruction of resonant unperturbed tori. Other parts of the theory, such as the Hamiltonian higher

Pacer cell response to periodic Zeitgebers

We consider perturbations of integrable Hamiltonian systems in the neighbourhood of normally parabolic invariant tori. Under appropriate transversality conditions the tori in the unperturbed system bifurcate according to a (generalized) cuspoid catastrophe. Combining techniques of KAM theory and singularity theory, we show that such bifurcation scenarios survive the perturbation on large Cantor sets. Applications to rigid body dynamics and forced oscillators are pointed out.

Analysis of a slow–fast system near a cusp singularity

Resonance tongues arise in bifurcations of discrete or continuous dynamical systems undergoing bifurcations of a fixed point or an equilibrium satisfying certain resonance conditions. They occur in several different contexts, depending, for example, on whether the dynamics is dissipative, conservative, or reversible. Generally, resonance tongues are domains in parameter space, with periodic dynamics of a specified type (regarding period of rotation number, stability, etc.). In each case, the tongue boundaries are part of the bifurcation set. We mention here several standard ways that resonance tongues appear.