Igor Hoveijn

Nonlinear dynamical systems and chaos

Symmetries in dynamical systems.- Symplccticity, reversibility and elliptic operators.- The Rolling Disc.- Testing for Sn-Symmetry with a Recursive Detective.- Normal forms of vector fields satisfying certain geometric conditions.- On symmetric ?-limit sets in reversible flows.- Symmetry Breaking in Dynamical Systems.- Invariant Cj functions and center manifold reduction.- Hopf bifurcation at k-fold resonances in conservative systems.- KAM theory and other perturbation theories.- Families of Quasi-Periodic Motions in Dynamical Systems Depending on Parameters.- Towards a Global Theory of Singularly Perturbed Dynamical Systems.- Equivariant Perturbations of the Euler Top.- On stability loss delay for a periodic trajectory.- Parametric and autoparametric resonance.- Global attractors and bifurcations.- Infinite dimensional systems.- Modulated waves in a perturbed Korteweg-de Vries equation.- Hamiltonian Perturbation Theory for Concentrated Structures in Inhomogeneous Media.- On instability of minimal foliations for a variational problem on T2.- Local and Global Existence of Multiple Waves Near Formal Approximations.- Time series analysis.- Estimation of dimension and order of time series.- Numerical continuation and bifurcation analysis.- On the computation of normally hyperbolic invariant manifolds.- The Computation of Unstable Manifolds Using Subdivision and Continuation.

Chaos in the 1:2:3 Hamiltonian normal form

The normal form of the Hamiltonian 1:2:3 resonance to degree 3 contains seven families of periodic solutions of which one can be complex unstable. Associated with this complex unstable solution is an invariant manifold N on which the dynamics can be characterised completely; one of the ingredients of N is a set of homoclinic orbits. In the normal form to degree 4 the set of homoclinic orbits breaks up, for certain parameter values, into one homoclinic orbit. This enables us to apply Silnikov-Devaney theory to prove, at this stage numerically, the existence of a horseshoe map in the system with the implication of non-integrability and chaos in the normal form.

A reversible bifurcation analysis of the inverted pendulum

The inverted pendulum with a periodic parametric forcing is considered as a bifurcation problem in the reversible setting. Parameters are given by the size of the forcing and the frequency ratio. Normal form theory provides an integrable approximation of the Poincare map generated by a planar vector field. Genericity of the model is studied by a perturbation analysis, where the spatial symmetry is optional. Here equivariant singularity theory is used.

Resonances in a spring-pendulum: algorithms for equivariant singularity theory

A spring-pendulum in resonance is a time-independent Hamiltonian model system for formal reduction to one degree of freedom, where some symmetry (reversibility) is maintained. The reduction is handled by equivariant singularity theory with a distinguished parameter, yielding an integrable approximation of the Poincare map. This makes a concise description of certain bifurcations possible. The computation of reparametrizations from normal form to the actual system is performed by Grobner basis techniques.

Normal forms and unfoldings of linear systems in eigenspaces of (anti)-automorphisms of order two

Abstract In this article we classify normal forms and unfoldings of linear maps in eigenspaces of (anti)-automorphisms of order two. Our main motivation is provided by applications to linear systems of ordinary differential equations, general and Hamiltonian, which have both time-preserving and time-reversing symmetries. However, the theory gives a uniform method to obtain normal forms and unfoldings for a wide variety of linear differential equations with additional structure. We give several examples and include a discussion of the phenomenon of orbit splitting. As a consequence of orbit splitting we observe passing and splitting of eigenvalues in unfoldings.

Singularities on the boundary of the stability domain near 1:1-resonance

Abstract We study the linear differential equation x ˙ = L x in 1 : 1 -resonance. That is, x ∈ R 4 and L is 4 × 4 matrix with a semi-simple double pair of imaginary eigenvalues ( i β , − i β , i β , − i β ) . We wish to find all perturbations of this linear system such that the perturbed system is stable. Since linear differential equations are in one-to-one correspondence with linear maps we translate this problem to gl ( 4 , R ) . In this setting our aim is to determine the stability domain and the singularities of its boundary. The dimension of gl ( 4 , R ) is 16, therefore we first reduce the dimension as far as possible. Here we use a versal unfolding of L , i.e. a transverse section of the orbit of L under the adjoint action of Gl ( 4 , R ) . Repeating a similar procedure in the versal unfolding we are able to reduce the dimension to 4. A 3-sphere in this 4-dimensional space contains all information about the neighborhood of L in gl ( 4 , R ) . Considering the 3-sphere as two 3-discs glued smoothly along their common boundary we find that the boundary of the stability domain is contained in two right conoids, one in each 3-disc. The singularities of this surface are transverse self-intersections, Whitney umbrellas and an intersection of self-intersections where the surface has a self-tangency. A Whitney stratification of the 3-sphere such that the eigenvalue configurations of corresponding matrices are constant on strata allows us to describe the neighborhood of L and in particular identify the stability domain.

Symplectic reversible maps, tiles and chaos

Abstract The symplectic map F(z) = R α z + eƒ(x)(-sin α, cos α) , where Rα is a rotation, produces a periodic tiling of the phase-plane for some values of α if ƒ is a periodic function. Due to the periodicity of the map, the chaotic regions of the hyperbolic fixed points of the appropriate iterate of F are connected, thereby allowing large scale diffusion in a two dimensional map. In the non periodic case large scale diffsion does not seem possible for all values of e. An approximating integrable system is constructed. We also consider the effect of nonsymplectic perturbations.

Pacer cell response to periodic Zeitgebers

Almost all organisms show some kind of time periodicity in their behavior. In mammals, the neurons of the suprachiasmatic nucleus form a biological clock regulating the activity–inactivity cycle of the animal. The main question is how this clock is able to entrain to the natural 24 h light–dark cycle by which it is stimulated. Such a system is usually modeled as a collection of mutually coupled two-state (active–inactive) phase oscillators with an external stimulus (Zeitgeber). In this article however, we investigate the entrainment of a single pacer cell to the ensemble of other pacer cells. Moreover the stimulus of the ensemble is taken to be periodic. The pacer cell interacts with its environment by phase delay at the end of its activity interval and phase advance at the end of its inactivity interval. We develop a mathematical model for this system, naturally leading to a circle map depending on parameters like the intrinsic period and phase delay and advance. The existence of resonance tongues in a circle map shows that an individual pacer cell is able to synchronize with the ensemble. We furthermore show how the parameters in the model can be related to biological observable quantities. Finally we give several directions of further research.

Periodic orbits in k -symmetric dynamical systems

Abstract A map L is called k-symmetric if itskth iterate Lk possesses more symmetry than L, for some value of k. In k-symmetric systems, there exists a notion of k-symmetric orbits. This paper deels with k-symmetric periodic orbits. We derive a relation between orbits that are k-symmetric with respect to reversing k-symmetries and symmetric orbits of Lk. With this relation we set out an efficient method for finding systematically all periodic orbits that are k-symmetric with respect to reversing k-symmetries. This k-symmetric fixed set iteration (FSI) method generalizes a celebrated method due to DeVogelaere that applies to symmetric periodic orbits in reversible dynamical systems. We use the FSI method to study k-symmetric periodic orbits of a map of the planeR2 possessing a crystallographic reversing k-symmetry group. The explicit findings illustrate a typically k-symmetric phenomenon, consisting of a nontrivial relation between the symmetry properties of periodic orbits and their periods.

Differentiability of the volume of a region enclosed by level sets

The level of a function f on R-n encloses a region. The volume of a region between two such levels depends on both levels. Fixing one of them the volume becomes a function of the remaining level. We show that if the function f is smooth, the volume function is again smooth for regular values of f. For critical values of f the volume function is only finitely differentiable. The initial motivation for this study comes from Radiotherapy, where such volume functions are used in an optimization process. Thus their differentiability properties become important