UK Nonlinear News, March 2003

Recent thesis

Higher-order Resonances in Dynamical Systems

Theo Tuwankotta, University of Utrecht, Netherlands

Supervisor: Ferdinand Verhulst, University of Utrecht, Netherlands

Higher-order resonances in dynamical systems are often neglected in the literature although they arise quite often in applications. One of the reasons is due to the fact that the dynamics of such systems usually is not as exciting as it is for lower-order resonances. Studying dynamical systems at higher-order resonances is still important because such a system can be produced from systems at low-order resonances in the presence of discrete symmetries such as mirror symmetry. In a perturbed system with purely imaginary and double zero eigenvalues at the origin, which can be viewed as an extreme type of higher-order resonances, exciting dynamics is really present, even in the first-order approximation of the system.

The analysis of the thesis is also presented in 

1. J. M. Tuwankotta, F. Verhulst, Symmetry and Resonance in Hamiltonian Systems, SIAM journal on Applied Mathematics, vol 61, number 4, 2000. 
2. J. M. Tuwankotta, G.R.W. Quispel, Geometric Numerical Integration Applied to the Elastic Pendulum at Higher Order Resonance, accepted for being published in Journal of Computational and Applied Mathematics. 
3. J. M. Tuwankotta, F. Verhulst, Hamiltonian Systems with Widely Separated Frequencies, Preprint Universiteit Utrecht nr. 1211, November 2001. 
4. J. M. Tuwankotta, Widely Separated Frequencies in Coupled Oscillators with Energy-Preserving Quadratic Nonlinearity, Preprint Universiteit Utrecht nr. 1245, July 2002.

Source: Ferdinand Verhulst (

Bifurcations in dynamical systems with parametric excitation

Siti Fatimah, University of Utrecht, Netherlands

Supervisors: Ferdinand Verhulst and Thijs Ruijgrok

 This thesis is a collection of studies on coupled dissipative oscillator systems which contain an oscillator with parametric excitation. The emphasis in this study is, on one hand, on the bifurcations of the simple solutions such as fixed points and periodic orbits, and on the other hand on identifying more complicated dynamics, such as chaotic solutions.

We study an autoparametric system, that is a vibrating system which consists of at least two subsystems: the oscillator and the excited subsystem. This system is governed by differential equations where the equations representing the oscillator are coupled to those representing the excited subsystem in a nonlinear way and such that the excited subsystem can be at rest while the oscillator is vibrating. We call this solution the semi-trivial solution. When this semi-trivial solution becomes unstable, non-trivial solutions can be initiated. In this study we consider the oscillator and the subsystem are in 1:1 internal resonance. The excited subsystem is in 1:2 parametric resonance with the external forcing. Using the method of averaging and numerical bifurcation continuation, we study the dynamics of this system. In particular, we consider the stability of the semi-trivial solutions, where the oscillator is at rest and the excited subsystem performs a periodic motion. We find various types of bifurcations, leading to non-trivial periodic or quasi-periodic solutions. We also find numerically sequences of period-doublings, leading to chaotic solutions. Finally, we mention that in the averaged system we encounter a codimension 2 bifurcation.

In a separate chapter we analytically study aspects of local dynamics and global dynamics of the system. The method of averaging is again used to yield a set of autonomous equation for approximation of the response of the system. We use two different methods to study this averaged system. First, center manifold theory is used to derive a codimension two bifurcation equation. The results we found in this equation are related to local dynamics in the full system. Second, we use a global perturbation technique developed by Kovacic and Wiggins to analyze the parameter range for which a Silnikov type homoclinic orbit exists. This orbit gives rise to well-described chaotic dynamics. We finally combine these results and draw conclusions for the full averaged system.

There is also a study on coupled oscillator systems with self excitation which is generated by flow-induced vibrations. In application the flow-induced model can describe, for instance, the fluid flow around structures that can cause destructive vibrations. These vibrations have become increasingly important in recent years because designers are using materials to their limits, causing structures to become progressively lighter and more flexible.

Suppressing flow-induced vibrations by using a conventional spring-mass absorber system has often been investigated and applied in practice. It is also well-known that self-excited vibrations can be suppressed by using different kinds of damping. However, only little attention has been paid to vibration suppression by using interaction of different types of excitation. In the monograph by Tondl, some results on the investigation of synchronization phenomena by means of parametric resonances have lead to the idea to apply a parametric excitation for suppressing self-excited vibrations. The conditions for full vibration suppression (also called quenching) were formulated.

We discuss the possibility of suppressing self-excited vibrations of mechanical systems using parametric excitation in two degrees of freedom. We consider a two-mass system of which the main mass is excited by a flow-induced, self-excited force. A single mass which acts as a dynamic absorber is attached to the main mass and, by varying the stiffness between the main mass and the absorber mass, represents a parametric excitation. It turns out that for certain parameter ranges full vibration cancellation is possible. Using the averaging method the fully non-linear system is investigated producing as non-trivial solutions stable periodic solutions and tori. In the case of a small absorber mass we have to carry out a second-order calculation.


S. Fatimah, and M. Ruijgrok, Bifurcation in Autoparametric System in 1:1 Internal Resonance with Parametric Excitation, Int. J. Non-Linear Mechanics, 37(2)(2002) 297-308.

S. Fatimah, and M. Ruijgrok, Global Bifurcations and Chaotic solutions of an Autoparametric System in 1:1 Internal Resonance with Parametric Excitation, to be submitted.

S. Fatimah and F. Verhulst, Bifurcation in Flow-induced Vibration, proceedings, SPT, Sardinia, 2002.

S. Fatimah and F. Verhulst, Suppressing flow-induced Vibration by Parametric Excitation, to appear in Nonlinear Dynamics, 2003.

Source: Ferdinand Verhulst (

Vector fields with heteroclinic networks

Manuela A.D. Aguiar, Universidade do Porto, Portugal

Supervisors: Sofia B.S.D. Castro and Isabel S. Labouriau, Universidade do Porto, Portugal

The starting point for the work in this thesis is a family of differential equations with very complicated dynamics presented and studied by Field (See Field - M.J. Field, 1996, Lectures on Bifurcations, Dynamics and Symmetry, Pitman Research Notes in Mathematics Series 356, Longman).

Based on his analytical and numerical studies, Field conjectures, for certain parameter values, the existence of a heteroclinic network involving equilibria and periodic trajectories in the dynamics of the system. He also conjectures, provided the invariant manifolds of dimension > 2 of the equilibria and periodic trajectories intersect transversely, that the existence of such a heteroclinic network would imply horseshoe dynamics in its neighbourhood.

In this thesis we pursue Field’s studies about the example. The work developed here indicates the existence of a Shilnikov heteroclinic network and proves the existence of horseshoe dynamics in the neighbourhood of such a Shilnikov heteroclinic network.

We use the symmetry of the system to define the quotient heteroclinic network. This suggests an approach to study the dynamics in the neighbourhood of the Shilnikov heteroclinic network. We codify the dynamics along the heteroclinic network and the local dynamics in the neigbourhood of the cycles in the quotient heteroclinic network. We use both codifications to charaterize the local chaotic dynamics in the neighbourhood of the network.

We construct simple examples with Shilnikov heteroclinic cycles topologically equivalent to quotient heteroclinic cycles in Field’s example. These examples, although exhibiting complex dynamics, are by construction easy to study analytically. For example, we prove analytically the transverse intersection of the invariant manifolds with dimension > 2, essential for Field’s conjecture.

The examples we construct help understand the chaotic behaviour in Field’s example. We prove the existence of horseshoe dynamics in the neighbourhood of heteroclinic cycles involving two saddle-foci. This proves the existence of horseshoe dynamics in the specific case of one of the examples that we construct.

Our codification gives an idea of the complexity of the dynamics in Field’s example. The final conclusion is that, under certain conditions, the dynamics in the neighbourhood of the Shilnikov heteroclinic network is a horsehoe of horsehoe dynamics.

Source: Ana Paula Dias (

Anti-integrability in Lagrangian Systems

Yi-Chiuan Chen, DAMTP, University of Cambridge

Supervisor: Claude Baesens

Three examples of the anti-integrability in Lagrangian systems proposed by Robert S. MacKay are proved.

The first example arises in adiabatically perturbed systems. Assuming the adiabatic Poincaré-Melnikov function has simple zeros, we constructed a variational functional whose critical points give homoclinic trajectories for the unperturbed Lagrangian but multi-bump trajectories under perturbations. The conjugacy of the Poincaré map to the Bernoulli shift was obtained.

The second example occurs in the Sinai billiards. The system becomes anti-integrable when the radius of the scatterer-discs is zero. The anti-integrable orbits are piecewise straight lines joining zero-radius discs to discs. Under some non-degeneracy conditions, all anti-integrable orbits can be continued to the small radius case. Given integer N, for sufficiently small radius, the billiard map restricted to an invariant set is conjugate to a subshift with N symbols.

In the third example when the scatterers are approximated by potentials, such as the Coulomb potential ε/r is the distance from the potential centre. The anti-integrable limit is the limit ε -> 0. The results in the Sinai billiards also hold here when ε > 0 but small. More general type of potentials were also investigated.

Source: Claude Baesens (

 Geometry and stability of multi-periodic surface wave patterns

Fiona Laine-Pearson, Department of Mathematics and Statistics, University of Surrey

Supervisor: Tom Bridges

Examiners: Dr. Peter E. Hydon (Surrey), Professor Frederic Dias (Ecole Normale Superieure de Cachan)

The thesis develops a general theory for the bifurcation and stability of multi-periodic patterns (i.e. periodic in space and time) for conservative systems, such as occur in oceanography and optics. The theoretical developments include a geometrical theory for one- two- and three-phase wavetrain solutions of multisymplectic systems. Applications include: a theory for instability of standing waves, and short-crested Stokes waves of the water wave problem; a theory for 1:2 resonant wavetrains with applications in optics; interaction between three waves and the implications for hexagonal wave patterns.

Source: Tom Bridges (

Regularity of Minimizers in the Calculus of Variations and Optimal Control

Delfim F. M. Torres, University of Aveiro, Portugal

Supervisor: Andrey Sarychev

In this Ph.D. thesis a new approach to Lipschitzian regularity of the minimizing trajectories to problems of the calculus of variations and optimal control is developed. The author also generalizes the classical Noether's symmetry theorem (first Noether theorem) to invariant optimal control problems, by extending the very concept of invariance of the problem. The result relates the quasi-symmetries of the problems with the conservation laws for the Pontryagin extremals. With the help of such conservation laws, the first results in the literature, regarding Lipschitzian regularity of the minimizing trajectories for optimal control problems with nonlinear dynamics, are obtained. Even for problems with linear dynamics, such as those of the calculus of variations, the results are new and able to cover new situations not treated before. Some results of the thesis are available in the English language from the author's web site  

Source: Delfim F. M. Torres (

Closed trajectories in planar piecewise-smooth systems of Liénard-type with a line of discontinuity

Karin Pliete, University of Cologne, Germany

Supervisor: T. Küpper

We consider planar piecewise-smooth systems of Liénard-type with a line of discontinuity. These systems arise from many applications in control theory, mechanics or engineering. We analyse such a system in terms of dynamical and bifurcation behaviour. For this, we determine all equilibria, periodic, heteroclinic and homoclinic solutions in dependency on parameters. Our main goal is the analytical determination of global behaviour. This is in particular possible if the system is peicewise-linear. When we additionally assume that the system is Z2-symmetric we obtain a complete characterisation. As one result we detect bifurcation phenomena which do not exist in this form for smooth systems.

Source: Karin Pliete (kpliete@MI.Uni-Koeln.DE)

Dynamics of Piecewise Isometric Systems with Particular Emphasis to the Goetz Map

Miguel Mendes, University of Surrey

Supervisor: Matthew Nicol

Examiners: David Broomhead and Jonathan Deane

No abstract was submitted. However, a description of miguel's work is at 

Source: Matthew Nicol (

<< Move to UK Nonlinear News Issue 31 Index Page.
Page Created: 1 March 2003.
Last Updated: 1 March 2003.