Mathematics

Nonlinear and Complex Systems Research Group

Research Programmes

The Nonlinear and Complex Systems Group is engaged in research in Dynamical Systems on a variety of topics including transitions to chaos; various aspects of Hamiltonian dynamics, including phase space transitions and integrability and non-integrability; delay-differential equations; arrays of coupled oscillators; dynamically evolving networks; and stochastic dynamics.

Programme: Collective Transport in Hamiltonian Dynamical Systems

The study of transport phenomena has attracted considerable interest over the years due to its relevance in many physical situations, the prototypical model being one-dimensional particle motion in a tilted spatially periodic potential. Corresponding experimental realisations include Josephson junctions, charge density waves, superionic conductors, rotation of dipoles in external fields, phase-locked loops and diffusion of dimers on surfaces to name but a few. In many of these aforementioned situations the particles, in addition to their motion in the periodic potential, interact, which may lead to cooperative effects not found in situations of individual particle motion.

The objective of the current work is to investigate the conditions under which it is possible to generate a directed flow along with collective motion in systems of coupled particles and in systems subject to driving forces.

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Programme: Nonlinear Mechanisms of Energy And Charge Transfer in Biomolecules

Many biological activities, such as photosynthesis, repair mechanism of DNA after radiation dam- age, metabolism, signal transduction in cells, enzymatic processes and respiration are driven by electron transfer (ET) reactions. In biomolecules ET is assumed to take place via single-step tunneling from donors to acceptors. Characteristic for biomolecules is that they exhibit a strong interplay between function and structure. In fact, structural elements such as the protein backbone can serve as effective molecular wires along which electrons tunnel between redox sites in proteins. So has it been shown that in certain protein ET systems the electron tunneling occurs along polypeptide strands with tunneling jumps via hydrogen bonds.

We investigate a possible ET scenario where supersonic acoustic solitons can capture and transfer self-trapping modes in biomolecular systems and ET can be mediated by supersonic solitons using realistic parameter values of biomolecular systems. The model represents typical polypeptides where neighboring peptide groups are bridged via hydrogen bonds.

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Programme: Integrability of geodesic flow on closed surfaces

Geodesics are a special class of curves which can be drawn on a surface and are a natural choice when examining motion on the surface; for example in General Relativity theory the paths of free particles are geodesics of the spacetime manifold. As the geodesic equations themselves are very nonlinear a qualitative understanding of their properties is sought. In particular, the integrability of the geodesic equations and the implications for the long term behaviour of the geodesics is the focus of this research program.

Classical results showing the integrability of the geodesic equations on surfaces of revolution and the ellipsoid are well known. Other work has focussed on surfaces defined in terms of Lie groups which are by definition very symmetric. The aim of the current work is to attempt to move away from symmetry, and seek a more direct examination of integrability by considering surfaces defined in polar form.

Examples include two parameter families of surfaces and the geodesics on these surfaces which are analysed both numerically, using Poincaré sections, and analytically, using differential Galois theory. In the first case the Poincaré sections provide evidence of a breakdown in integrability as we deform the surface away from the sphere, as well as expose the rich structure of periodic orbits or closed geodesics. In the second case we linearise around a particular solution and separate out the normal variational equation. The Morales-Ramis theorem tells us that if the geodesic equations are integrable, then the normal variational equation must be solvable, in the sense of differential Galois theory. For certain classes of surfaces the normal variational equation is of Fuchsian type (regular singular points only) and analysing the monodromy group leads to rigorous results on the non-integrability of the geodesic equations.

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Programme: Surmounting thresholds, passing through bottlenecks and escape processes

The cornerstone work by Kramers has instigated an ever ongoing interest in the dynamics of escape processes of single particles, of coupled degrees of freedom or of chains of coupled objects out of metastable states. In undergoing an escape the objects considered manage to overcome an energetic barrier, separating the local minimum from a neighbouring attracting domain.

A much studied situation in statistical physics is that of a stochastic escape for which the total energy remains a constant on average only. The latter circumstance assumes the existence of a thermal bath, causing dissipation and local energy fluctuations. Thus, in this situation the escape necessitates the creation of an optimal fluctuation triggering the escape. Put differently, when such an optimal fluctuation is transferred to the chain it provides sufficient energy to the chain to statistically overcome the energetic bottle-neck. Characteristic time-scales of these processes are determined by the calculation of corresponding rates of escape out of the corresponding domain of attraction.

Recently interest has been focused on a different scenario of the possible exit from a metastable domain of attraction. The underlying mechanism is based on the assistance of a strongly nonlinear deterministic dynamics. Macroscopic discrete, coupled nonlinear oscillator chains are considered. An efficient deterministic escape that is driven in absence of noise is particularly important when dealing with low temperatures for which the activated escape becomes far too slow, or also for situations with many coupled nonlinear units in presence of non-thermal intrinsic noise that scales inversely with the square root of the system size.

An escape is related with a crossing of a saddle point in configuration space, corresponding to bottlenecks. To this end energy needs to be concentrated at the critical mode. It has been shown that the latter can be reached in the microcanonical situation spontaneously. Hence we encounter a self-organized creation of critical states in clear contrast to noise activated escape. Strikingly the mechanism of nonlinear energy localization may promote a faster escape dynamics as compared to the noise-assisted situation.

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Programme: Control-based Continuation in Experiments

The aim of this research is to extend continuation methods, which are a well-known and successful tool for numerical bifurcation analysis, to feedback-controllable real-life experiments. This will enable experimenters to observe phenomena that remain hidden in conventional experiments due to their dynamical instability, or their sensitivity to disturbances. Since control-based continuation does not require the ability to initialise the system's state `at will', or to do numerical computations in real-time, it is in principle suitable for complex experiments such as inclined cables, fast rotating machinery, or dyamically clamped neurons (these are planned experiments).

The environment of a controlled experiment features much larger disturbances than the well-researched numerical roundoff error (2 significant digits at best, instead of 16). This requires a redesign of the current numerical methods for solving boundary-value problems. The algorithms will be developed and tested using computer simulations, tunable electrical circuits and table-top sized mechanical experiments.

The first prototype experiment is a vertically excited pendulum. When the pivot of the pendulum is excited the pendulum can rotate periodically. In a conventional experiment one could observe only the stable part of the branch of rotations. Using a combination of feedback control and Newton iterations we have tracked the rotations through a region that is too sensitive to detect in a conventional experiment into the dynamically unstable part of the branch.

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Programme: Delay and Discontinuity

Balancing a long stick is easier than balancing a short stick. This well-known fact is due to the human reaction time, which introduces the effect of a time delay into this control problem. This reaction time is about 100 ms for eye-hand coordination, which is not negligible for a short stick (about 30 cm) since the inherent time scale for the stick motion is of the same order. Delay-induced instabilities have been studied extensively in other systems with delay, for example, in coupled neurons, lasers subject to optical feedback due to external reflections, and in cutting processes.

Mathematically, dynamical systems with delay are often modelled using delay-differential equations (DDEs), which have an infinite-dimensional phase space. In applications one is typically interested in bifurcation diagrams, charting the long-time behaviour of the system depending on system parameters. We recently proved (based on a construction by R. Szalai) that it is possible to construct a low-dimensional characteristic matrix for periodic linear delay-differential equations is singular if and only if lambda is a Floquet multiplier of the periodic problem. The technique is potentially applicable also to nonlinear problems with state-dependent delays where questions about smooth dependence of periodic orbits on system parameters are still open.

A related area of research is the interaction between delay and discontinuities of the right-hand-side as occurs, for example, in system with delayed switching. Paradoxically, the study of dynamics near periodic orbits becomes simpler as long as all switching events satisfy certain transversality conditions. When these conditions are violated (at so-called discontinuity-induced bifurcations) new phenomena may occur, for example, instant transition from stable periodic orbits to chaos, period-adding cascades, or emergence of invariant polygons.

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Programme: Renormalisation in Dynamical Systems

Certain families of dynamical systems display a period-doubling cascade along the route to chaos which possesses remarkable universal quantitative features. This universality was explained, in renormalization terms, by Feigenbaum and others in the late 1970s. The quantitative universality is determined by the stability properties of a critical fixed point (function) of an operator acting on a certain space of functions. The universal constants observed are given in terms of the fixed point and the eigenvalues of the linearised renormalisation operator there. Similar ideas are used to understand period- doubling in Hamiltonian systems, the breakup of quasiperiodic orbits in dynamical systems including dissipative systems (exemplified by circle maps), Hamiltonian systems (KAM tori), and the iteration of complex analytic maps (Siegel discs). A number of questions have been settled via rigorous computer-assisted proofs, which extend the concept of interval arithmetic to the algebra of operators on the appropriate spaces, allowing rigorous proof of the existence and properties of renormalisation fixed points.

The group has studied universality in the Feigenbaum period-doubling scenario. Using Herglotz function techniques, we have recently shown the existence of period-two points of the period-doubling operator, thereby explaining the behaviour of unimodal maps of the interval with asymmetric critical points. We have also used these ideas to analyse the asymptotics with degree dependence of the universal functions (and scaling constants).

The special case of the discrete Schrödinger equation known as the Harper equation (also known as the almost Mathieu equation) is important in the study the localization transition in incommensurate systems. In the strong coupling regime the exponentially localized eigenstates possess universal self-similar fluctuations. Renormalization explains these fluctuations, and we rigorously verify the existence and properties of the underlying golden mean renormalization operator fixed point.

In a generalised Harper equation which includes the effect of next-nearest neighbour interactions the strong-coupling regime is characterised by a universal strange attractor (the Ketoja-Satija orchid) for the above recurrence. Current research is aimed at rigorously determining the nature of this attractor. The generalisation to cases other than the golden mean is also being undertaken.

Strange nonchaotic attractors, barrier billards and quantum two-level systems.

Strange nonchaotic attractors occur in certain quasiperiodically forced systems. Renormalization explains universal characteristics of the onset of a strange nonchaotic attractor. There are important links here with the theory of localization in solid-state physics as exemplified by the Harper equation (see above).

The autocorrelations in a strange nonchaotic attractor display self-similar fluctuations and these are understood in terms of piecewise constant periodic orbits of the functional recurrence above. These exact same orbits also serve to explain the self-similarity of autocorrelations in barrier billards. In addition, the additive version of this recurrence helps us understand autocorrelations in a quantum two-level system subject to quasiperiodically modulated periodic forcing.

Siegel discs

Recent work has revolved around rigorous (computer-assisted) proofs of the existence of a renormalization fixed point for quadratic golden mean Siegel discs. Properties of Siegel disc boundaries have been explored as a by-product.

Julia sets

In collaboration with Pierre Moussa (Saclay) and Simonetta Abenda (Bologna), the development of a thermodynamical formalism for nearly-circular Julia sets. As a result we have been able to calculate rigorous perturbative expansions for the Hausdorff dimension and the multifractal dimension spectrum.

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Programme: Algebraic Methods in Dynamical Systems

In the realm of Dynamical Systems, an important problem is to decide whether a given system is `solvable' in a reasonably simple manner. If so, the system is usually called integrable. Otherwise, its evolution with respect to time is generally unpredictable and very sensible to initial conditions -- a phenomenon commonly known as chaos. One concept is not intrinsically antonymous with the other, but there seems to be indeed an inverse correlation between the two in practical examples. If the system is Hamiltonian, as are most problems in Mechanics, this "chaos vs solvability" disjunctive is doubly advantageous. On one hand, the techniques of symplectic geometry may be adapted to our situation. On the other hand, in virtue of the empirical studies and the Liouville-Arnold theory, the notion of `integrability' has been rendered equivalent to a very specific, and thus observable, condition: the existence of a customary amount of independent first integrals in pairwise involution.

The algebraic approaches by Ziglin, Morales-Ruiz and Ramis are a major breakthrough in the study of Hamiltonian integrability. It is based on the study of the invariants of a given matrix group -- be it the monodromy or the differential Galois group. Each of these invariants arises from one of the first integrals of the original dynamical system. Said matrix group is linked to the first-order variational equations along a given integral curve. A second step forward was done by Morales-Ruiz, Ramis and Simo in order to extend the preceeding theoretical framework to the Galois groups of the corresponding (linearized) higher-order variational systems. Using this theory on the first-order variational equations, we have so far proven the non-integrability of a certain number of problems in Celestial Mechanics: Hill's Problem, the Three-Body Problem (already proven non-integrable by other means by Tsygvintsev, Boucher and Weil) and the equal-mass N-Body Problem, as well as established necessary conditions on the existence of additional first integrals for Hamiltonians of particular forms with homogeneous potentials. Since the N-Body Problem fits this profile, these conditions implied the absence of an additional first integral for the Three-Body Problem and the equal-mass 4,5,6-Body Problems.

We currently study the higher-order Galois groups and the link between their inverse limit and the Galois groupoid of the system as defined by Malgrange and Umemura. Stokes and monodromy matrices are studied separately within every group. This is being done for homogeneous potentials so far, the more general case being the next logical step.

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