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| Current BCANM seminars | |||||||||
Wednesday 7 June 2006 at 4pm in 1.59 QB (note change of day & location)
Bifurcations with co-dimension n>1 occurring in piecewise-smooth systems are still far away from being understood completely. The central part of our work is given by a special type of co-dimension three bifurcation, which we have detected in several maps with discontinuous system function. They represent organizing centers of periodic dynamics. Unfolding these bifurcations, we are on the one hand able to explain several bifurcation structures, which are often observed in 1D and 2D parameter subspaces. On the other hand, for one of the investigated systems we track the paths of these bifurcations in 4D parameter space. This leads us to the assumption, that the overall structure of the 4D parameter space is dominated by a single discontinuity induced co-dimension four bifurcation.
A general theory for computing the phase space structures that govern classical reaction dynamics in systems with an arbitrary (finite) number of degrees of freedom is presented. The theory is based on a dynamical systems approach to transition state theory which was invented originally by Wigner and Eyring in the 1930's to compute chemical reaction rates. The theory presented is algorithmic in nature and provides, among other things, a solution to the long-standing problem of how to construct a dividing surface for multidimensional systems which has all of the properties that are crucial for reaction-rate computations, i.e., the dividing surface locally divides the energy surface into two disjoint components that correspond to "reactants" and "products," it is crossed exactly once by trajectories in order to react from one component to the other (it is locally a "surface of no return"), and it is of minimal (directional) flux. The theory also provides a procedure to compute the global phase space transition pathways that trajectories must follow in order to react. The latter are enclosed by the stable and unstable manifolds of a so-called normally hyperbolic invariant manifold (NHIM). A description of the geometrical structures and the resulting constraints on reaction dynamics is given.
The theory is based on the exact Hamiltonian dynamics and allows one to study fundamental questions in reaction rate theory, such as the violation of ergodicity assumptions, and non-Markovian behaviour, in a rigorous and computationally efficient way. Moreover, the classical phase space structures form a skeleton for quantum mechanical wavefunctions. A quantum mechanical version of the algorithm that leads to the construction of the classical phase space structures provides an efficient procedure to compute quantum reaction rates, the associated Gamov-Siegert resonances, and the corresponding scattering and resonance states that become accessible in current high-resolution experiments.
A new method to design asymptotically stabilizing and adaptive control laws for nonlinear systems is presented. The method relies upon the notions of system immersion and manifold invariance and does not require the knowledge of a (control) Lyapunov function. The construction of the stabilizing control laws resembles the construction used in nonlinear regulator theory to derive the (invariant) output zeroing manifold and its friend. The method is well suited in situations where we know a stabilizing controller of a nominal reduced order model, which we would like to robustify with respect to high order dynamics. This is achieved by designing a control law that immerses the full system dynamics into the reduced order one. We also show that in this new framework the adaptive control problem can be formulated from a new perspective that, under some suitable structural assumptions, allows to modify the classical certainty equivalent controller and derive parameter update laws such that stabilization is achieved. It is interesting to note that our construction does not require a linear parameterization, furthermore, viewed from a Lyapunov perspective, it provides a procedure to add cross terms between the parameter estimates and the plant states. Finally, it is shown that the proposed approach yields new stabilizing control laws for systems in feedback and feedforward form. We illustrate the method with several practical examples, including a mechanical system with flexibility modes, an electromechanical system with parasitic actuator dynamics and an adaptive nonlinearly parameterized visual servoing application.
High-speed milling is a common manufacturing technology in industry, which is used to produce complicated parts with high accuracy. However, at some parameters this process shows instabilities resulting inaccuracies in the machined part. Traditionally stability charts are computed to determine paramater regions, where vibrations do not occur. For computing these charts one uses linear approximation of the mathematical model, leaving nonlinear vibrations unrevealed. However investigating local bifurcations it turns out that nonlinear vibrations coexist with lineraly stable stationary motions. By using numerical continuation techniques we find that stable high amplitude oscillations cover quite large parameter regions explaining the presence of undesired vibrations in the stable part of the stability chart. To illustrate the accuracy of our model numerical data is compared with our findings.
Chaotic motions also arise in high-speed milling due to the piecewise smooth dynamics of the tool's motion when it leaves and enters the workpiece. We model this phenomena by a two dimensional discrete-time map and find that the dynamics itself is conjugate to a modified version of Smale's horseshoe map.
Helium is a fascinating substance, exhibiting some exotic properties. Most notably, its Helium-4 isotope form can give rise to a Bosonic superfluid. Superfluid nanodroplets of Helium-4, formed at ultra-low temperatures, offer some exciting prospects both for probing the structure of matter and for creating novel molecular arrangements and chemical species.
Experimental progress in recent years has resulted in the successful creation of Helium-4 nanodroplets, built one atom at a time, around single dopant molecules (and small molecular clusters). We will discuss the transition from isolated (dopant) molecule to "solvated" molecule; Quantum Solvation. We will describe work-in-progress to understand the dynamics of this transition in terms of angular momentum coupling between dopant and quantum solvent and will outline some future directions.
In modern experimental sciences, "parallel" measurement techniques provide huge amounts of "multivariate data". Though, the tools for modeling/uncovering the systems behind these measurements have remained underdeveloped. I propose a modeling/inference approach which assumes a complex network of dynamical systems as reference model for the data. Under this assumption, I discuss the possibility of inferring the structural/connectivity properties of the underlying network from two typical neuroscience measurements: the EEG and Extracellular Spike Recordings. In both cases, the conceptual solution for this inferential problem is provided by combining nonlinear dynamical system theory and identification methods. After validation, the two methods are applied on real signals providing new insights.
A general calculation method of calculating the spectrum of Lyapunov exponents is presented for n-dimensional nonlinear nonsmooth systems by using the Poincare map method. The Poincare map is constructed by means of local maps to avoid calculating the Jacobian matrices at non-smooth points. The calculation of the spectrum of Lyapunov exponents for impact-vibrating systems with rigid constraints is given in detail. This method can be generally applied to systems with rigid or flexible (that is, perfectly elastic) constraints. In order to show the validity of this method, the spectrums of Lyapunov exponents are calculated in a large range of parameters for two given dynamical systems with rigid constraints and used to predict the dynamic behaviour.
