Computational Nonlinear Dynamics and Optimization

Luis Vázquez, Carlos Aguirre, Salvador Jiménez, Pedro Pascual (from left to right)

Members of the Group (from left to right):
Luis Vázquez, Carlos Aguirre, Salvador Jiménez, Pedro Pascual.

 

Fields of Research

Complex Systems, Nonlinear Wave Propagation, Stochastic and Fractal Perturbations, Numerical Simulations, Algorithms, Parallel Computing, Small-World Networks Dynamics, Wavelet Analysis, Optimization, Data Mining

 

Current Projects

Dynamics of Nonlinear Scalar and Electromagnetic Waves in Different Media

Study of several models with the joint effect of nonlinearity and different types of perturbations: spatially non-local, periodic, fractal, stochastic and Dirac delta like impurities.

Numerical simulations as well as analytical studies of the Schrödinger and Maxwell-Bloch systems related to optical solitons propagation and laser physics.

 

Applications of Wave Propagation in Disordered Media

Numerical simulations and analytical study of several systems where nonlinear effects and different time and space scales are combined by means of non-local, periodic, fractal and stochastic perturbations.

Systems studied are Maxwell, Klein-Gordon and Schrödinger.

 

Diffusion on Nonlinear Systems

Study, both numerically and analytically, of some dynamical systems under stochastic perturbations, starting from the Liouville and Fokker-Planck equations. We focus our attention on anomalous diffusion and the basic mechanisms that give rise to it.

 

Control, Synchronization and Characterization of Spatially Extended Nonlinear Systems

Dynamical chaos in low-dimensional systems was one of the most exciting discovery of physics research during the late of seventies and early eighties. A particular development that took place in the early nineties was the discovery of the control of chaos, a term that means that by applying very small (but judicious) perturbations one can control a low-dimensional chaotic system and force it to exhibit one of the many periodic motions that underlie its chaotic attractor.

The goal is to shift the focus of research on spatio-temporal complexity from the fundamental (but relatively unsuccessful) attempt to find universal aspects and scenarios for the onset of spatio-temporal complexity, to the useful issues of control, synchronization and adaptation of extended chaotic systems.

The systems that we have in mind include discretized extended systems, continuous nonlinear media, nonlinear optical fields and their interaction with matter, turbulence in fluids with its large scale structures, and biological systems.

 

Small-World Networks Dynamics

Small-World networks are intermediate between the regular lattice and a random network, the dynamics of the diffusion or the transmission of a particle (whatever particle could mean) over the network is usually governed by a nonlinear function, presenting usually a chaotic behaviour. We study by means of computer simulations, the behaviour of a single particle or a set of particles, that are transmitted or diffused through different network topologies, ranging from regular to random. Our work has been applied to the problem of package routing in Internet, or obtaining the optimal network topology in terms of reliability and cost.

 

Wavelet Analysis

Wavelet analysis is a powerful tool in data analysis. Wavelets extract some characteristics of the data much better than traditional methods do, like Fourier or statistical methods. For example, wavelets allow the study of a signal at different time scales. We make use of wavelets for the analysis of economical data. The value of company shares present a strong stochastic component, but when the behaviour of the market value of shares for several well known companies are analyzed and compared at different time scales, some very interesting common patterns appear. Wavelet analysis allows to detect, for example, what time scale produces the significative changes in the value of a share.

Wavelets has been used for analysis of clinical images from PET or NMR. Neural activity changes with time, this means that to explore neural activity and its response to a determined stimulus is necessary to study the evolution in time of the set of neurons that are activated by the stimulus.

 

Results

Development of new algorithms and parallel implementation to simulate nonlinear wave equations and complex systems.

Bound state of three solitons of the nonlinear Schrödinger equation

Bound state of three solitons of the nonlinear Schrödinger equation

A new iterative method to solve linear systems which depends on the eigenvalues of the associated matrix but not directly on its size.

Introduction of a new diffusion equation which incorporates the fractional calculus and the internal degrees of freedom in the Dirac framework.

A new substrate for modeling some real-life networks such as Internet or hierarchical networks.  Optimal topologies in terms of cost/performance for some general dynamics over the network.

A proposed substrate for network modelling

A proposed substrate for network modelling

Simulation of the full Maxwell equations to show the formation of electromagnetic shocks occurring on the optical cycle, that were conjectured theoretically.

To show that the diffraction of the above shocking optical pulses creates optical missiles which are localized radiation field whose amplitude and energy decays slower than the classical ones. At the same time, the focusing of the shocks produces singular intensity and energy density at the focal point.

A method using wavelets for the classification of neural activity movies generated by different initial stimulus. The method also allows the detection of the dominant neuron coupling frequency generated by the stimulus.

Inferior Olive neurons activity for two different stimuli

Inferior Olive neurons activity for two different stimuli

Wavelet analysis of the above movies, showing the differences between them

Wavelet analysis of the above movies, showing the differences between them

A new mechanical approach to solve the Primal Problem associated to the linear optimization problem. The number of iterations is dim(X)-1, independently of the number of inequality constraints and being X the unknowns.

Solution of finding the maximum of a linear function (in blue, darker values are greater) in a convex domain (green boundary) by a mechanical approach (red trayectory)

Solution of finding the maximum of a linear function (in blue, darker values are greater) in a convex domain (green boundary) by a mechanical approach (red trayectory)

 

References

ZiF Publications

L. Vázquez, Fractional diffusion equations with internal degrees of freedom, ZiF Preprint 2001/043, to be published in Journal of Computational Mathematics

C. Aguirre, F. Corbacho, R. Huerta, A realistic substrate for small-world networks modeling, ZiF Preprint 2001/060

G. Turchetti, D. Usero, L. Vázquez, Hamiltonian systems with fractional time derivative, ZiF Preprint 2001/061

L. Vázquez, J.L. Vázquez-Poletti, A mechanical solver for linear programming, ZiF Preprint 2001/066

 

Recent Publications

L. Gilles, S.C. Hagness and L. Vázquez, Comparison between staggered and unstaggered finite-difference time-domain grids for few-cycle temporal optical soliton propagation, Journal of Computational Physics 161, 379-400 (2000)

M.A. Porras, F. Salazar-Bloise and L. Vázquez, Creation of localized optical waves that do not obey the radiation condition at infinity, Physical Review Letters 85, 2104-2107 (2000)

M.A. Porras, F. Salazar-Bloise and L. Vázquez, Focusing properties of shocking optical pulses, Optics Letters 26, 376-378 (2001)

L. Vázquez and J.L. Vázquez-Poletti, A new approach to solve systems of linear equations, Journal of Computational Mathematics 19, 445-448 (2001)

L. Vázquez and S. Jiménez, Analysis of a mechanical solver for linear systems of equations, Journal of Computational Mathematics 19, 9-14 (2001)

S. Jiménez, I.M. Llorente, A.M. Mancho, V.M. Pérez-García and L. Vázquez, A numerical scheme for the simulation of blow-up in the nonlinear Schrödinger equation, Univ. of Madeira CCM-Preprint Nr. 45/01 (pdf or ps file), to be published in Applied Mathematics and Computation

P.J. Pascual, E. Ortiz and L. Vázquez, Simulation and visualization of electromagnetic shock waves, UCM Preprint (2001)

L. Vázquez, C. Aguirre, S. Jiménez and P.J. Pascual, A mechanical solver for some nonlinear programming problems, in preparation (2001)

P. Varona, C. Aguirre, J.J. Torres, H.D.J. Abarbanel, M.I. Rabinovich, Spatio-temporal patterns of network activity in the inferior olive, to be published in the Proceedings of CNS'01 San Francisco (2001)