arXiv subjects

I. G. Kevrekidis

Publications and source records attributed to I. G. Kevrekidis.

64 records · Page 4Linked to original sources

arXiv2003

Coarse Bifurcation Diagrams via Microscopic Simulators: A State-Feedback Control-Based Approach

The arc-length continuation framework is used for the design of state feedback control laws that enable a microscopic simulator trace its own open-loop coarse bifurcation diagram. The steering of the system along solution branches is achieved through the manipulation of the bifurcation parameter, which becomes our actuator. The design approach is based on the assumption that the eigenvalues of the linearized system can be decomposed into two well separated clusters: one containing eigenvalues with large negative real parts and one containing (possibly unstable) eigenvalues close to the origin.

nlin.AO↗

arXiv2003

An exploding glass ?

We propose a connection between self-similar, focusing dynamics in nonlinear partial differential equations (PDEs) and macroscopic dynamic features of the glass transition. In particular, we explore the divergence of the appropriate relaxation times in the case of hard spheres as the limit of random close packing is approached. We illustrate the analogy in the critical case, and suggest a ``normal form'' that can capture the onset of dynamic self-similarity in both phenomena.

cond-mat↗

arXiv2002

Coarse Brownian Dynamics for Nematic Liquid Crystals: Bifurcation Diagrams via Stochastic Simulation

We demonstrate how time-integration of stochastic differential equations (i.e. Brownian dynamics simulations) can be combined with continuum numerical bifurcation analysis techniques to analyze the dynamics of liquid crystalline polymers (LCPs). Sidestepping the necessity of obtaining explicit closures, the approach analyzes the (unavailable in closed form) coarse macroscopic equations, estimating the necessary quantities through appropriately initialized, short bursts of Brownian dynamics simulation. Through this approach, both stable and unstable branches of the equilibrium bifurcation diagram are obtained for the Doi model of LCPs and their coarse stability is estimated. Additional macroscopic computational tasks enabled through this approach, such as coarse projective integration and coarse stabilizing controller design, are also demonstrated.

cond-mat↗

arXiv2002

Microscopic/stochastic timesteppers and coarse control: a kinetic Monte Carlo example

Coarse timesteppers provide a bridge between microscopic / stochastic system descriptions and macroscopic tasks such as coarse stability/bifurcation computations. Exploiting this computational enabling technology, we present a framework for designing observers and controllers based on microscopic simulations, that can be used for their coarse control. The proposed methodology provides a bridge between traditional numerical analysis and control theory on the one hand and microscopic simulation on the other.

nlin.CG↗

arXiv2002

Focusing Revisited: an MN-dynamics Approach

The nonlinear Schr{ö}dinger (NLS) equation is a ubiquitous example of an envelope wave equation for conservative, dispersive systems. We revisit here the problem of self-similar focusing of waves in the case of the focusing NLS equation through the prism of a dynamic renormalization technique (MN dynamics) that factors out self-similarity and yields a bifurcation view of the onset of focusing. As a result, identifying the focusing self-similar solution becomes a steady state problem. The discretized steady states are subsequently obtained and their linear stability is numerically examined. The calculations are performed in the setting of variable index of refraction, in which the onset of focusing appears as a supercritical bifurcation of a novel type of mixed Hamiltonian-dissipative dynamical system (reminiscent, to some extent, of a pitchfork bifurcation).

nlin.PS↗

arXiv2002

Adaptive Detection of Instabilities: An Experimental Feasibility Study

We present an example of the practical implementation of a protocol for experimental bifurcation detection based on on-line identification and feedback control ideas. The idea is to couple the experiment with an on-line computer-assisted identification/feedback protocol so that the closed-loop system will converge to the open-loop bifurcation points. We demonstrate the applicability of this instability detection method by real-time, computer-assisted detection of period doubling bifurcations of an electronic circuit; the circuit implements an analog realization of the Roessler system. The method succeeds in locating the bifurcation points even in the presence of modest experimental uncertainties, noise and limited resolution. The results presented here include bifurcation detection experiments that rely on measurements of a single state variable and delay-based phase space reconstruction, as well as an example of tracing entire segments of a codimension-1 bifurcation boundary in two parameter space.

nlin.CD↗

arXiv2001

Going with the Flow: a Lagrangian approach to self-similar dynamics and its consequences

We present a systematic computational approach to the study of self-similar dynamics. The approach, through the use of what can be thought of as a ``dynamic pinning condition" factors out self-similarity, and yields a transformed, non-local evolution equation. The approach, which is capable of treating both first and second kind self-similar solutions, yields as a byproduct the self-similarity exponents of the original dynamics. We illustrate the approach through the porous medium equation, showing how both the Barenblatt (first kind) and the Graveleau (second kind) self-similar solutions arise in this framework. We also discuss certain implications of the dynamics of the transformed equation (which we will name "MN-dynamics"); in particular we discuss the discrete-time implementation of the approach, and connections with time-stepper based methods for the "coarse" integration/bifurcation analysis of microscopic simulators.

nlin.AO↗

arXiv2001

Coarse Bifurcation Studies of Bubble Flow Microscopic Simulations

The parametric behavior of regular periodic arrays of rising bubbles is investigated with the aid of 2-dimensional BGK Lattice-Boltzmann (LB) simulators. The Recursive Projection Method is implemented and coupled to the LB simulators, accelerating their convergence towards what we term coarse steady states. Efficient stability/bifurcation analysis is performed by computing the leading eigenvalues/eigenvectors of the coarse time stepper. Our approach constitutes the basis for system-level analysis of processes modeled through microscopic simulations.

nlin.PS↗

arXiv2001

Pulse Bifurcations and Instabilities in an Excitable Medium: Computations in Finite Ring Domains

We investigate the instabilities and bifurcations of traveling pulses in a model excitable medium; in particular we discuss three different scenarios for the loss of stability resp. the disappearance of stable pulses. In numerical simulations beyond the instabilities we observe replication of pulses (backfiring) resulting in complex periodic or spatiotemporally chaotic dynamics as well as modulated traveling pulses. We approximate the linear stability of traveling pulses through computations in a finite albeit large domain with periodic boundary conditions. The critical eigenmodes at the onset of the instabilities are related to the resulting spatiotemporal dynamics and act upon the back of the pulses. The first scenario has been analyzed earlier for high excitability resp. low excitation threshold: it involves the collision of a stable pulse branch with an unstable pulse branch in a so called T-point.

nlin.PS↗

arXiv1993

Global Bifurcations in Rayleigh-Benard Convection: Experiments, Empirical Maps and Numerical Bifurcation Analysis

We use nonlinear signal processing techniques, based on artificial neural networks, to construct an empirical mapping from experimental Rayleigh-Benard convection data in the quasiperiodic regime. The data, in the form of a one-parameter sequence of Poincare sections in the interior of a mode-locked region (resonance horn), are indicative of a complicated interplay of local and global bifurcations with respect to the experimentally varied Rayleigh number. The dynamic phenomena apparent in the data include period doublings, complex intermittent behavior, secondary Hopf bifurcations, and chaotic dynamics. We use the fitted map to reconstruct the experimental dynamics and to explore the associated local and global bifurcation structures in phase space. Using numerical bifurcation techniques we locate the stable and unstable periodic solutions, calculate eigenvalues, approximate invariant manifolds of saddle type solutions and identify bifurcation points. This approach constitutes a promising data post-processing procedure for investigating phase space and parameter space of real experimental systems; it allows us to infer phase space structures which the experiments can only probe with limited measurement precision and only at a discrete number of operating parameter settings.

comp-gas↗