4 research outputs found
On Hyperbolic Attractors in Complex Shimizu -- Morioka Model
We present a modified complex-valued Shimizu -- Morioka system with uniformly
hyperbolic attractor. The numerically observed attractor in Poincar\'{e}
cross-section is topologically close to Smale -- Williams solenoid. The
arguments of the complex variables undergo Bernoulli-type map, essential for
Smale -- Williams attractor, due to the geometrical arrangement of the phase
space and an additional perturbation term. The transformation of the phase
space near the saddle equilibrium "scatters" trajectories to new angles, then
trajectories run from the saddle and return to it for the next "scatter". We
provide the results of numerical simulations of the model and demonstrate
typical features of the appearing hyperbolic attractor of Smale -- Williams
type. Importantly, we show in numerical tests the transversality of tangent
subspaces -- a pivotal property of uniformly hyperbolic attractor.Comment: 9 pages, 8 figure
Homoclinic puzzles and chaos in a nonlinear laser model
We present a case study elaborating on the multiplicity and self-similarity
of homoclinic and heteroclinic bifurcation structures in the 2D and 3D
parameter spaces of a nonlinear laser model with a Lorenz-like chaotic
attractor. In a symbiotic approach combining the traditional parameter
continuation methods using MatCont and a newly developed technique called the
Deterministic Chaos Prospector (DCP) utilizing symbolic dynamics on fast
parallel computing hardware with graphics processing units (GPUs), we exhibit
how specific codimension-two bifurcations originate and pattern regions of
chaotic and simple dynamics in this classical model. We show detailed
computational reconstructions of key bifurcation structures such as Bykov
T-point spirals and inclination flips in 2D parameter space, as well as the
spatial organization and 3D embedding of bifurcation surfaces, parametric
saddles, and isolated closed curves (isolas).Comment: 28 pages, 23 figure
Complex Dynamics in Dedicated / Multifunctional Neural Networks and Chaotic Nonlinear Systems
We study complex behaviors arising in neuroscience and other nonlinear systems by combining dynamical systems analysis with modern computational approaches including GPU parallelization and unsupervised machine learning. To gain insights into the behaviors of brain networks and complex central pattern generators (CPGs), it is important to understand the dynamical principles regulating individual neurons as well as the basic structural and functional building blocks of neural networks. In the first section, we discuss how symbolic methods can help us analyze neural dynamics such as bursting, tonic spiking and chaotic mixed-mode oscillations in various models of individual neurons, the bifurcations that underlie transitions between activity types, as well as emergent network phenomena through synergistic interactions seen in realistic neural circuits, such as network bursting from non-intrinsic bursters. The second section is focused on the origin and coexistence of multistable rhythms in oscillatory neural networks of inhibitory coupled cells. We discuss how network connectivity and intrinsic properties of the cells affect the dynamics, and how even simple circuits can exhibit a variety of mono/multi-stable rhythms including pacemakers, half-center oscillators, multiple traveling-waves, fully synchronous states, as well as various chimeras. Our analyses can help generate verifiable hypotheses for neurophysiological experiments on central pattern generators. In the last section, we demonstrate the inter-disciplinary nature of this research through the applications of these techniques to identify the universal principles governing both simple and complex dynamics, and chaotic structure in diverse nonlinear systems. Using a classical example from nonlinear laser optics, we elaborate on the multiplicity and self-similarity of key organizing structures in 2D parameter space such as homoclinic and heteroclinic bifurcation curves, Bykov T-point spirals, and inclination flips. This is followed by detailed computational reconstructions of the spatial organization and 3D embedding of bifurcation surfaces, parametric saddles, and isolated closed curves (isolas). The generality of our modeling approaches could lead to novel methodologies and nonlinear science applications in biological, medical and engineering systems
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Generalised nonlinear stability of stratified shear flows: adjoint-based optimisation, Koopman modes, and reduced models
In this thesis I investigate a number of problems in the nonlinear stability of density stratified plane Couette flow. I begin by describing the history of transient growth phenomena, and in particular the recent application of adjoint based optimisation to find nonlinear optimal perturbations and associated minimal seeds for turbulence, the smallest amplitude perturbations that are able to trigger transition to turbulence. I extend the work of Rabin et al. (2012) in unstratified plane Couette flow to find minimal seeds in both vertically and horizontally sheared stratified plane Couette flow. I find that the coherent states visited by such minimal seed trajectories are significantly altered by the stratification, and so proceed to investigate these states both with generalised Koopman mode analysis and by stratifying the self-sustaining process described by Waleffe (1997). I conclude with an introductory problem I considered that investigates the linear Taylor instability of layered stratified plane Couette flow, and show that the nonlinear evolution of the primary Taylor instability is not coupled to the form of the linearly unstable mode, in contrast to the Kelvin-Helmholtz instability, for example. I also include an appendix in which I describe joint work conducted with Professor Neil Balmforth of UBC during the 2015 WHOI Geophysical Fluid Dynamics summer programme, investigating stochastic homoclinic bifurcations