641 research outputs found
Competition of spatial and temporal instabilities under time delay near codimension-two Turing-Hopf bifurcations
Competition of spatial and temporal instabilities under time delay near the
codimension-two Turing-Hopf bifurcations is studied in a reaction-diffusion
equation. The time delay changes remarkably the oscillation frequency, the
intrinsic wave vector, and the intensities of both Turing and Hopf modes. The
application of appropriate time delay can control the competition between the
Turing and Hopf modes. Analysis shows that individual or both feedbacks can
realize the control of the transformation between the Turing and Hopf patterns.
Two dimensional numerical simulations validate the analytical results.Comment: 13 pages, 6 figure
Turing patterns in parabolic systems of conservation laws and numerically observed stability of periodic waves
Turing patterns on unbounded domains have been widely studied in systems of
reaction-diffusion equations. However, up to now, they have not been studied
for systems of conservation laws. Here, we (i) derive conditions for Turing
instability in conservation laws and (ii) use these conditions to find families
of periodic solutions bifurcating from uniform states, numerically continuing
these families into the large-amplitude regime. For the examples studied,
numerical stability analysis suggests that stable periodic waves can emerge
either from supercritical Turing bifurcations or, via secondary bifurcation as
amplitude is increased, from sub-critical Turing bifurcations. This answers in
the affirmative a question of Oh-Zumbrun whether stable periodic solutions of
conservation laws can occur. Determination of a full small-amplitude stability
diagram-- specifically, determination of rigorous Eckhaus-type stability
conditions-- remains an interesting open problem.Comment: 12 pages, 20 figure
Analysis of pattern dynamics for a nonlinear model of the human cortex via bifurcation theories
This thesis examines the bifurcations, i.e., the emergent behaviours, for the Waikato cortical model under the influence of the gap-junction inhibitory diffusion D₂ (identified as the Turing bifurcation parameter) and the time-to-peak for hyperpolarising GABA response γi (i.e., inhibitory rate-constant, identified as the Hopf bifurcation parameter). The cortical model simplifies the entire cortex to a cylindrical macrocolumn (∼ 1 mm³) containing ∼ 10⁵ neurons (85% excitatory, 15% inhibitory) communicating via both chemical and electrical (gap-junction) synapses. The linear stability analysis of the model equations predict the emergence of a Turing instability (in which separated areas of the cortex become activated) when gap-junction diffusivity is increased above a critical level. In addition, a Hopf bifurcation (oscillation) occurs when the inhibitory rate-constant is sufficiently small. Nonlinear interaction between these instabilities leads to spontaneous cortical patterns of neuronal activities evolving in space and time. Such model dynamics of delicately balanced interplay between Turing and Hopf instabilities may be of direct relevance to clinically observed brain dynamics such as epileptic seizure EEG spikes, deep-sleep slow-wave oscillations and cognitive gamma-waves.
The relationship between the modelled brain patterns and model equations can normally be inferred from the eigenvalue dispersion curve, i.e., linear stability analysis. Sometimes we experienced mismatches between the linear stability analysis and the formed cortical patterns, which hampers us in identifying the type of instability corresponding to the emergent patterns. In this thesis, I investigate the pattern-forming mechanism of the Waikato cortical model to better understand the model nonlinearities. I first study the pattern dynamics via analysis of a simple pattern-forming system, the Brusselator model, which has a similar model structure and bifurcation phenomena as the cortical model. I apply both linear and nonlinear perturbation methods to analyse the near-bifurcation behaviour of the Brusselator in order to precisely capture the dominant mode that contributes the most to the final formed-patterns. My nonlinear analysis of the Brusselator model yields Ginzburg-Landau type amplitude equations that describe the dynamics of the most unstable mode, i.e., the dominant mode, in the vicinity of a bifurcation point. The amplitude equations at a Turing point unfold three characteristic spatial structures: honeycomb Hπ, stripes, and reentrant honeycomb H₀. A codimension-2 Turing–Hopf point (CTHP) predicts three mixed instabilities: stable Turing–Hopf (TH), chaotic TH, and bistable TH. The amplitude equations precisely determine the bifurcation conditions for these instabilities and explain the pattern-competition mechanism once the bifurcation parameters cross the thresholds, whilst driving the system into a nonlinear region where the linear stability analysis may not be applicable.
Then, I apply the bifurcation theories to the cortical model for its pattern predictions. Analogous to the Brusselator model, I find cortical Turing pattens in Hπ, stripes and H₀ spatial structures. Moreover, I develop the amplitude equations for the cortical model, with which I derive the envelope frequency for the beating-waves of a stable TH mode; and propose ideas regarding emergence of the cortical chaotic mode. Apart from these pattern dynamics that the cortical model shares with the Brusselator system, the cortical model also exhibits “eye-blinking” TH patterns latticed in hexagons with localised oscillations. Although we have not found biological significance of these model pattens, the developed bifurcation theories and investigated pattern-forming mechanism may enrich our modelling strategies and help us to further improve model performance.
In the last chapter of this thesis, I introduce a Turing–Hopf mechanism for the anaesthetic slow-waves, and predict a coherence drop of such slow-waves with the induction of propofol anaesthesia. To test this hypothesis, I developed an EEG coherence analysing algorithm, EEG coherence, to automatically examine the clinical EEG recordings across multiple subjects. The result shows significantly decreased coherence along the fronto-occipital axis, and increased coherence along the left- and right-temporal axis. As the Waikato cortical model is spatially homogenous, i.e., there are no explicit front-to-back or right-to-left directions, it is unable to produce different coherence changes for different regions. It appears that the Waikato cortical model best represents the cortical dynamics in the frontal region. The theory of pattern dynamics suggests that a mode transition from wave–Turing–wave to Turing–wave–Turing introduces pattern coherence changes in both positive and negative directions. Thus, a further modelling improvement may be the introduction of a cortical bistable mode where Turing and wave coexist
Turing pattern formation in the Brusselator system with nonlinear diffusion
In this work we investigate the effect of density dependent nonlinear
diffusion on pattern formation in the Brusselator system. Through linear
stability analysis of the basic solution we determine the Turing and the
oscillatory instability boundaries. A comparison with the classical linear
diffusion shows how nonlinear diffusion favors the occurrence of Turing pattern
formation. We study the process of pattern formation both in 1D and 2D spatial
domains. Through a weakly nonlinear multiple scales analysis we derive the
equations for the amplitude of the stationary patterns. The analysis of the
amplitude equations shows the occurrence of a number of different phenomena,
including stable supercritical and subcritical Turing patterns with multiple
branches of stable solutions leading to hysteresis. Moreover we consider
traveling patterning waves: when the domain size is large, the pattern forms
sequentially and traveling wavefronts are the precursors to patterning. We
derive the Ginzburg-Landau equation and describe the traveling front enveloping
a pattern which invades the domain. We show the emergence of radially symmetric
target patterns, and through a matching procedure we construct the outer
amplitude equation and the inner core solution.Comment: Physical Review E, 201
Spatiotemporal pattern formation in a three-variable CO oxidation reaction model
The spatiotemporal pattern formation is studied in the catalytic carbon
monoxide oxidation reaction that takes into account the diffusion processes
over the Pt(110) surface, which may contain structurally different areas. These
areas are formed during CO-induced transition from a reconstructed phase with
geometry of the overlayer to a bulk-like () phase with
square atomic arrangement. Despite the CO oxidation reaction being
non-autocatalytic, we have shown that the analytic conditions of the existence
of the Turing and the Hopf bifurcations can be satisfied in such systems. Thus,
the system may lose its stability in two ways --- either through the Hopf
bifurcation leading to the formation of temporal patterns in the system or
through the Turing bifurcation leading to the formation of regular spatial
patterns. At a simultaneous implementation of both scenarios, spatiotemporal
patterns for CO and oxygen coverages are obtained in the system.Comment: 11 pages, 6 figures, 1 tabl
Forced patterns near a Turing-Hopf bifurcation
We study time-periodic forcing of spatially-extended patterns near a
Turing-Hopf bifurcation point. A symmetry-based normal form analysis yields
several predictions, including that (i) weak forcing near the intrinsic Hopf
frequency enhances or suppresses the Turing amplitude by an amount that scales
quadratically with the forcing strength, and (ii) the strongest effect is seen
for forcing that is detuned from the Hopf frequency. To apply our results to
specific models, we perform a perturbation analysis on general two-component
reaction-diffusion systems, which reveals whether the forcing suppresses or
enhances the spatial pattern. For the suppressing case, our results explain
features of previous experiments on the CDIMA chemical reaction. However, we
also find examples of the enhancing case, which has not yet been observed in
experiment. Numerical simulations verify the predicted dependence on the
forcing parameters.Comment: 4 pages, 4 figure
Mesa-type patterns in the one-dimensional Brusselator and their stability
The Brusselator is a generic reaction-diffusion model for a tri-molecular
chemical reaction. We consider the case when the input and output reactions are
slow. In this limit, we show the existence of -periodic, spatially bi-stable
structures, \emph{mesas}, and study their stability. Using singular
perturbation techniques, we find a threshold for the stability of mesas.
This threshold occurs in the regime where the exponentially small tails of the
localized structures start to interact. By comparing our results with Turing
analysis, we show that in the generic case, a Turing instability is followed by
a slow coarsening process whereby logarithmically many mesas are annihilated
before the system reaches a steady equilibrium state. We also study a
``breather''-type instability of a mesa, which occurs due to a Hopf
bifurcation. Full numerical simulations are shown to confirm the analytical
results.Comment: to appear, Physica
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