Computational dynamical systems analysis : Bogdanov-Takens points and an economic model

The subject of this thesis is the bifurcation analysis of dynamical systems (ordinary differential equations and iterated maps). A primary aim is to study the branch of homoclinic solutions that emerges from a Bogdanov-Takens point. The problem of approximating such branch has been studied intensively but neither an exact solution was ever found nor a higher-order approximation has been obtained. We use the classical ``blow-up'' technique to reduce an appropriate normal form near a Bogdanov-Takens bifurcation in a generic smooth autonomous ordinary differential equations to a perturbed Hamiltonian system. With a regular perturbation method and a generalization of the Lindstedt-Poincare' perturbation method, we derive two explicit third-order corrections of the unperturbed homoclinic orbit and parameter value. We prove that both methods lead to the same homoclinic parameter value as the classical Melnikov technique and the branching method. We show that the regular perturbation method leads to a ``parasitic turn'' near the saddle point while the Lindstedt-Poincare' solution does not have this turn, making it more suitable for numerical implementation. To obtain the normal form on the center manifold, we apply the standard parameter dependent center manifold reduction combined with the normalization, using the Fredholm solvability of the homological equation. By systematically solving all linear systems appearing from the homological equation, we correct the parameter transformation existing in the literature. The generic homoclinic predictors are applied to explicitly compute the homoclinic solutions in the Gray-Scott kinetic model. The actual implementation of both predictors in the MATLAB continuation package MatCont and five numerical examples illustrating its efficiency are discussed. Besides, the thesis discusses the possibility to use the derived homoclinic predictor of generic ordinary differential equations to continue the branches of homoclinic tangencies in the Bogdanov-Takens map. The second part of this thesis is devoted to the application of bifurcation theory to analyze the dynamic and chaotic behaviors of a nonlinear economic model. The thesis studies the monopoly model with cubic price and quadratic marginal cost functions. We present fundamental corrections to the earlier studies of the model and a complete discussion of the existence of cycles of period 4. A numerical continuation method is used to compute branches of solutions of period 5, 10, 13 and 17 and to determine the stability regions of these solutions. General formulas for solutions of period 4 are derived analytically. We show that the solutions of period 4 are never linearly asymptotically stable. A nonlinear stability criterion is combined with basin of attraction analysis and simulation to determine the stability region of the 4-cycles. This corrects the erroneous linear stability analysis in previous studies of the model. The chaotic and periodic behavior of the monopoly model are further analyzed by computing the largest Lyapunov exponents, and this confirms the above mentioned results

Assessment and control of transition to turbulence in plane Couette flow

Transition to turbulence in shear flows is a puzzling problem regarding the motion of fluids flowing, for example, through the pipe (pipe flow), as in oil pipelines or blood vessels, or confined between two counter-moving walls (plane Couette flow). In this kind of flows, the initially laminar (ordered and layered) state of fluid motion is linearly stable, but turbulent (disordered and swirling) flows can also be observed if a suitable perturbation is imposed. This thesis concerns the assessment of transitional properties of such flows in the uncontrolled and controlled environments allowing for the quantitative comparisons of control strategies aimed at suppressing or trigerring transition to turbulence. Efficient finite-amplitude perturbations typically take the form of small patches of turbulence embedded in the laminar flow and called turbulent spots. Using direct numerical simulations, the nonlinear dynamics of turbulent spots, modelled as exact solutions, is investigated in the transitional regime of plane Couette flow and a detailed map of dynamics encompassing the main features found in transitional shear flows (self-sustained cycles, front propagation and spot splitting) is built. The map represents a quantitative assessment of transient dynamics of turbulent spots as a dependence of the relaminarisation time, i.e. the time it takes for a finite-amplitude perturbation, added to the laminar flow, to decay, on the Reynolds number and the width of a localised perturbation. By applying a simple passive control strategy, sinusoidal wall oscillations, the change in the spot dynamics with respect to the amplitude and frequency of the wall oscillations is assessed by the re-evaluation of the relaminarisation time for few selected localised initial conditions. Finally, a probabilistic protocol for the assessment of transition to turbulence and its control is suggested. The protocol is based on the calculation of the laminarisation probability, i.e. the probability that a random perturbation decays as a function of its energy. It is used to assess the robustness of the laminar flow to finite-amplitude perturbations in transitional plane Couette flow in a small computational domain in the absence of control and under the action of sinusoidal wall oscillations. The protocol is expected to be useful for a wide range of nonlinear systems exhibiting finite-amplitude instability

Generalization of hyperbolic perturbation solution for heteroclinic orbits of strongly nonlinear self-excited oscillator

A generalized hyperbolic perturbation method for heteroclinic solutions is presented for strongly nonlinear self-excited oscillators in the more general form of x⋅⋅+g(x)=ɛf(μ,x,x⋅)x··+g(x)=ɛf(μ,x,x·). The advantage of this work is that heteroclinic solutions for more complicated and strong nonlinearities can be analytically derived, and the previous hyperbolic perturbation solutions for Duffing type oscillator can be just regarded as a special case of the present method. The applications to cases with quadratic-cubic nonlinearities and with quintic-septic nonlinearities are presented. Comparisons with other methods are performed to assess the effectiveness of the present method.postprin

Advection and autocatalysis as organizing principles for banded vegetation patterns

We motivate and analyze a simple model for the formation of banded vegetation patterns. The model incorporates a minimal number of ingredients for vegetation growth in semi-arid landscapes. It allows for comprehensive analysis and sheds new light onto phenomena such as the migration of vegetation bands and the interplay between their upper and lower edges. The key ingredient is the formulation as a closed reaction-diffusion system, thus introducing a conservation law that both allows for analysis and provides ready intuition and understanding through analogies with characteristic speeds of propagation and shock waves.Comment: 25

Dynamical systems and their applications in neuroscience

This thesis deals with dynamical systems, numerical software for the continuation study of dynamical systems, and some important neurobiological applications. First there are two introductory chapters, in which a background is given in dynamical systems and neuroscience. We elucidate what the problems are with some existing classifications of neural models, and suggest an improved version. We introduce the Phase Response Curve (PRC), which is a curve that describes the effect of an input on a periodic orbit. We derive an efficient method to compute this PRC. The extended functionalities of MatCont, a software package for the study of dynamical systems and their bifurcations, are explained: the user can compute the PRC of a limit cycle and its derivative, he can detect and continue homoclinic bifurcations, initiate these curves from different bifurcations and detect many codim 2 bifurcations on these curves. The speed of the software was improved by introducing C-code among the matlab-routines. We have for the first time made a complete bifurcation diagram of the Morris-Lecar neural model. We show that PRCs can be used to determine the synchronizing and/or phase-locking abilities of neural networks, and how the connection delay plays a role in this, and demonstrate some phenomena to do with PRCs and bifurcations. In collaboration with biologists at the University of Bristol, we have built detailed models of the neurons in the spinal cord of the hatchling Xenopus laevis. The biological background and the equations and parameters for the models of individual neurons and synapses are listed elaborately. These models are used to construct biologically realistic networks of neurons. The first network was used to simulate the swimming behaviour of the tadpole and to show that to disregard some important differences in the models for different neurons, will result in breakdown of the good network output. Then we have used the individual models to study a hypothesis regarding synaptogenesis, which states that the specificity in connection between neurons could be purely based on the anatomical organization of the neurons, instead of the ability of growing synapses to make a distinction between the different neurons