Nonlinear Oscillations: Exact Solutions and their Approximations

Multistability refers to the coexistence of different stable states in nonlinear dynamical systems. This phenomenon has been observed in laboratory experiments and in nature. In this introduction, we briefly introduce the classes of dynamical systems in which this phenomenon has been found and discuss the extension to new system classes. Furthermore, we introduce the concept of critical transitions and discuss approaches to distinguish them according to their characteristics. Finally, we present some specific applications in physics, neuroscience, biology, ecology, and climate science.

Multistability and tipping: From mathematics and physics to climate and brain-Minireview and preface to the focus issue.

Introduction An introduction to shooting methods Some boundary value problems for the Painleve transcendents Periodic solutions of a higher order system A linear example Homoclinic orbits of the FitzHugh-Nagumo equations Singular perturbation problems--rigorous matching Asymptotics beyond all orders Some solutions of the Falkner-Skan equation Poiseuille flow: Perturbation and decay Bending of a tapered rod variational methods and shooting Uniqueness and multiplicity Shooting with more parameters Some problems of A. C. Lazer Chaotic motion of a pendulum Layers and spikes in reaction-diffusion equations, I Uniform expansions for a class of second order problems Layers and spikes in reaction-diffusion equations, II Three unsolved problems Bibliography Index

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Classical Methods in Ordinary Differential Equations: With Applications to Boundary Value Problems

This paper presents a similarity solution for Marangoni flow over a flat surface for both the momentum equations and the energy equation assuming a developing boundary layer along a surface The analysis also shows how the heat transfer and flow rate vary with Prandtl number Since the Predicted boundary layer thickness would be less than the diameter of a typical Vapor bubble during nucleate boiling, the curvature effects can be neglected and this analysis can be used as a first estimate of the effect of Marangoni flow around a vapor bubble

Prandtl number effects for marangoni convection over a flat surface

A conditional Gaussian framework for understanding and predicting complex multiscale nonlinear stochastic systems is developed. Despite the conditional Gaussianity, such systems are nevertheless highly nonlinear and are able to capture the non-Gaussian features of nature. The special structure of the system allows closed analytical formulae for solving the conditional statistics and is thus computationally efficient. A rich gallery of examples of conditional Gaussian systems are illustrated here, which includes data-driven physics-constrained nonlinear stochastic models, stochastically coupled reaction-diffusion models in neuroscience and ecology, and large-scale dynamical models in turbulence, fluids and geophysical flows. Making use of the conditional Gaussian structure, efficient statistically accurate algorithms involving a novel hybrid strategy for different subspaces, a judicious block decomposition and statistical symmetry are developed for solving the Fokker-Planck equation in large dimensions. The conditional Gaussian framework is also applied to develop extremely cheap multiscale data assimilation schemes, such as the stochastic superparameterization, which use particle filters to capture the non-Gaussian statistics on the large-scale part whose dimension is small whereas the statistics of the small-scale part are conditional Gaussian given the large-scale part. Other topics of the conditional Gaussian systems studied here include designing new parameter estimation schemes and understanding model errors.

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Conditional Gaussian Systems for Multiscale Nonlinear Stochastic Systems: Prediction, State Estimation and Uncertainty Quantification.

The effect of demographic stochasticity, in the form of Gaussian white noise, in a predator-prey model with one fast and two slow variables is studied. We derive the stochastic differential equations (SDEs) from a discrete model. For suitable parameter values, the deterministic drift part of the model admits a folded node singularity and exhibits a singular Hopf bifurcation. We focus on the parameter regime near the Hopf bifurcation, where small amplitude oscillations exist as stable dynamics in the absence of noise. In this regime, the stochastic model admits noise-driven mixed-mode oscillations (MMOs), which capture the intermediate dynamics between two cycles of population outbreaks. We perform numerical simulations to calculate the distribution of the random number of small oscillations between successive spikes for varying noise intensities and distance to the Hopf bifurcation. We also study the effect of noise on a suitable Poincare map. Finally, we prove that the stochastic model can be transformed into a normal form near the folded node, which can be linked to recent results on the interplay between deterministic and stochastic small amplitude oscillations. The normal form can also be used to study the parameter influence on the noise level near folded singularities.

Stochastic mixed-mode oscillations in a three-species predator-prey model

The entry-exit theorem and relaxation oscillations in slow-fast planar systems

Uncertainty and predictability in population dynamics of a bitrophic ecological model: Mixed-mode oscillations, bistability and sensitivity to parameters

The effect of stochasticity, in the form of Gaussian white noise, in a predator-prey model with two distinct time-scales is presented. A supercritical singular Hopf bifurcation yields a Type II excitability in the deterministic model. We explore the effect of stochasticity in the excitable regime, leading to dynamics that are not anticipated by its deterministic counterpart. The stochastic model admits several kinds of noise-driven mixed-mode oscillations which capture the intermediate dynamics between two cycles of population outbreaks. Depending on the strength of noise, the prey population exhibits intermediate to high-amplitude fluctuations (related to moderate or severe outbreaks respectively). We classify these fluctuations as isolated or intermittent or as clusters depending on their recurrences. We study the distribution of the random variable N, representing the number of small oscillations between successive spikes, as a function of the noise intensity and the distance to the Hopf bifurcation. The distribution of N is "asymptotically geometric" with the corresponding parameter related to the principal eigenvalue of a substochastic Markov chain. Finally, the stochastic model is transformed into its "normal form" which is used to obtain an estimate of the probability of repeated outbreaks.

Stochasticity induced mixed-mode oscillations and distribution of recurrent outbreaks in an ecosystem.

We consider an ecological model consisting of two species of predators competing for their common prey with explicit interference competition. With a proper rescaling, the model is portrayed as a singularly perturbed system with one-fast (prey dynamics) and two-slow variables (dynamics of the predators). The model exhibits variety of rich and interesting dynamics, including, but not limited to mixed mode oscillations (MMOs), featuring concatenation of small and large amplitude oscillations, relaxation oscillations and bistability between a semi-trivial equilibrium state and a coexistence oscillatory state. Existence of co-dimenison two bifurcations such as fold-Hopf and generalized Hopf bifurcations make the system further intriguing. More interestingly, in a neighborhood of {\emph{singular Hopf}} bifurcation, long lasting transient dynamics in form of chaotic MMOs or relaxation oscillations are observed as the system approaches the periodic attractor born out of supercritical Hopf bifurcation or a semi-trivial equilibrium state respectively. The transient dynamics could persist for hundreds or thousands of generations before the ecosystem experiences a regime shift. The time series of population cycles with different types of irregular oscillations arising in this model stem from a biological realistic feature, namely, by the variation in the intraspecific competition amongst the predators. To explain these oscillations, we use bifurcation analysis and methods from {\emph{geometric singular perturbation theory}}.

Susmita Sadhu

Papers

Stochastic mixed-mode oscillations in a three-species predator-prey model

The entry-exit theorem and relaxation oscillations in slow-fast planar systems

Uncertainty and predictability in population dynamics of a bitrophic ecological model: Mixed-mode oscillations, bistability and sensitivity to parameters

Stochasticity induced mixed-mode oscillations and distribution of recurrent outbreaks in an ecosystem.

Complex oscillatory patterns near singular Hopf bifurcation in a two time-scale ecosystem