Computer-Assisted Methods for Analyzing Periodic Orbits in Vibrating Gravitational Billiards

Using rigorous numerical methods, we prove the existence of 608 isolated periodic orbits in a gravitational billiard in a vibrating unbounded parabolic domain. We then perform pseudo-arclength continuation in the amplitude of the parabolic surface’s oscillation to compute large, global branches of periodic orbits. These branches are themselves proven rigorously using computer-assisted methods. Our numerical investigations strongly suggest the existence of multiple pitchfork bifurcations in the billiard model. Based on the numerics, physical intuition and existing results for a simplified model, we conjecture that for any pair [Formula: see text], there is a constant [Formula: see text] for which periodic orbits consisting of [Formula: see text] impacts per period [Formula: see text] cannot be sustained for amplitudes of oscillation below [Formula: see text]. We compute a verified upper bound for the conjectured critical amplitude for [Formula: see text] using our rigorous pseudo-arclength continuation.

Finite Cascades of Pitchfork Bifurcations and Multistability in Generalized Lorenz-96 Models

In this paper, we study a family of dynamical systems with circulant symmetry, which are obtained from the Lorenz-96 model by modifying its nonlinear terms. For each member of this family, the dimension n can be arbitrarily chosen and a forcing parameter F acts as a bifurcation parameter. The primary focus in this paper is on the occurrence of finite cascades of pitchfork bifurcations, where the length of such a cascade depends on the divisibility properties of the dimension n. A particularly intriguing aspect of this phenomenon is that the parameter values F of the pitchfork bifurcations seem to satisfy the Feigenbaum scaling law. Further bifurcations can lead to the coexistence of periodic or chaotic attractors. We also describe scenarios in which the number of coexisting attractors can be reduced through collisions with an equilibrium.

Dynamics of a Predator–Prey Model with Hunting Cooperation and Allee Effects in Predators

With both hunting cooperation and Allee effects in predators, a predator–prey system was modeled as a planar cubic differential system with three parameters. The known work numerically plots the horizontal isocline and the vertical one with appropriately chosen parameter values to show the cases of two, one and no coexisting equilibria. Transitions among those cases with the rise of limit cycle and homoclinic loop were exhibited by numerical simulations. Although it is hard to obtain the explicit expression of coordinates, in this paper, we give the distribution of equilibria qualitatively, discuss all cases of coexisting equilibria, and obtain the Bogdanov–Takens bifurcation diagram to show analytical parameter conditions for those transitions. Our results give analytical conditions for not only the observed saddle-node bifurcation, Hopf bifurcation and homoclinic bifurcation but also the transcritical and pitchfork bifurcations at the predator-extinction equilibrium, which were not considered in the known work. Our analytic conditions provide a quantitative instruction to reduce the risk of predator extinction and promote the ecosystem diversity.

Amplitude Equations and Bifurcation Diagrams for Multifrequency Synchronization of Canonical-Dissipative Oscillators

Coupled systems of two canonical-dissipative limit cycle oscillators are considered in the general case and for the case of monofrequency and multifrequency synchronization. Specifically, the oscillator frequency ratios of 1:1, 1:2, and 1:3 are examined and modeled by a hybrid Rayleigh–van der Pol oscillator and an oscillator model suggested by Holt as well as an oscillator model suggested by Fokas and Lagerstrom. It is shown that all three systems exhibit a unique bifurcation diagram that describes limit cycle attractors of monofrequency and multifrequency synchronization. In particular, the relative phase describing the lag between the two oscillators both in the monofrequency and multifrequency case can be tuned by an appropriately defined bifurcation parameter [Formula: see text]. For [Formula: see text] two limit cycle attractors with different relative phases exist that merge at [Formula: see text] into pitchfork bifurcations and give rise to single limit cycle attractors that continue to exist for [Formula: see text]. Similarities and differences to bifurcation diagrams published in previous work of a similar coupled oscillator system, one based on Smorodinsky–Winternitz potentials and exhibiting 1:1 synchronization, are noted.

Effect of Structural Nonlinearity on a Piecewise Linear Aeroelastic System

Aeroelasticity is the study of the interaction of aerodynamic, elastic and inertia forces. When flexible structures, such as an airfoil, undergo wind excitation, divergence or flutter instability may arise. We study the dynamics of a two-degree-of-freedom (pitch and plunge) aeroelastic system with cubic structural nonlinearities. The aerodynamic forces are modeled as a piecewise linear function of the effective angle of attack. Stability and bifurcations of equilibria are analyzed. The effect of the structural nonlinearity is investigated. We find border collision, rapid, Hopf, saddle-node and pitchfork bifurcations. Bifurcation diagrams of the system were calculated utilizing MatCont and Mathematica.

Random switching near bifurcations

The interplay between bifurcations and random switching processes of vector fields is studied. More precisely, we provide a classification of piecewise-deterministic Markov processes arising from stochastic switching dynamics near fold, Hopf, transcritical and pitchfork bifurcations. We prove the existence of invariant measures for different switching rates. We also study when the invariant measures are unique, when multiple measures occur, when measures have smooth densities, and under which conditions finite-time blow-up occurs. We demonstrate the applicability of our results for three nonlinear models arising in applications.

Probabilistic Analysis of Bifurcations in Stochastic Nonlinear Dynamical Systems

Bifurcation diagrams are limited most often to deterministic dynamical systems. However, stochastic dynamics can substantially affect the interpretation of such diagrams because the deterministic diagram often is not simply the mean of the probabilistic diagram. We present an approach based on the Fokker-Planck equation (FPE) to obtain probabilistic bifurcation diagrams for stochastic nonlinear dynamical systems. We propose a systematic approach to expand the analysis of nonlinear and linear dynamical systems from deterministic to stochastic when the states or the parameters of the system are noisy. We find stationary solutions of the FPE numerically. Then, marginal probability density function (MPDF) is used to track changes in the shape of probability distributions as well as determining the probability of finding the system at each point on the bifurcation diagram. Using MPDFs is necessary for multidimensional dynamical systems and allows direct visual comparison of deterministic bifurcation diagrams with the proposed probabilistic bifurcation diagrams. Hence, we explore how the deterministic bifurcation diagrams of different dynamical systems of different dimensions are affected by noise. For example, we show that additive noise can lead to an earlier bifurcation in one-dimensional (1D) subcritical pitchfork bifurcation. We further show that multiplicative noise can have dramatic changes such as changing 1D subcritical pitchfork bifurcations into supercritical pitchfork bifurcations or annihilating the bifurcation altogether. We demonstrate how the joint probability density function (PDF) can show the presence of limit cycles in the FitzHugh–Nagumo (FHN) neuron model or chaotic behavior in the Lorenz system. Moreover, we reveal that the Lorenz system has chaotic behavior earlier in the presence of noise. We study coupled Brusselators to show how our approach can be used to construct bifurcation diagrams for higher dimensional systems.