Singularly perturbed dynamics of the tippedisk

The tippedisk is a mathematical-mechanical archetype for a peculiar friction-induced instability phenomenon leading to the inversion of an unbalanced spinning disc, being reminiscent of (but different from) the well-known inversion of the tippetop. A reduced model of the tippedisk, in the form of a three-dimensional ordinary differential equation, has been derived recently, followed by a preliminary local stability analysis of stationary spinning solutions. In the current paper, a global analysis of the reduced system is pursued using the framework of singular perturbation theory. It is shown how the presence of friction leads to slow–fast dynamics and the creation of a two-dimensional slow manifold. Furthermore, it is revealed that a bifurcation scenario involving a homoclinic bifurcation and a Hopf bifurcation leads to an explanation of the inversion phenomenon. In particular, a closed-form condition for the critical spinning speed for the inversion phenomenon is derived. Hence, the tippedisk forms an excellent mathematical-mechanical problem for the analysis of global bifurcations in singularly perturbed dynamics.

Local stability analysis of a low-dissipation cyclic refrigerator

We study the local stability near the maximum figure of merit for the low-dissipation cyclic refrigerator, where the irreversible dissipation occurs not only in the thermal contacts but also the adiabatic strokes. We find that the bounds of the coefficient of performance at maximum figure of merit or maximum cooling rate in presence of internal dissipation are identical to corresponding those in absence of internal dissipation. Using two different scenarios, we prove the existence of a single stable steady state for the refrigerator, and clarify the role of internal dissipation on the stability of thermodynamic steady state, showing that the speed of system evolution to the steady state decreases due to internal dissipation.

Dynamics in two competing predators-one prey system with two types of Holling and fear effect

In this article, it is formulated a predator-prey model of two predators consuming a single limited prey resource. On the other hand, two predators have to compete with each other for survival. The predation function for two predators is assumed to be different where one predator uses Holling type I while the other uses Holling type II. It is also assumed that the fear effect is considered in this model as indirect influence evoked by both predators. Non-negativity and boundedness is written to show the biological justification of the model. Here, it is found that the model has five equilibrium points existed under certain condition. We also perform the local stability analysis on the equilibrium points with three equilibrium points are stable under certain conditions and two equilibrium points are unstable. Hopf bifurcation is obtained by choosing the consumption rate of the second predator as the bifurcation parameter. In the last part, several numerical solutions are given to support the analysis results.

A Control Theoretical Approach to Spectral Factorization is Unstable

Local stability analysis of a recently proposed recursive feedback-based approach to spectral factorization is performed. The method is found not to give stability guarantees. Interestingly enough, its global behavior often allows one to obtain reasonable approximations of spectral factorizations if a suitable stopping criterion is employed.