Bifurcation In Presence Research Articles

Bifurcation analysis of a vaccination mathematical model with application to COVID-19 pandemic

In this research, we propose a delayed vaccination model with the application for predicting the evolution of infectious cases related to COVID-19 disease. The main purpose of this paper is to show the existence of Hopf bifurcation that can explain the multiple waves that the world witnessed this recent times. Therefore, it can be used the length between the doses for the vaccine that considered for different vaccines and its effect on the evolution of the infectious cases. It has been shown that the investigated model can undergo Hopf bifurcation in presence of delay time lags to the vaccine against a COVID-19, and can lead to the persistence of the disease. The obtained mathematical findings are checked using graphical representations with proper interpretations on the manner of controlling the outbreak of COVID-19 disease.

Optimization-based prediction of fold bifurcations in nonlinear ODE models

A complex nonlinear system might exhibit a rich variety of dynamics. Detecting and classifying these dynamics is crucial to understand and ultimately control the system. Starting from an ODE model, this analysis can be done using standard bifurcation/continuation methods. However, when we start from measured data to infer the equations governing the dynamics (parameters and topology) we need tools to detect bifurcations in presence of parameter and even structural uncertainty. This is particularly the case in the field of systems biology, where we often deal with limited mechanistic knowledge (including potential conflicting hypotheses) and scarce experimental data.Here we present a method to efficiently predict the appearance of fold bifurcations in general systems of ordinary differential equations. The approach formulates the search of the bifurcation condition through a Mixed Integer Nonlinear Programming optimization strategy. In this way, we can simultaneously explore parameter and topology spaces searching for the desired bifurcation behaviour. We illustrate the capabilities and efficiency of our method through an example of biological relevance, the transcriptional network controlling adipogenesis, finding the conditions for an irreversible switch that causes the conversion of preadipocytes into adipocytes responsible of lipid accumulation.

Hopf bifurcation of a delayed diffusive predator-prey model with strong Allee effect

The paper is concerned with a delayed diffusive predator-prey system where the growth of prey population is governed by Allee effect and the predator population consumes the prey according to Beddington-DeAngelis type functional response. The situation of bi-stability and the existence of two coexisting equilibria for the proposed model system are addressed. The stability of the steady state together with its dependence on the magnitude of time delay has been obtained. The conditions that guarantee the occurrence of the Hopf bifurcation in presence of delay are demonstrated. Furthermore, the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem. Finally, some numerical simulations have been carried out in order to validate the assumptions of the model.

Bifurcation analysis of a modified Leslie–Gower predator–prey model with Beddington–DeAngelis functional response and strong Allee effect

The paper is concerned with a modified Leslie–Gower delayed predator–prey system where the growth of prey population is governed by Allee effect and the predator population consumes the prey according to Beddington–DeAngelis type functional response. The situation of bi-stability and existence of two interior equilibrium points for the proposed model system are addressed. The stability of the steady state together with its dependence on the magnitude of time delay has been obtained. The conditions that guarantee the occurrence of the Hopf bifurcation in presence of delay are demonstrated. Furthermore, the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem. It is shown that time delay is incapable of avoiding the situation of extinction of the prey species. Finally, some numerical simulations have been carried out in order to validate the assumptions of the model.

DEGENERATE BIFURCATIONS AND BORDER COLLISIONS IN PIECEWISE SMOOTH 1D AND 2D MAPS

We recall three well-known theorems related to the simplest codimension-one bifurcations occurring in discrete time dynamical systems, such as the fold, flip and Neimark–Sacker bifurcations, and analyze these bifurcations in presence of certain degeneracy conditions, when the above mentioned theorems are not applied. The occurrence of such degenerate bifurcations is particularly important in piecewise smooth maps, for which it is not possible to specify in general the result of the bifurcation, as it strongly depends on the global properties of the map. In fact, the degenerate bifurcations mainly occur in piecewise smooth maps defined in some subspace of the phase space by a linear or linear-fractional function, although not necessarily only by such functions. We also discuss the relation between degenerate bifurcations and border-collision bifurcations.

Hopf Bifurcation in Presence of 1 : 3 Resonance

Abstract Assume that the linear part at a singular point p0 of a C4 differential system Y0 in ℝ4 has eigenvalues ±αi and ±βi such that β/α = 1/3. In the main result of the paper we exhibit a one-parameter family of systems Yε for ε ∈ (-δ0;+δ0) where is shown that the original vector field around p0 can bifurcate in 0, 1, 2, 3 or 4 one-parameter families of periodic orbits. The tool for proving such a result is the averaging theory for non-C1 differentiable system. Moreover, assuming now that Yε is a one-parameter family of ℤ2- reversible polynomial vector fields of degree 5, we show that it can bifurcate in 0 or 2 one-parameter families of periodic orbits.

Non-axisymmetric subcritical bifurcations in viscoelastic Taylor–Couette flow

Recently, an account of the linear and nonlinear analysis of the viscoelastic Taylor–Couette flow between independently rotating cylinders against axisymmetric disturbances was presented (Avgousti & Beris 1993 a ). However, more recent linear stability analysis has shown that for the range of geometric and kinematic parameters studied and for high enough values of flow elasticity, the critical instabilities are caused by non-axisymmetric, time-dependent disturbances (Avgousti & Beris 1993 b ). In this work, we calculate the bifurcating families corresponding to each one of the two possible non-axisymmetric patterns emerging at the point of criticality, namely the spirals and ribbons and determine their stability. It is shown that for a narrow gap size, for upper convected Maxwell and Oldroyd-B fluids, at least one of the non-axisymmetric families bifurcates subcritically. This result, in conjunction with the theoretical analysis of Hopf bifurcation in presence of symmetries (Golubitsky et al . 1988), implies that neither of the bifurcating families is stable. Consequently, there is a finite transition corresponding to infinitesimal changes of the flow parameters in the vicinity of the Hopf bifurcation point. Although a change in the ratio of the Deborah and Reynolds numbers has not been found to qualitatively affect this behaviour, calculations with a wider gap size have shown that both bifurcating families become supercritical. There, a Ginzburg–Landau analysis shows that the ribbons are the stable pattern. This behaviour is qualitatively similar to that seen for the newtonian fluid, but for counterrotating cylinders, albeit there, spirals have been found to be stable (Golubitsky & Langford 1988).