Controlling the Dynamical Spread of Coronavirus Disease (COVID-19) in a Population

In the paper, a model governed by a system of ordinary differential equations was considered; the whole population was divided into Susceptible individuals (S), Exposed individuals (E), Infected individuals (I), Quarantined individuals (Q) and Recovered individuals (R). The well-posedness of the model was investigated by the theory of positivity and boundedness. Analytically, the equilibrium solutions were examined. A key threshold which measures the potential spread of the Coronavirus in the population is derived using the next generation method. Bifurcation analysis and global stability of the model were carried out using centre manifold theory and Lyapunov functions respectively. The effects of some parameters such as Progression rate of exposed class to infectious class, Effective contact rate, Modification parameter, Quarantine rate of infectious class, Recovery rate of infectious class and Recovery rate of quarantined class on R0 were explored through sensitivity analysis. Numerical simulations were carried out to support the theoretical results, to reduce the burden of COVID 19 disease in the population and significant in the spread of it in the population.

The Lyapunov exponents and the neighbourhood of periodic orbits

ABSTRACT We show that the Lyapunov exponents of a periodic orbit can be easily obtained from the eigenvalues of the monodromy matrix. It turns out that the Lyapunov exponents of simply stable periodic orbits are all zero, simply unstable periodic orbits have only one positive Lyapunov exponent, doubly unstable periodic orbits have two different positive Lyapunov exponents, and the two positive Lyapunov exponents of complex unstable periodic orbits are equal. We present a numerical example for periodic orbits in a realistic galactic potential. Moreover, the centre manifold theorem allowed us to show that stable, simply unstable, and doubly unstable periodic orbits are the mothers of families of, respectively, regular, partially, and fully chaotic orbits in their neighbourhood.

Pattern formation in a slowly flattening spherical cap: delayed bifurcation

This article describes a reduction of a non-autonomous Brusselator reaction–diffusion system of partial differential equations on a spherical cap with time-dependent curvature using the method of centre manifold reduction. Parameter values are chosen such that the change in curvature would cross critical values which would change the stability of the patternless solution in the constant domain case. The evolving domain functions and quasi-patternless solutions are derived as well as a method to obtain this non-autonomous normal form. The coefficients of such a normal form are computed and the reduction solutions are compared to numerical solutions.

Method of Centre Manifold from Bifurcation Theory

The aim of this paper is to introduce tools from bifurcation theory is necessary in ways in our life particularly in the study of neural field equations set in the primary visual cortex. So we deal with saddle-node, trans- critical, pitchfork and Hopf. Bifurcations as an elementary bifurcation; directly related to the center manifold theory which is a canonical way to write differential equations. We conclude this paper with an overview of bifurcations with symmetry by solving some problems and giving Branching Lemma as the equivariant result

Finite-dimensional coagulation-fragmentation dynamics

We examine a finite-dimensional truncation of the discrete coagulation-fragmentation equations that is designed to allow mass to escape from the system into clusters larger than those in the truncated problem. The aim is to model within a finite system the process of gelation, which is a type of phase transition observed in aerosols, colloids, etc. The main result is a centre manifold calculation that gives the asymptotic behaviour of the truncated model as time [Formula: see text]. Detailed numerical results show that truncated system solutions are often very close to this centre manifold, and the range of validity of the truncated system as a model of the full infinite problem is explored for systems with and without gelation. The latter cases are mass conserving, and we provide an estimate using quantities from the centre manifold calculations of the time period and the truncated system can be used for before loss of mass which is apparent. We also include some observations on how numerical approximation can be made more reliable and efficient.

Periodic waves of the Lugiato–Lefever equation at the onset of Turing instability

We study the existence and the stability of periodic steady waves for a nonlinear model, the Lugiato–Lefever equation, arising in optics. Starting from a detailed description of the stability properties of constant solutions, we then focus on the periodic steady waves which bifurcate at the onset of Turing instability. Using a centre manifold reduction, we analyse these Turing bifurcations, and prove the existence of periodic steady waves. This approach also allows us to conclude on the nonlinear orbital stability of these waves for co-periodic perturbations, i.e. for periodic perturbations which have the same period as the wave. This stability result is completed by a spectral stability result for general bounded perturbations. In particular, this spectral analysis shows that instabilities are always due to co-periodic perturbations. This article is part of the theme issue ‘Stability of nonlinear waves and patterns and related topics’.

On the analysis of the double Hopf bifurcation in machining processes via centre manifold reduction

The single-degree-of-freedom model of orthogonal cutting is investigated to study machine tool vibrations in the vicinity of a double Hopf bifurcation point. Centre manifold reduction and normal form calculations are performed to investigate the long-term dynamics of the cutting process. The normal form of the four-dimensional centre subsystem is derived analytically, and the possible topologies in the infinite-dimensional phase space of the system are revealed. It is shown that bistable parameter regions exist where unstable periodic and, in certain cases, unstable quasi-periodic motions coexist with the equilibrium. Taking into account the non-smoothness caused by loss of contact between the tool and the workpiece, the boundary of the bistable region is also derived analytically. The results are verified by numerical continuation. The possibility of (transient) chaotic motions in the global non-smooth dynamics is shown.