Fractional amplitude and phase dynamics in super-diffusive reaction–diffusion systems

Study of weakly non-linear dynamics of a reaction–super-diffusion system near a Hopf bifurcation by means of fractional analogues of complex Ginzburg-Landau and Kuramoto-Sivashinsky equations is presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Pacemaker dynamics in the full Morris–Lecar model

This article reports the finding of pacemaker dynamics in certain region of the parameter space of the three-dimensional version of the Morris–Lecar model for the voltage oscillations of a muscle cell. This means that the cell membrane potential displays sustained oscillations in the absence of an external electrical stimulation. The development of this dynamic behavior is shown to be tied to the strength of the leak current contained in the model. The approach followed is mostly based on the use of linear stability analysis and numerical continuation techniques. In this way it is shown that the oscillatory dynamics is associated to the existence of two Hopf bifurcations, one subcritical and other supercritical. Moreover, it is explained that in the region of parameter values most commonly studied for this model such pacemaker dynamics is not displayed because of the development of two fold bifurcations, with the increase of the strength of the leak current, whose interaction with the Hopf bifurcations destroys the oscillatory dynamics.     

Well-posedness of Hamilton–Jacobi equations in population dynamics and applications to large deviations

We prove Freidlin–Wentzell type large deviation principles for various rescaled models in populations dynamics that have immigration and possibly harvesting: birth–death processes, Galton–Watson trees, epidemic SI models, and prey–predator models.The proofs are carried out using a general analytic approach based on the well-posedness of a class of associated Hamilton–Jacobi equations. The notable feature for these Hamilton–Jacobi equations is that the Hamiltonian can be discontinuous at the boundary. We prove a well-posedness result for a large class of Hamilton–Jacobi equations corresponding to one-dimensional models, and give partial results for the multi-dimensional setting.     

Low-energy vortex dynamics in the self-dual Chern–Simons–Higgs model

The relativistic Chern–Simons–Higgs theory finds application in anyonic superconductivity and contains topological vortices whose dynamics are poorly understood. The gauge fields are defined by a set of nonlinear constraint equations that can be accurately solved with effective Green’s functions, spectral methods, and a discretization scheme using lattice gauge techniques. Simulations show that low-energy two-vortex interactions are elastic with final scattering angles sensitive to vortex velocity; furthermore, vortex pairs form rotating breather states for certain impact parameters. In this study, a function that reproduces scattering angles in the adiabatic limit for nontangential collisions is presented. Simulation results are discussed in the context of analytical methods that extract vortex dynamics from low-energy effective Lagrangians, and a numerical method to calculate the effective Lagrangian is suggested. The numerical techniques used can be applied to the study of other Chern–Simon theories.     

Optimal control of the freezing time in the Hegselmann–Krause dynamics

We study the optimal control problem of minimizing the freezing time in the discrete Hegselmann–Krause (HK) model of opinion dynamics. The underlying model is extended with a set of strategic agents that can freely place their opinion at every time step. Indeed, if suitably coordinated, the strategic agents can significantly lower the freezing time of an instance of the HK model. We give several lower and upper worst-case bounds for the freezing time of a HK system with a given number of strategic agents, while still leaving some gaps for future research.     

Mobile-phone-based classroom response systems: Students’ perceptions of engagement and learning in a large undergraduate course

Using a Classroom Response System (CRS) has been associated with positive educational outcomes, by fostering student engagement and by allowing immediate feedback to both students and instructors. This study examined a low-cost CRS (VotApedia) in a large first-year class, where students responded to questions using their mobile phones. This study explored whether the use of VotApedia retained the advantages of other CRS, overcame some of the challenges of other CRS, and whether new challenges were introduced by using VotApedia. These issues were studied within three themes: students’ perceptions of using VotApedia; the impact of VotApedia on their engagement; and the impact of VotApedia on their learning. Data were collected from an online survey, focus groups and student feedback on teaching and course content. The results indicated that using VotApedia retains the pedagogical advantages of other CRS, while overcoming some of the challenges presented by using other CRS, without introducing any new challenges.     

On the existence of dynamics in Wheeler–Feynman electromagnetism

Wheeler–Feynman electrodynamics (WF) is an action-at-a-distance theory about world-lines of charges that in contrary to the textbook formulation of classical electrodynamics is free of ultraviolet singularities and is capable of explaining the irreversible nature of radiation. In WF, the world-lines of charges obey the so-called Fokker–Schwarzschild–Tetrode (FST) equations, a coupled set of nonlinear and neutral differential equations that involve time-like advanced as well as retarded arguments of unbounded delay. Using a reformulation of this theory in terms of Maxwell–Lorentz electrodynamics without self-interaction that we have introduced in a preceding work, we are able to establish the existence of conditional solutions. These conditional solutions solve the FST equations on any finite time interval with prescribed continuations outside of this interval. As a byproduct, we also prove existence and uniqueness of solutions to the Synge equations on the time half-line for a given history of charge world-lines.     

Existence of a solution ‘in the large’ for the 3D large-scale ocean dynamics equations

For the 3D system of equations describing large-scale ocean dynamics in the Cartesian coordinate system existence and uniqueness of a solution on an arbitrary time interval $[0,T]$ is proved and the norm $6{\stackrel{?}{\mathbf{u}}}_{x}6$ is shown to be continuous in time on $[0,T]$. To cite this article: G.M. Kobelkov, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Schrödinger–Poisson systems in the 3-sphere

We investigate nonlinear Schrödinger–Poisson systems in the 3-sphere. We prove existence results for these systems and discuss the question of the stability of the systems with respect to their phases. While, in the subcritical case, we prove that all phases are stable, we prove in the critical case that there exists a sharp explicit threshold below which all phases are stable and above which resonant frequencies and multi-spikes blowing-up solutions can be constructed. Solutions of the Schrödinger–Poisson systems are standing waves solutions of the electrostatic Maxwell–Schrödinger system. Stable phases imply the existence of a priori bounds on the amplitudes of standing waves solutions. Unstable phases give rise to resonant states.     

Vortex dynamics in the presence of excess energy for the Landau–Lifshitz–Gilbert equation

We study the Landau–Lifshitz–Gilbert equation for the dynamics of a magnetic vortex system. We present a PDE-based method for proving vortex dynamics that does not rely on strong well-preparedness of the initial data and allows for instantaneous changes in the strength of the gyrovector force due to bubbling events. The main tools are estimates of the Hodge decomposition of the supercurrent and an analysis of the defect measure of weak convergence of the stress energy tensor. Ginzburg–Landau equations with mixed dynamics in the presence of excess energy are also discussed.     

Existence of limit cycles in general planar piecewise linear systems of saddle–saddle dynamics

The existence and number of limit cycles in a class of general planar piecewise linear systems constituted by two linear subsystems with saddle–saddle dynamics are investigated. Using the Liénard-like canonical form with seven parameters, the parametric regions of the existence of limit cycles are given by constructing proper Poincaré maps. In particular, the existence of at least two limit cycles is proved and some parameter regions where two nested limit cycles exist are given.     

On the Marcinkiewicz–Zygmund law of large numbers in Banach lattices

We strengthen the well-known Marcinkiewicz–Zygmund law of large numbers in the case of Banach lattices. Examples of applications to empirical distributions are presented.     

Evolution in predator–prey systems

We study the adaptive dynamics of predator–prey systems modeled by a dynamical system in which the traits of predators and prey are allowed to evolve by small mutations. When only the prey are allowed to evolve, and the size of the mutational change tends to 0, the system does not exhibit long term prey coexistence and the trait of the resident prey type converges to the solution of an ODE. When only the predators are allowed to evolve, coexistence of predators occurs. In this case, depending on the parameters being varied, we see that (i) the number of coexisting predators remains tight and the differences in traits from a reference species converge in distribution to a limit, or (ii) the number of coexisting predators tends to infinity, and we calculate the asymptotic rate at which the traits of the least and most “fit” predators in the population increase. This last result is obtained by comparison with a branching random walk killed to the left of a linear boundary and a finite branching–selection particle system.     

On the dynamics of the Rayleigh–Duffing oscillator

We give a complete algebraic characterization of the first integrals of the Rayleigh–Duffing oscillator. We prove the non existence of centers of such system and we study the form of the singular first integrals at the origin.     

Large-time behavior for the solution to the generalized Benjamin–Bona–Mahony–Burgers equation with large initial data in the whole-space

In this paper, we consider the global existence as well as the optimal decay estimates of the Cauchy problem for the multi-dimensional Benjamin–Bona–Mahony–Burgers equation with large initial data in the whole-space. And these results are obtained by Green?s function method, Fourier analysis method, energy estimates method combined with the time-frequency decomposition method.     

Modeling the Influence of Public’s Memory on the Corruption–Popularity Dilemma in Politics

We consider the problem of a rational politician who gains benefit from both being popular and corrupt. In 1994, Feichtinger and Wirl studied this trade-off by means of an infinite-horizon optimal control approach. We reconsider the problem over a finite time horizon, to model the dilemma of a politician who stays in office for a limited period of time and wishes to be reelected. We also include explicitly the possibility that awareness of politician’s conduct to the general population is delayed and the effect of such delay on the politician’s optimal behavior is analyzed. We show the outcomes under several different scenarios, with particular reference to the role of the politician’s communication skills. One general conclusion is that the synergy of a good public’s memory and immediate knowledge of corruption may often help the people to protect themselves from political corruption. Moreover, this synergy may greatly enhance the popularity of a politician with poor communication skills, provided that he/she aims at maximizing benefit from popularity during his/her mandate.     

Modeling and analysis of a prey–predator system with disease in the prey

A three dimensional ecoepidemiological model consisting of susceptible prey, infected prey and predator is proposed and analysed in the present work. The parameter delay is introduced in the model system for considering the time taken by a susceptible prey to become infected. Mathematically we analyze the dynamics of the system such as, boundedness of the solutions, existence of non-negative equilibria, local and global stability of interior equilibrium point. Next we choose delay as a bifurcation parameter to examine the existence of the Hopf bifurcation of the system around its interior equilibrium. Moreover we use the normal form method and center manifold theorem to investigate the direction of the Hopf bifurcation and stability of the bifurcating limit cycle. Some numerical simulations are carried out to support the analytical results.