Software for ordinary and delay differential equations: Accurate discrete approximate solutions are not enough

Numerical methods for both ordinary differential equations (ODEs) and delay differential equations (DDEs) are traditionally developed and assessed on the basis of how well the accuracy of the approximate solution is related to the specified error tolerance on an adaptively-chosen, discrete mesh. This may not be appropriate in numerical investigations that require visualization of an approximate solution on a continuous interval of interest (rather than at a small set of discrete points) or in investigations that require the determination of the ‘average’ values or the ‘extreme’ values of some solution components.In this paper we will identify modest changes in the standard error-control and stepsize-selection strategies that make it easier to develop, assess and use methods which effectively deliver approximations to differential equations (both ODEs and DDEs) that are more appropriate for these type of investigations. The required changes will typically increase the cost per step by up to 40%, but the improvements and advantages gained will be significant. Numerical results will be presented for these modified methods applied to two example investigations (one ODE and one DDE).     

Inertial ranges for turbulent solutions of complex Ginzburg-Landau equations

We consider the spatially periodic, complex Ginzburg-Landau (CGL) equation in regimes close to that of a critical or supercritical focusing non-linear Schrödinger (NLS) equation, which is known to have solutions that exhibit self-similar blow-up. We use the NLS blow-up solutions as a template to develop a theory of how nearly self-similar intermittent burst events can create a power-law inertial range in the time-averaged wave-number spectrum of CGL solutions. Numerical experiments in one dimension with a quintic (critical) and septant (supercritical) non-linearity show a that power-law inertial range emerges which differs from that predicted by the theory. However, as one approaches the NLS limit in the supercritical case, a second power-law inertial range is seen to emerge that agrees with the theory.     

Simulated test of electric activity of neurons by using Josephson junction based on synchronization scheme

The chaotic circuit of resistive–capacitive–inductive-shunted Josephson junction is used to simulate behavior of Hindmarsh–Rose neuronal discharges. Based on tracking control theory, the controller contains two gain coefficients was constructed to control the chaotic system of Josephson junction to synchronize the chaotic Hindmarsh–Rose system, and the single controller was approached analytically. The results confirmed that the controller with appropriate gain coefficients was effective to reach complete synchronization (the amplitudes and rhythms of two systems are identical), phase synchronization (rhythms of two systems are identical) of Josephson junction and Hindmarsh–Rose neurons, respectively. The power consumption is estimated in a feasible way. As a result, the electric activities of Hindmarsh–Rose neurons could be simulated by using Josephson junction model completely.     

On cascades of bifurcations leading to chaos in several nonlinear dissipative systems of ODEs

This paper considers several nonlinear dissipative systems of ordinary differential equations. The studied systems undergo a full analysis of corresponding singular points on a whole set of parameters’ values variation. Specifically, types of singular points, boarders of stability regions, as well as presented local bifurcations, are determined. By using numerical methods a consideration of scenarios of transition to chaos in these systems with one bifurcation parameter variation is held. The aim of this research is a confirmation of a Feigenbaum–Sharkovskii–Magnitskii mechanism of transition to chaos unique for all dissipative systems of ODEs. As the result of analysis of one of the systems the lack of any chaotic behavior is shown with the help of Poincare sections.     

Elimination of spiral waves in excitable media by magnetic induction

Nonlinear Dynamics - The formation of spiral waves in excitable media is a fascinating example of the beauty of nonlinear dynamics in spatiotemporal systems. Apart from the beauty of the patterns,...     

Transport properties of the antiferromagnetic systems with fluctuating valence

We present the calculation of the DC resistivity (conductivity) for the antiferromagnetic, two-band, extended s–f model. The influence on the finite bandwidth of the narrow 4f (5f) band on the transport properties of the model is investigated. We notice that the increase of the 4f (5f) bandwidth destroys the antiferromagnetic order and the system becomes paramagnetic in all temperatures. A systematic review of the DC resistivity (conductivity) is presented in the form of the 3D plots including different average occupation numbers of electrons per site (n=0.5, 0.75, 1, 1.25, 1.5, 1.75, 2), different relative positions of the 4f (5f) band and different 4f (5f) bandwidths. The calculated temperature dependence of the DC resistivity (conductivity) shows great similarities to the experimental results for many intermediate-valence rare-earth and actinide-based materials.     

Bifurcation analysis of the corona discharge on a negative electrode

A bifurcation analysis of the cylindrically symmetric solution obtained in the framework of the steady-state model of the negative corona discharge in air at atmospheric pressure has been carried out. No bifurcations have been found, which indicates that current contraction probably does not occur in the considered model. It follows that the current-free stripes on the corona cathode surface, observed by previous authors in two- and three-dimensional computer simulation and in experiment, are due to the specific discharge geometry rather than current contraction.     

Remarques sur les mesures de Wigner

We compute the Wigner measures associated to solutions of semi-classical nonlinear Schrödinger equations. These solutions focus at a point. Outside the caustic, the measures “smooth” the nonlinearity. In the critical cases, a scattering operator describes the jump of Wigner measures at the focus. We show that the problem is ill-posed in terms of Wigner measures.     

On Some SEIRS Epidemic Models

We discuss a few variations of the SEIRS epidemic model. How basic dynamical properties of the models can be derived by using some tools of the computer algebra system Mathematica is shown, and how invariant surfaces of the system can be found by using computer algebra system Singular is explained. Some numerical simulations are presented as well.