We study the planar FitzHugh–Nagumo system with an attracting periodic orbit that surrounds a repelling focus equilibrium. When the associated oscillation of the system is perturbed, in a given direction and with a given amplitude, there will generally be a change in phase of the perturbed oscillation with respect to the unperturbed one. This is recorded by the phase transition curve (PTC), which relates the old phase (along the periodic orbit) to the new phase (after perturbation). We take a geometric point of view and consider the phase-resetting surface comprising all PTCs as a function of the perturbation amplitude. This surface has a singularity when the perturbation maps a point on the periodic orbit exactly onto the repelling focus, which is the only point that does not return to stable oscillation. We also consider the PTC as a function of the direction of the perturbation and present how the corresponding phase-resetting surface changes with increasing perturbation amplitude. In this way, we provide a complete geometric interpretation of how the PTC changes for any perturbation direction. Unlike other examples discussed in the literature so far, the FitzHugh–Nagumo system is a generic example and, hence, representative for planar vector fields.

Macroscopically, a Darcian unsaturated moisture flow in the top soil is usually represented by an one-dimensional volume scale of evaporation from a static water table. On the microscale, simple pore-level models posit bundles of small-radius capillary tubes of a constant circular cross-section, fully occupied by mobile water moving in the Hagen–Poiseuille (HP) regime, while large-diameter pores are occupied by stagnant air. In our paper, cross-sections of cylindrical pores are polygonal. Steady, laminar, fully developed two-dimensional flows of Newtonian water in prismatic conduits, driven by a constant pressure gradient along a pore gradient, are more complex than the HP formula; this is based on the fact that the pores are only partially occupied by water and immobile air. The Poisson equation in a circular tetragon, with no-slip or mixed (no-shear-stress) boundary conditions on the two adjacent pore walls and two menisci, is solved by the methods of complex analysis. The velocity distribution is obtained via the Keldysh–Sedov type of singular integrals, and the flow rate is evaluated for several sets of meniscus radii by integrating the velocity over the corresponding tetragons.

We consider a pair of identical theta neurons in the active regime, each coupled to the other via a delayed Dirac delta function. The network can support periodic solutions and we concentrate on solutions for which the neurons are half a period out of phase with one another, and also solutions for which the neurons are perfectly synchronous. The dynamics are analytically solvable, so we can derive explicit expressions for the existence and stability of both types of solutions. We find two branches of solutions, connected by symmetry-broken solutions which arise when the period of a solution as a function of delay is at a maximum or a minimum.

In this paper, we derive simple analytical bounds for solutions of $x - \ln x = y -\ln y$, and use them for estimating trajectories following Lotka–Volterra-type integrals. We show how our results give estimates for the Lambert W function as well as for trajectories of general predator–prey systems, including, for example, Rosenzweig–MacArthur equations.

This study explores the dynamics of a simple mechanical oscillator involving a magnet on a spring constrained to an axis; this magnet is additionally subject to the attractive force from a second magnet, which is placed on a parallel offset axis. The moments of both magnets remain aligned. The dynamics of the first magnet is first analysed in isolation for an unforced situation in which the second magnet is static and its position is taken as a parameter. We find codimension-1 saddle-node bifurcations, as well as a codimension-2 cusp bifurcation. The system has a region of bistability which increases in size with increasing force ratio. Next, the parametrically forced situation is considered, in which the second magnet moves sinusoidally. A comprehensive analysis of the forced oscillator behaviour is presented from the dynamical-systems standpoint. The solutions are shown to include periodic, quasiperiodic and chaotic trajectories. Resonances are shown to exist and the effect of weak damping is explored. Layered stroboscopic maps are used to produce cross-sections of the chaotic attractor as the parametric forcing frequency is varied. The strange attractor is found to disappear for a narrow window of forcing frequencies near the natural frequency of the spring.

While constructing mathematical models, scientists usually consider biotic factors, but it is crystal-clear that abiotic factors, such as wind, are also important as biotic factors. From this point of view, this paper is devoted to the investigation of some bifurcation properties of a fractional-order prey–predator model under the effect of wind. Using fractional calculus is very popular in modelling, since it is more effective than classical calculus in predicting the system’s future state and also discretization is one of the most powerful tools to study the behaviour of the models. In this paper, first of all, the model is discretized by using a piecewise discretization approach. Then, the local stability of fixed points is considered. We show using the centre manifold theorem and bifurcation theory that the system experiences a flip bifurcation and a Neimark–Sacker bifurcation at a positive fixed point. Finally, numerical simulations are given to demonstrate our results.

We conduct a theoretical analysis of the performance of $\beta $-encoders. The $\beta $-encoders are A/D (analogue-to-digital) encoders, the design of which is based on the expansion of real numbers with noninteger radix. For the practical use of such encoders, it is important to have theoretical upper bounds of their errors. We investigate the generating function of the Perron–Frobenius operator of the corresponding one-dimensional map and deduce the invariant measure of it. Using this, we derive an approximate value of the upper bound of the mean squared error of the quantization process of such encoders. We also discuss the results from a numerical viewpoint.

We are concerned with the micro-macro Parareal algorithm for the simulation of initial-value problems. In this algorithm, a coarse (fast) solver is applied sequentially over the time domain and a fine (time-consuming) solver is applied as a corrector in parallel over smaller chunks of the time interval. Moreover, the coarse solver acts on a reduced state variable, which is coupled with the fine state variable through appropriate coupling operators. We first provide a contribution to the convergence analysis of the micro-macro Parareal method for multiscale linear ordinary differential equations. Then, we extend a variant of the micro-macro Parareal algorithm for scalar stochastic differential equations (SDEs) to higher-dimensional SDEs.

This study aims to formulate a highly accurate numerical method, specifically a seventh-order Hermite technique with an error term of sixth order, to solve the Fisher and Burgers–Fisher equations. This technique employs a combination of orthogonal collocation on the finite element method and hepta Hermite basis functions. By ensuring continuity of the dependent variable and its first three derivatives across the entire solution domain, it achieves a remarkable level of accuracy and smoothness. The space discretization is handled through the application of hepta Hermite polynomials, while the time discretization is managed by the Crank–Nicholson scheme. The stability and convergence analysis of the scheme are discussed in detail. To validate the accuracy of the proposed technique, three examples are taken. The results obtained from these examples are thoroughly analysed and compared against the exact solutions and reliable data from the existing literature. It is established that the proposed technique is easy to implement and gives better results as compared with existing ones.

Conformal image registration has always been an area of interest among modern researchers, particularly in the field of medical imaging. The idea of image registration is not new. In fact, it was coined nearly 100 years ago by the pioneer D’Arcy Wentworth Thompson, who conjectured the idea of image registration among the biological forms. According to him, several images of different species are related by a conformal transformations. Thompson’s examples motivated us to explore his claim using image registration. In this paper, we present a conformal image registration (for the two-dimensional grey scaled images) along with a penalty term. This penalty term, which is based on the Cauchy–Riemann equations, aims to enforce the conformality.