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A review is presented of the one-parameter, nonsmooth bifurcations that occur in a variety of continuous-time piecewise-smooth dynamical systems. Motivated by applications, a pragmatic approach is taken to defining a discontinuity-induced bifurcation (DIB) as a nontrivial interaction of a limit set with respect to a codimension-one discontinuity boundary in phase space. Only DIBs that are local are considered, that is, bifurcations involving equilibria or a single point of boundary interaction along a limit cycle for flows. Three classes of systems are considered, involving either state jumps, jumps in the vector field, or jumps in some derivative of the vector field. A rich array of dynamics are revealed, involving the sudden creation or disappearance of attractors, jumps to chaos, bifurcation diagrams with sharp corners, and cascades of period adding. For each kind of bifurcation identified, where possible, a kind of \"normal form\" or discontinuity mapping (DM) is given, together with a canonical example and an application. The goal is always to explain dynamics that may be observed in simulations of systems which include friction oscillators, impact oscillators, DC-DC converters, and problems in control theory.

This paper investigates bifurcations analysis and resonances in a discrete-time prey-predator model analytically and numerically as well. The local stability conditions of all the fixed points in the system are determined. Here, codim-1 and codim-2 bifurcation including multiple and generic bifurcations in the discrete model are explored. The model undergoes fold bifurcation, flip bifurcation, Neimark–Sacker bifurcation and resonances 1:2, 1:3, 1:4 at different fixed points. Using the critical normal form theorem and bifurcation theory, normal form coefficients are calculated for each bifurcation. The different bifurcation curves of fixed points are drawn which validate the analytical findings. The numerical simulation gives a wide range of periodic cycles including codim-1 bifurcation and resonance curves in the system. The results in this manuscript reveal that the dynamics of the discrete-time model in both single-parameter and two-parameter spaces are inherently rich and complex. The resonance bifurcation in the discrete-time map indicates that both species coincide till order 4 in stable periodic cycles near some critical parametric values.

We demonstrate how the pitchfork, transcritical and saddle-node bifurcations of steady states observed in dynamical systems with a finite number of isolated equilibrium points occur in systems with lines of equilibria. The exploration is carried out by using the numerical simulation and linear stability analysis applied to a model of a memristor-based circuit. All the discussed bifurcation scenarios are considered in the context of models with the piecewise-smooth memristor current-voltage characteristic (Chua’s memristor), as well as on examples of oscillators with the memristor nonlinearity that is smooth everywhere. Finally, we compare the dynamics of ideal-memristor-based oscillators with the behavior of models taking into account the memristor forgetting effect. The presented results are obtained for electronic circuit models, but the studied bifurcation phenomena can be exhibited by systems with lines of equilibria of any nature.

The nature of fibre motion is much more complex than the motion of spherical particles. A complex model must be used for tracking of the fibre carried by the flow. The model must take into account the orientation of the fibre against the flow and solve the consequent rotation of the fibre. Description of such model can be found in this article. It is then used for simulation and analysis of fibre movement in a single planar bifurcation.

A series of bifurcations from period-1 bursting to period-1 spiking in a complex (or simple) process were observed with increasing extra-cellular potassium concentration during biological experiments on different neural pacemakers. This complex process is composed of three parts: period-adding sequences of burstings, chaotic bursting to chaotic spiking, and an inverse period-doubling bifurcation of spiking patterns. Six cases of bifurcations with complex processes distinguished by period-adding sequences with stochastic or chaotic burstings that can reach different bursting patterns, and three cases of bifurcations with simple processes, without the transition from chaotic bursting to chaotic spiking, were identified. It reveals the structures closely matching those simulated in a two-dimensional parameter space of the Hindmarsh–Rose model, by increasing one parameter I and fixing another parameter r at different values. The experimental bifurcations also resembled those simulated in a physiologically based model, the Chay model. The experimental observations not only reveal the nonlinear dynamics of the firing patterns of neural pacemakers but also provide experimental evidence of the existence of bifurcations from bursting to spiking simulated in the theoretical models.

A non-contact salinity sensor using a bifurcated fiber bundle sensing is proposed. The sensor was operated based on intensity modulation due to the variation of displacement between the tip of the probe with the surface of the sample. The light that reflected back into the receiving fiber bundle is analysed and correlation with the salinity of the samples has been performed. This allows the performance of the sensor towards salinity to be quantified based on the front slope, back slope and peak intensity. The proposed configuration has been calibrated and then experimentally validated. The calibration curve can be plotted based on three parameters: peak intensity, front slope and back slope. The peak intensity curve exhibits the highest sensitivity of 1.52×10 5 a.u./M with the resolution of 4.2×10 -2 M. The ubiquitousness of sensor configuration and non-contact measurement make the proposed sensor a viable alternative to standard commercial devices.

We use direct numerical simulations to study convection in rotating Rayleigh-B & eacute;nard convection in horizontally confined geometries of a given aspect ratio, with the walls held at fixed temperatures. We show that this arrangement is unconditionally unstable to flow that takes the form of wall-adjacent convection rolls. For wall temperatures close to the temperatures of the upper or lower boundaries, we show that the base state undergoes a Hopf bifurcation to a state comprised of spatiotemporal oscillations - 'wall modes' - precessing in a retrograde direction. We study the saturated nonlinear state of these modes, and show that the velocity boundary conditions at the upper and lower boundaries are crucial to the formation and propagation of the wall modes: asymmetric velocity boundary conditions at the upper and lower boundaries can lead to prograde wall modes, while stress-free boundary conditions at both walls can lead to wall modes that have no preferred direction of propagation.

In this paper, we propose a reaction-diffusion genetic regulatory network with Neumann boundary conditions. We incorporate the diffusion factors into the delayed mathematical model of genetic networks, and with the time delays, we demonstrate the local stability and Hopf bifurcation in accordance with the Hopf bifurcation theory. When the sum of the delays exceeds the critical value, a Hopf bifurcation occurs. Finally, we select the appropriate system parameters and present a numerical simulation to illustrate the theoretical analysis.

In order to prevent unpleasant incidents, preservation high-speed railway vehicle stability has vital importance. For this purpose, railway vehicle is modeled using a 38-DOF. This paper presents an investigation on limit cycle and bifurcation of high speed railway behavior. It is revealed that the critical hunting speed decreases by increasing the wheel conicity. Also the critical hunting speed and hunting frequency calculated via the linear elastic rail is higher than that derived using a nonlinear model. Index Terms- railway vehicle dynamics, heuristic nonlinear creep model, critical hunting speed, numerical simulation, bifurcation analysis.

In this paper, the complicated heteroclinic and codimension-four bifurcations of a triple SD (smooth and discontinuous) oscillator are investigated by analyzing the bifurcation sets in three-dimensional parameter space. The structure of the transition set including the equilibrium bifurcation set and a special kind of heteroclinic orbit bifurcation set is constructed comprising of a catastrophe point of the fifth order, the catastrophe curves of third order and also the catastrophe surfaces of the first order, respectively, according to the restoring forces and also the potentials, respectively. Also, a theorem of structural stability of heteroclinic orbit in 2-dimensional Hamilton system is introduced to find the heteroclinic bifurcation set. The equilibria and the phase structures are classified and shown in details on the transition set and the enclosed structurally stable areas for smooth and discontinuous cases, respectively. The normal forms for each bifurcation surface are built up showing the complex supercritical subcritical pitchfork bifurcations and also the double saddle-node bifurcations, along with the bifurcations of homoclinic and heteroclinic orbit. Taken one of the bifurcation surfaces as an example, the complicated bifurcation is investigated by employing subharmonic Melnikov functions including Hopf, double Hopf, the closed orbit and also the homoclinic/heteroclinic bifurcations. The results presented herein this paper enriched the complex dynamic behavior for the geometrical nonlinear systems.