Bifurcation of Dividing Surfaces Constructed from a Pitchfork Bifurcation of Periodic Orbits in a Symmetric Potential Energy Surface with a Post-Transition-State Bifurcation

In this work, we analyze the bifurcation of dividing surfaces that occurs as a result of a pitchfork bifurcation of periodic orbits in a two degrees of freedom Hamiltonian System. The potential energy surface of the system that we consider has four critical points: two minima, a high energy saddle and a lower energy saddle separating two wells (minima). In this paper, we study the structure, the range, and the minimum and maximum extent of the periodic orbit dividing surfaces of the family of periodic orbits of the lower saddle as a function of the total energy.

Complex Dynamics of a Four-Dimensional Circuit System

Combining qualitative analysis and numerical technique, the present work revisits a four-dimensional circuit system in [Ma et al., 2016] and mainly reveals some of its rich dynamics not yet investigated: pitchfork bifurcation, Hopf bifurcation, singularly degenerate heteroclinic cycle, globally exponentially attractive set, invariant algebraic surface and heteroclinic orbit. The main contributions of the work are summarized as follows: Firstly, it is proved that there exists a globally exponentially attractive set with three different exponential rates by constructing a suitable Lyapunov function. Secondly, the existence of a pair of heteroclinic orbits is also proved by utilizing two different Lyapunov functions. Finally, numerical simulations not only are consistent with theoretical results, but also illustrate potential existence of hidden attractors in its Lorenz-type subsystem, singularly degenerate heteroclinic cycles with distinct geometrical structures and nearby hyperchaotic attractors in the case of small [Formula: see text], i.e. hyperchaotic attractors and nearby pseudo singularly degenerate heteroclinic cycles, i.e. a short-duration transient of singularly degenerate heteroclinic cycles approaching infinity, or the true ones consisting of normally hyperbolic saddle-foci (or saddle-nodes) and stable node-foci, giving some kind of forming mechanism of hyperchaos.

Pattern Transition on Inertial Focusing of Neutrally Buoyant Particles Suspended in Rectangular Duct Flows

The continuous separation and filtration of particles immersed in fluid flows are important interests in various applications. Although the inertial focusing of particles suspended in a duct flow is promising in microfluidics, predicting the focusing positions depending on the parameters, such as the shape of the duct cross-section and the Reynolds number (Re) has not been achieved owing to the diversity of the inertial-focusing phenomena. In this study, we aimed to elucidate the variation of the inertial focusing depending on Re in rectangular duct flows. We performed a numerical simulation of the lift force exerted on a spherical particle flowing in a rectangular duct and determined the lift-force map within the duct cross-section over a wide range of Re. We estimated the particle trajectories based on the lift map and Stokes drag, and identified the particle-focusing points appeared in the cross-section. For an aspect ratio of the duct cross-section of 2, we found that the blockage ratio changes transition structure of particle focusing. For blockage ratios smaller than 0.3, particles focus near the centres of the long sides of the cross-section at low Re and near the centres of both the long and short sides at relatively higher Re. This transition is expressed as a subcritical pitchfork bifurcation. For blockage ratio larger than 0.3, another focusing pattern appears between these two focusing regimes, where particles are focused on the centres of the long sides and at intermediate positions near the corners. Thus, there are three regimes; the transition between adjacent regimes at lower Re is found to be expressed as a saddle-node bifurcation and the other transition as a supercritical pitchfork bifurcation.

The Dynamics of Sokol-Howell Prey-Predator Model Involving Strong Allee Effect

In this paper, a Sokol-Howell prey-predator model involving strong Allee effect is proposed and analyzed. The existence, uniqueness, and boundedness are studied. All the five possible equilibria have been are obtained and their local stability conditions are established. Using Sotomayor's theorem, the conditions of local saddle-node and transcritical and pitchfork bifurcation are derived and drawn. Numerical simulations are performed to clarify the analytical results

On the genetic divergence of two adjacent populations living in a homogeneous habitat

The purpose is to study the mechanisms leading to the genetic divergence, i.e. stable genetic differences between two adjacent populations coupled by migration of individuals. We considered the case when the fitness of individuals is strictly determined genetically by a single diallelic locus with alleles A and a, the population is panmictic and Mendel's laws of inheritance hold. The dynamic model contains three phase variables: concentration of allele A in each population and fraction (weight) of the first population in the total population size. We assume that the numbers of coupled populations change independently or strictly synchronously. In the first case, the growth rates are determined by fitness of homo- and heterozygotes, the mean fitness of the each population and the initial concentrations of alleles. In the second case, the growth rates are the same. Methods. To study the model, we used the qualitative theory of differential equations studies, including the construction of parametric and phase portraits, basins of attraction and bifurcation diagrams. We studied local bifurcations that provide the fundamental possibility of genetic divergence. Results. If heterozygote fitness is higher than homozygotes, then both populations are polymorphic with the same concentration of homologous alleles. If the heterozygotes fitness is reduced, then over time the populations will have the same monomorphism in one allele, regardless of the type of population changes. In this case, the dynamics is bistable. We showed that the divergence in the model is a result of subcritical pitchfork bifurcation of an unstable polymorphic state. As a result, the genetic divergent state is unstable and exists as part of the transient process to one of monomorphic state. Conclusion. Divergence is stable only for populations that maintain a population ratio in a certain way. In this case, it is preceded by a saddle-node bifurcation and dynamics is quad-stable, i.e. depending on the initial conditions, two types of stable monomorphism and divergence are possible simultaneously.

Parametric Frequency Analysis of Mathieu–Duffing Equation

The classic linear Mathieu equation is one of the archetypical differential equations which has been studied frequently by employing different analytical and numerical methods. The Mathieu equation with cubic nonlinear term, also known as Mathieu–Duffing equation, is one of the many extensions of the classic Mathieu equation. Nonlinear characteristics of such equation have been investigated in many papers. Specifically, the method of multiple scale has been used to demonstrate the pitchfork bifurcation associated with stability change around the first unstable tongue and Lie transform has been used to demonstrate the subharmonic bifurcation for relatively small values of the undamped natural frequency. In these works, the resulting bifurcation diagram is represented in the parameter space of the undamped natural frequency where a constant value is allocated to the parametric frequency. Alternatively, this paper demonstrates how the Poincaré–Lindstedt method can be used to formulate pitchfork bifurcation around the first unstable tongue. Further, it is shown how higher order terms can be included in the perturbation analysis to formulate pitchfork bifurcation around the second tongue, and also subharmonic bifurcations for relatively high values of parametric frequency. This approach enables us to demonstrate the resulting global bifurcation diagram in the parameter space of parametric frequency, which is beneficial in the bifurcation analysis of systems with constant undamped natural frequency, when the frequency of the parametric force can vary. Finally, the analytical approximations are verified by employing the numerical integration along with Poincaré map and phase portraits.

Analysis of a mass-spring-relay system with periodic forcing

The dynamics of a hysteretic relay oscillator with harmonic forcing is investigated. Periodic excitation of the system results in periodic, quasi-periodic, chaotic and unbounded behavior. An explicit Poincaré map is constructed with an implicit constraint on the switching time. The stability of the fixed points of the Poincaré map corresponding to period-one solutions is investigated. By varying the forcing parameters, we observed a saddle-center and a pitchfork bifurcation of two centers and a saddle-type fixed point. The global dynamics of the system exhibits discontinuity induced bifurcations of the fixed points.

Double grazing bifurcations of the non-smooth railway wheelset systems

There are numerous non-smooth factors in railway vehicle systems, such as flange impact, dry friction, creep force, and so on. Such non-smooth factors heavily affect the dynamical behavior of the railway systems. In this paper, we investigate and mathematically analyze the double grazing bifurcations of the railway wheelset systems with flange contact. Two types of models of flange impact are considered, one is a rigid impact model and the other is a soft impact model. First, we derive Poincaré maps near the grazing trajectory by the Poincaré-section discontinuity mapping (PDM) approach for the two impact models. Then, we analyze and compare the near grazing dynamics of the two models. It is shown that in the rigid impact model the stable periodic motion of the railway wheelset system translates into a chaotic motion after the gazing bifurcation, while in the soft impact model a pitchfork bifurcation occurs and the system tends to the chaotic state through a period doubling bifurcation. Our results also extend the applicability of the PDM of one constraint surface to that of two constraint surfaces for autonomous systems.

Normal Form of Bifurcation for Caputo-Hadamard Fractional Differential System With a Parameter

This paper focuses on the normal form computation of the pitchfork bifurcation for the Caputo-Hadamard fractional differential system with a parameter. By using Taylor’s expansion and Implicit Function Theorem, we derive the normal form of the pitchfork bifurcation for the Caputo-Hadamard fractional differential system with a parameter.