Pitchfork Bifurcation Research Articles

The bifurcation analysis of an eco-toxicant model with anti-predator behavior

In this study, the mathematical model of four differential equations for organisms that describe the effect of anti-predation behavior, age stage and toxicity have been analyzed. Local bifurcation and Hopf bifurcation have been studied by changing a parameter of a model to study the dynamic behavior determined by bifurcation curves and the occurrence states of bifurcation saddle node, transcritical and pitch fork bifurcation. The potential equilibrium point at which Hopf bifurcation occurs has been determined and the results of the bifurcation behavior analysis have been fully presented using numerical simulation.

Hopf bifurcations in the full SKT model and where to find them

<p style='text-indent:20px;'>In this paper, we consider the Shigesada–Kawasaki–Teramoto (SKT) model, which presents cross-diffusion terms describing competition pressure effects. Even though the reaction part does not present the activator–inhibitor structure, cross-diffusion can destabilise the homogeneous equilibrium. However, in the full cross-diffusion system and weak competition regime, the cross-diffusion terms have an opposite effect and the bifurcation structure of the system modifies as the interspecific competition pressure increases. The major changes in the bifurcation structure, the type of pitchfork bifurcations on the homogeneous branch, as well as the presence of Hopf bifurcation points are here investigated. Through weakly nonlinear analysis, we can predict the type of pitchfork bifurcation. Increasing the additional cross-diffusion coefficients, the first two pitchfork bifurcation points from super-critical become sub-critical, leading to the appearance of a multi-stability region. The interspecific competition pressure also influences the possible appearance of stable time-period spatial patterns appearing through a Hopf bifurcation point.</p>

Bifurcations and Dynamical Behavior of 2D Coupled Chaotic Sine Maps

The main characteristics of a dynamical system are determined by the bifurcation theory. In particular, in this paper we examined the properties of the discrete dynamical system of a two coupled maps, i.e. the maps with an invariant unidimensional submanifold. The study of coupled chaotic systems shows rich and complex dynamic behaviors, particularly through structures of bifurcations or chaotic synchronization. A bifurcation is a qualitative change of the system behavior under the influence of control parameters. This change may correspond to the disppearance or appearance of new singularities or a change in the nature of singularities. We can define different kinds of bifurcations for fixed points and period two cycles as, saddle-node, period doubling, transcritical or pitchfork bifurcations. The study of the sequence of bifurcations permits to understand the mechanisms that lead to chaos. The phenomena of synchronization and antisynchronization in coupled chaotic systems is very important because its applications in several areas, such as secure communication or biology. In this paper, we study bifurcation properties of a two-dimensional coupled map T with three parameters. The first objective is to locate the bifurcation curves and their evolution in the parametric plane (<i>a,b</i>), when a third parameter <i>c</i> varies. The equations of some bifurcation curves are given analytically; cusp points and co-dimension two points on these bifurcation curves are determined. The second is related to the study of the chaotic synchronization and antisynchronization in the phase space (<i>x,y</i>).

The chaotic mechanisms in some jerk systems

&lt;abstract&gt;&lt;p&gt;In this work, a five-parameter jerk system with one hyperbolic sine nonlinearity is proposed, in which $ \varepsilon $ is a small parameter, and $ a $, $ b $, $ c $, $ d $ are some other parameters. For $ \varepsilon = 0 $, the system is $ Z_{2} $ symmetric. For $ \varepsilon \neq {0} $, the system loses the symmetry. For the symmetrical case, the pitchfork bifurcation and Hopf bifurcation of the origin are studied analytically by Sotomayor's theorem and Hassard's formulas, respectively. These bifurcations can be either supercritical or subcritical depending on the governing parameters. In comparison, it is much more restrictive for the origin of the Lorenz system: Only a supercritical pitchfork bifurcation is available. Thus, the symmetrical system can exhibit very rich local bifurcation structures. The continuation of local bifurcations leads to the main contribution of this work, i.e., the discovery of two basic mechanisms of chaotic motions for the jerk systems. For four typical cases, Cases A–D, by varying the parameter $ a $, the mechanisms are identified by means of bifurcation diagrams. Cases A and B are $ Z_{2} $ symmetric, while Cases C and D are asymmetric (caused by constant terms). The forward period-doubling routes to chaos are observed for Cases A and C; meanwhile, the backward period-doubling routes to chaos are observed for Cases B and D. The dynamical behaviors of these cases are studied via phase portraits, two-sided Poincaré sections and Lyapunov exponents. Using Power Simulation (PSIM), a circuit simulation model for a chaotic jerk system is created. The circuit simulations match the results of numerical simulations, which further validate the dynamical behavior of the jerk system.&lt;/p&gt;&lt;/abstract&gt;

Irregular spots on body surfaces of vertebrates induced by supercritical pitchfork bifurcations.

The classical Turing mechanism containing a long-range inhibition and a short-range self-enhancement provides a type of explanation for the formation of patterns on body surfaces of some vertebrates, e.g., zebras, giraffes, and cheetahs. For other type of patterns (irregular spots) on body surfaces of some vertebrates, e.g., loaches, finless eels, and dalmatian dogs, the classical Turing mechanism no longer applies. Here, we propose a mechanism, i.e., the supercritical pitchfork bifurcation, which may explain the formation of this type of irregular spots, and present a method to quantify the similarity of such patterns. We assume that, under certain conditions, the only stable state of "morphogen" loses its stability and transitions to two newly generated stable states with the influence of external noise, thus producing such ruleless piebald patterns in space. The difference between the competitiveness of these two states may affect the resulting pattern. Moreover, we propose a mathematical model based on this conjecture and obtain this type of irregular patterns by numerical simulation. Furthermore, we also study the influence of parameters in the model on pattern structures and obtain the corresponding pattern structures of some vertebrates in nature, which verifies our conjecture.

Nonlinear Dynamics and Bifurcation Analysis of a Hinged-Clamped Beam Under Parametric and Internal Resonances

The present study examines the stability and bifurcation of axially moving hinged-clamped beams subjected to parametric excitation originating due to speed alterations. Attention is paid to the principal parametric resonance amid 3:1 internal resonance in the subcritical speed regime. The direct method of multiple scales is applied to predict the nonlinear behavior of the beam modeled in the form of the nonlinear integro-partial differential equation. A continuation algorithm is used to solve numerical examples to illustrate the stability and bifurcations of periodic solutions for the given set of system parameters. The fixed point solution is different for different modes. Three kinds of equilibrium solution curves like trivial, two-mode, and single-mode solutions are obtained from numerical computations. The first two kinds are available for both first and second modes while the third one is available for the second mode only. The introduction of material damping converts the isolated two-mode solution into an isolated closed-loop form. The system displays saddle-node and Hopf bifurcation points in both modes while super and subcritical pitchfork bifurcation points in the second mode only. Decreasing viscous damping strengthens the effect of internal resonance. The numerically simulated results are unique, interesting, and are not available in the existing literature.

Modal Analysis for Localization Phenomena in Two-Pendulum Arrays Subjected to Vertical Excitation

When an array of two identical pendula connected by a linear, torsion spring is subjected to vertical excitation, the two pendula may oscillate with different amplitudes. This is referred to as a localization phenomenon. In this paper, modal analysis is performed to investigate the cause of the phenomenon. The equations of motion for the two pendula are derived in the physical coordinate. First, in the linearized equations of motion, the natural frequencies and the corresponding modal vectors are determined. Then, the equations of motion in the physical coordinates are transformed to those in the modal coordinates. Van der Pol’s method is employed to obtain the frequency response curves for the principal parametric resonance in the modal coordinates, and the response curves for the physical coordinates are determined using the coordinate transformation. These theoretical response curves are validated by being compared to the time histories and their FFT results. The numerical results show that as the excitation frequency decreases, a branch of the response curves for the first mode undergoes a pitchfork bifurcation, which induces the second mode owing to the nonlinear coupling with the first mode, and the two modes appear simultaneously at a specific excitation frequency range. This is the reason why localization phenomena are observed in the physical coordinates.

Bifurcation Points of Periodic Triangular Patterns Obtained in Reaction-Diffusion System with Anisotropic Diffusion

Turing demonstrated that spatially heterogeneous patterns can be self-organized, when the two substances interact locally and diffuse randomly. Turing systems have been applied not only to explain patterns observed within the biological and chemical fields, but also to develop image information processing tools. In a twin study, to evaluate the V-shaped bundle of the inner ear outer hair, we developed a method that utilizes a reaction-diffusion system with anisotropic diffusion that exhibited triangular patterns with the introduction of a certain anisotropy strength. In this study, we explored the parameter range over which these periodic triangular patterns were obtained. First, we defined an index for triangular clearness, TC. Triangular patterns can be obtained by introducing a large anisotropy δ, but the range of δ depends on the diffusion coefficient. We found an explanatory variable that can explain the change in TC based on a heuristic argument of the relative distance of the pitchfork bifurcation point between the maximum and minimum anisotropic diffusion function values. Clear periodic triangular patterns were obtained when the distance between the minimum anisotropic function value and pitchfork bifurcation point was over 2.5 times the distance to the anisotropic diffusion function maximum value. By changing the diffusion coefficients or the reaction terms, we further confirmed the accuracy of this condition using computer simulation. Its relevance to diffusion instability has also been discussed.

Stabilization and pattern formation in chemotaxis models with acceleration and logistic source.

We consider the following chemotaxis-growth system with an acceleration assumption, \begin{align*} \begin{cases} u_t= \Delta u -\nabla \cdot\left(u \bw \right)+\gamma\xkh{u-u^\alpha}, & x\in\Omega,\ t>0,\\ v_t=\Delta v- v+u, & x\in\Omega,\ t>0,\\ \bw_t= \Delta \bw -\bw +\chi\nabla v, & x\in\Omega,\ t>0, \end{cases} \end{align*} under the homogeneous Neumann boundary condition for $u,v$ and the homogeneous Dirichlet boundary condition for $\bw$ in a smooth bounded domain $\Omega\subset\R^{n}$ ($n\geq1$) with given parameters $\chi>0$, $\gamma\geq0$ and $\alpha>1$. It is proved that for reasonable initial data with either $n\leq3$, $\gamma\geq0$, $\alpha>1$ or $n\geq4,\ \gamma>0,\ \alpha>\frac12+\frac n4$, the system admits global bounded solutions, which significantly differs from the classical chemotaxis model that may have blow-up solutions in two and three dimensions. For given $\gamma$ and $\alpha$, the obtained global bounded solutions are shown to convergence exponentially to the spatially homogeneous steady state $(m,m,\mathbf 0$) in the large time limit for appropriately small $\chi$, where $m=\frac1{|\Omega|}\jfo u_0(x)$ if $\gamma=0$ and $m=1$ if $\gamma>0$. Outside the stable parameter regime, we conduct linear analysis to specify possible patterning regimes. In weakly nonlinear parameter regimes, with a standard perturbation expansion approach, we show that the above asymmetric model can generate pitchfork bifurcations which occur generically in symmetric systems. Moreover, our numerical simulations demonstrate that the model can generate rich aggregation patterns, including stationary, single merging aggregation, merging and emerging chaotic, and spatially inhomogeneous time-periodic. Some open questions for further research are discussed.

Effect of Static Bifurcation on Logical Stochastic Resonance in a Symmetric Bistable System

In previous research works, logical stochastic resonance (LSR) was reported to frequently occur in an asymmetric bistable system, where the bias parameter is the key factor to make the LSR appear. In this work, we investigate the effect of different anharmonic periodic signals on the pitchfork and saddle-node bifurcations in a symmetric bistable system. We focus on the relationship between the static bifurcation and LSR. We use both numerical and circuit simulations to analyze some interesting phenomena. Like the bias parameter, some anharmonic periodic signals also break the symmetry of the symmetric bistable system and lead to the saddle-node bifurcation. The anharmonic periodic signal with a constant term in its expanded Fourier series induces a reliable LSR in the symmetric bistable system. The key factor of LSR is the saddle-node bifurcation which implies the asymmetry of the system. Here, we replace the bias parameter by choosing an anharmonic periodic signal and make the LSR occur in a different way.

Transition to chaos in a reduced-order model of a shear layer

The present work studies the nonlinear dynamics of a shear layer, driven by a body force and confined between parallel walls, a simplified setting to study transitional and turbulent shear layers. It was introduced by Nogueira &amp; Cavalieri (J. Fluid Mech., vol. 907, 2021, A32), and is here studied using a reduced-order model based on a Galerkin projection of the Navier–Stokes system. By considering a confined shear layer with free-slip boundary conditions on the walls, periodic boundary conditions in streamwise and spanwise directions may be used, simplifying the system and enabling the use of methods of dynamical systems theory. A basis of eight modes is used in the Galerkin projection, representing the mean flow, Kelvin–Helmholtz vortices, rolls, streaks and oblique waves, structures observed in the cited work, and also present in shear layers and jets. A dynamical system is obtained, and its transition to chaos is studied. Increasing Reynolds number $Re$ leads to pitchfork and Hopf bifurcations, and the latter leads to a limit cycle with amplitude modulation of vortices, as in the direct numerical simulations by Nogueira &amp; Cavalieri. Further increase of $Re$ leads to the appearance of a chaotic saddle, followed by the emergence of quasi-periodic and chaotic attractors. The chaotic attractors suffer a merging crisis for higher $Re$ , leading to a chaotic dynamics with amplitude modulation and phase jumps of vortices. This is reminiscent of observations of coherent structures in turbulent jets, suggesting that the model represents a dynamics consistent with features of shear layers and jets.

Role of phase-dependent influence function in the Winfree model of coupled oscillators.

We consider a globally coupled Winfree model comprised of a phase-dependent influence function and sensitive function, and unravel the impact of offset and integer parameters, characterizing the shape of the influence function, on the phase diagram of the Winfree model. The decreasing value of the offset parameter decreases the degree of positive phase shift among the oscillators by promoting the negative phase shift, which indeed favors the onset of multistability among the synchronous oscillatory state and asynchronous stable steady states in a large region of the phase diagram. Further, large integer parameters lead to brief pulses of the influence function, which again enhances the effect of the offset parameter. There is an explosive transition to a synchronous oscillatory state from an asynchronous steady state via a Hopf bifurcation. Dynamical transitions and multistability emerge through saddle-node, pitchfork, and homoclinic bifurcations in the phase diagram. We deduce two ordinary differential equations corresponding to the two macroscopic variables from the population of globally coupled Winfree oscillators using the Ott-Antonsen ansatz. We also deduce various bifurcation curves analytically from the reduced low-dimensional macroscopic variables for the exactly solvable case. The analytical curves exactly match the simulation boundaries in the phase diagram.

Influence of Controller’s Parameters on Static Bifurcation of Magnetic-Liquid Double Suspension Bearing

Magnetic-Liquid Double Suspension Bearing (MLDSB) is composed of an electromagnetic supporting and a hydrostatic supporting system. Due to greater supporting capacity and static stiffness, it is appropriate for occasions of middle speed, overloading, and frequent starting. Because of the complicated structure of the supporting system, the probability and degree of static bifurcation of MLDSB can be increased by the coupling of hydrostatic force and electromagnetism force, and then the supporting capacity and operation stability are reduced. As the key part of MLDSB, the controller makes an important impact on its supporting capacity, operation stability, and reliability. Firstly, the mathematical model of MLDSB is established in the paper. Secondly, the static bifurcation point of MLDSB is determined, and the influence of parameters of the controller on singular point characteristics is analyzed. Finally, the influence of parameters of the controller on phase trajectories and basin of attraction is analyzed. The result showed that the pitchfork bifurcation will occur as proportional feedback coefficient Kp increases, and the static bifurcation point is Kp = −60.55. When Kp &lt; −60.55, the supporting system only has one stable node (0, 0). When Kp &gt; −60.55, the supporting system has one unstable saddle (0, 0) and two stable non-null focuses or nodes. The shape of the basin of attraction changed greatly as Kp increases from −60.55 to 30, while the outline of the basin of attraction is basically fixed as Kp increases from 30 to 80. Differential feedback coefficient Kd has no effect on the static bifurcation of MLDSB. The rotor phase trajectory obtained from theoretical simulation and experimental tests are basically consistent, and the error is due to the leakage and damping effect of the hydrostatic system within the allowable range of the engineering. The research in the paper can provide theoretical reference for static bifurcation analysis of MLDSB.

Two-color optically addressed spatial light modulator as a generic spatiotemporal system.

Nonlinear spatiotemporal systems are the basis for countless physical phenomena in such diverse fields as ecology, optics, electronics, and neuroscience. The canonical approach to unify models originating from different fields is the normal form description, which determines the generic dynamical aspects and different bifurcation scenarios. Realizing different types of dynamical systems via one experimental platform that enables continuous transition between normal forms through tuning accessible system parameters is, therefore, highly relevant. Here, we show that a transmissive, optically addressed spatial light modulator under coherent optical illumination and optical feedback coupling allows tuning between pitchfork, transcritical, and saddle-node bifurcations of steady states. We demonstrate this by analytically deriving the system's dynamical equations in correspondence to the normal forms of the associated bifurcations and confirm these results via extensive numerical simulations. Our model describes a nematic liquid crystal device using nano-dimensional dichalcogenide (a-As 2S 3) glassy thin films as photo sensors and alignment layers, and we use device parameters obtained from experimental characterization. Optical coupling, for example, using diffraction, holography, or integrated unitary maps allows implementing a variety of system topologies of technological relevance for neural networks and potentially Ising or XY-Hamiltonian models with ultralow energy consumption.

The focus case of a nonsmooth Rayleigh–Duffing oscillator

In this paper, we study the global dynamics of a nonsmooth Rayleigh–Duffing equation $$\ddot{x}+a{\dot{x}}+b{\dot{x}}|{\dot{x}}|+cx+{\mathrm {d}}x^3=0$$ for the case $$d>0$$ , i.e., the focus case. The global dynamics of this nonsmooth Rayleigh–Duffing oscillator for the case $$d<0$$ , i.e., the saddle case, has been studied in the companion volume (Wang and Chen in Int J Non-Linear Mech 129: 103657, 2021). The research for the focus case is more complex than the saddle case, such as the appearance of five limit cycles and the gluing bifurcation which means that two double limit cycle bifurcation curves and one homoclinic bifurcation curve are very adjacent. We present bifurcation diagram, including one pitchfork bifurcation curve, two Hopf bifurcation curves, two double limit cycle bifurcation curves and one homoclinic bifurcation curve. Finally, numerical phase portraits illustrate our theoretical results.

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.

Bifurcation tearing in a forced Duffing equation

The structure of solution branches of the equation ε 2 u x x = u 3 − λ u + δ g ( x ) , for x ∈ R is studied, for real constants, ε , λ and δ , where g ( x ) is periodic in x and u satisfies Neumann boundary conditions. For α : = u ( 0 ) bifurcation structure in the α – δ plane occurs that for small ε leads to families of singular asymptotic solutions with interior transition layers between outer solutions. In contrast, bifurcation in the α – λ plane produces families of oscillatory solutions that for small ε have large numbers, n , of oscillations between envelope outer solutions. The n label of the solution branch is defined by its local bifurcation at ε 2 n 2 = λ > 0 . Further the bifurcation structure on the n branch is classified by the integer n being either even or odd. When n is even the pitchfork bifurcation in the α – δ plane leads to connected branches of solutions in the α – λ – δ space. But when n is odd the presence of g ( x ) leads to cusp catastrophes (for small ε ) in the α – δ plane and these give rise to disconnected (or isolated) branches of solution in the α – λ plane that appear to be torn from the main solution branch. The integer n acts as a count of the number of oscillations on these solution branches. As ε 2 λ − 1 → 0 we see layer behaviour occurring about the top and bottom outer solutions and as ε 2 λ − 1 → ∞ we find convergence to the middle outer solution.

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.

Bifurcation study on a degenerate double van der Waals cirque potential energy surface using Lagrangian descriptors

In this paper, we explore the dynamics of a Hamiltonian system after a double van der Waals potential energy surface degenerates into a single well. The energy of the system is increased from the bottom of the potential well up to the dissociation energy, which occurs when the system becomes open. In particular, we study the bifurcations of the basic families of periodic orbits of this system as the energy increases using Lagrangian descriptors and Poincaré maps. We investigate the capability of Lagrangian descriptors to find periodic orbits of bifurcating families for the case of resonant, saddle–node and pitchfork bifurcations.

Ordered hexagonal patterns via notch–delta signaling

Many developmental processes in biology utilize notch–delta signaling to construct an ordered pattern of cellular differentiation. This signaling modality is based on nearest-neighbor contact, as opposed to the more familiar mechanism driven by the release of diffusible ligands. Here, exploiting this ‘juxtacrine’ property, we present an exact treatment of the pattern formation problem via a system of nine coupled ordinary differential equations. The possible patterns that are realized for realistic parameters can be analyzed by considering a co-dimension 2 pitchfork bifurcation of this system. This analysis explains the observed prevalence of hexagonal patterns with high delta at their center, as opposed to those with central high notch levels (referred to as anti-hexagons). We show that outside this range of parameters, in particular for low cis-coupling, a novel kind of pattern is produced, where high delta cells have high notch as well. It also suggests that the biological system is only weakly first order, so that an additional mechanism is required to generate the observed defect-free patterns. We construct a simple strategy for producing such defect-free patterns.