Pitchfork Bifurcation Research Articles

Bifurcation and Chaos Control of Mixed Rayleigh‐LiéNard Oscillator With Two Periodic Excitations and Mixed Delays

ABSTRACTThis paper investigates the bifurcation, chaos, and active control of a mixed Rayleigh‐Liénard oscillator with mixed time delays. First, the effects of system parameters on the supercritical pitchfork bifurcations are discussed in detail by applying the fast‐slow separation method. Second, it is rigorously proved by the Melnikov method that chaotic vibration exists when the parameters of the uncontrolled system are selected above the threshold of chaos occurrence. By fine‐tuning the system parameters, a criterion for designing the control parameters to make the Melnikov function non‐zero is derived. In addition, the routes to chaos in controlled system are explored by bifurcation diagrams, largest Lyapunov exponents, phase portraits, Poincaré maps, basins of attraction, frequency spectra, and displacement time series. The results indicate that by properly adjusting the displacement feedback coefficient and the amplitude of parameter excitation, the chaotic motion caused by increasing of the amplitude of external excitation and strength of distributed time delay can be effectively suppressed. This research result can provide theoretical support for exploring the potential chaotic motion of other types of oscillators.

Dynamic Modeling and Configuration Transformation of Origami with Soft Creases

Origami has attracted much attention from scientists and engineers in recent years. Classic origami only permits rotations around creases, while origami-like structures in nature can also undergo a certain degree of extension at creases. In this paper, an earwig-inspired origami with rotational symmetry is proposed and analyzed by introducing soft creases. Such a soft crease is equivalent to a combination of a rotational spring and an extensional spring. The motion of the earwig-inspired origami is described using spherical triangles. Based on Lagrange's Equation, the nonlinear dynamic equation of the system is formulated. It is then solved by using the fourth-order Runge-Kutta method. The results of theoretical calculations are consistent with those of simulations in ADAMS. With the established framework, the bifurcation behaviors of the equilibria of the proposed system, including supercritical pitchfork and saddle-node bifurcations, are investigated. Such origami can realize both mono-stable and bi-stable mechanisms, while the corresponding design parameters are demonstrated in the design map. In particular, the properties of the bi-stable origami can vary with different equilibria. The configuration transformation could be achieved using a continuous excitation. However, it is sometimes sensitive to the initial conditions. Inspired by the working mechanism of earwig wings, a simple control approach for origami configuration transformation is proposed. The key is to stop exciting the origami when its deformation crosses an energy barrier. This work lays a foundation for the design and study of novel multi-stable and morphing structures and provides an efficient approach for their configuration transformation.

Fast-slow dynamics with non-zero sliding characteristics related to pitchfork bifurcation delay in the parametrically excited Duffing system with frequency switching

Fast-slow dynamics with non-zero sliding characteristics related to pitchfork bifurcation delay in the parametrically excited Duffing system with frequency switching

HARMONIC RESONANCE OF SHORT-CRESTED GRAVITY WAVES ON DEEP WATER: ON THEIR PERSISTENCY

Abstract Three-dimensional short-crested water waves are known to host harmonic resonances (HRs). Their existence depends on their sporadicity versus their persistency. Previous studies, using a unique yet hybrid solution, suggested that HRs exhibit sporadic instability, with the domain of instability exhibiting a bubble-like structure which experiences a loss of stability followed by a re-stabilization. Through the calculation of their complete multiple solution structures and normal forms, we discuss the particular harmonic resonance (2,6). The (2,6) resonance was chosen, not only because it is of lower order, and thus more likely to be significant, but also because it is representative of a fully developed three-dimensional water wave field. Its appearance, growth rate and persistency are discussed. On our converged solutions, we show that, at an incidence angle for which HR (2,6) occurs, the associated superharmonic instability is no longer sporadic. It was also found that the multiple solution operates a subcritical pitchfork bifurcation, so regardless of the value of the control parameter, the wave steepness, a stable branch of the solution always exists. As a result, the analysis reveals two competing processes that either provoke and enhance HRs, or inhibit their appearance and development.

Geometrically exact post-buckling and post-flutter of standing cantilevered pipe conveying fluid

Although the nonlinear dynamics of hanging cantilevered pipes conveying fluid have been extensively scrutinized, there is limited research on the nonlinear behavior of standing ones. Hence, the objective of this study is to examine the geometrically exact nonlinear static and dynamic responses of cantilevered pipes conveying fluid in a standing position. The geometrically exact rotation-based model, combined with the shooting method and the Galerkin technique, is applied to assess the nonlinear static behavior of system and its stability characteristics. Moreover, to compute the nonlinear dynamics of system, the geometrically exact quaternion-based model, together with the Galerkin technique, is employed. It is revealed that the system may undergo buckling through either a supercritical or subcritical pitchfork bifurcation, depending on the gravity parameter, which may give rise to extremely large-amplitude responses. The system may also experience flutter instability due to a supercritical Hopf bifurcation, which brings about self-excited periodic oscillations. The generic behavior of system for a specific range of the gravity parameter is investigated across four distinct scenarios, which vary based on the gravity parameter and mass ratio. Notably, only one of these scenarios is analogous to the situation that comes to pass for the hanging case.

Global dynamics of a generalized arbitrary order Van der Pol–Duffing Oscillator

We study the global bifurcation diagram and corresponding global phase portraits in the Poincaré disc for a generalized van der Pol-Duffing oscillator, which has four nonlinear terms with arbitrary orders. This nonlinear oscillator possesses more diverse and complicated dynamical behaviours, including the heteroclinic bifurcation, generalized Hopf bifurcation and pitchfork bifurcation. Moreover, theoretical results are exhibited via numerical simulations.

Pattern dynamics of the nonreciprocal Swift-Hohenberg model.

We investigate the pattern dynamics of the one-dimensional nonreciprocal Swift-Hohenberg model. Characteristic spatiotemporal patterns such as disordered, aligned, swap, chiral-swap, and chiral phases emerge depending on the parameters. We classify the characteristic spatiotemporal patterns obtained in numerical simulation by focusing on the spatiotemporal Fourier spectrum of the order parameters. We derive a reduced dynamical system by using the spatial Fourier series expansion. We analyze the bifurcation structure around the fixed points corresponding to the aligned and chiral phases, and explain the transitions between them. The disordered phase is destabilized either to the aligned phase by the Turing bifurcation or to the chiral phase by the wave bifurcation, while the aligned phase and the chiral phase are connected by the pitchfork bifurcation.

Competing Bifurcations Determine Symmetry Breaking During Droplet Snaps on Smooth Patterned Surfaces.

The shape and stability of a droplet in contact with a solid surface is affected by the chemical composition and topography of the solid, and crucially, by the droplet's size. During a variation in size, most often observed during evaporation, droplets on smooth patterned surfaces can undergo sudden shape and position changes. Such changes, called snaps, are prompted by the surface pattern and arise from fold and pitchfork bifurcations which respectively cause symmetric and asymmetric motions. Yet, which type of snap is likely to be observed is an open fundamental question that has relevance in the rational design of surfaces for managing droplets. Here we show that the likelihood of observing symmetric or asymmetric snaps depends on the distance between fold and pitchfork bifurcation points and on how this distance varies for droplets that grow or shrink in size on surfaces patterned with a smooth topography. Our results can help develop strategies to control droplets by exploiting smooth surface patterns but also have broader relevance in situations where different types of bifurcations compete in determining the stability of a system, for instance in snap-through instabilities observed in elastic media.

Pattern formation on regular polygons and circles

We investigate the formation of Turing patterns on regular polygonal domains, as the number of edges grow, leading to the limiting case of the circle. Using linear and weakly nonlinear analysis, and evidence by simulations, we demonstrate how the domain shape can fundamentally change the expected bifurcation structure. Specifically, on the square domain we are able to derive pitchfork bifurcations for stripe and spot solutions, as well as show that both branches cannot bifurcate to produce stable patterns. This compares with the case of the equilateral triangle domain that causes the Turing bifurcation to be generically transcritical and, in some cases, none of the bifurcating branches are stable. Moreover, we find a monotonically increasing, but nonlinear relationship, between the minimal bifurcation area and the number of edges. Thus, patterns can occur on triangles with much smaller areas than circles. Overall, this work raises questions for researchers who are simulating applications on domains with simple shapes. Specifically, even small changes to domain geometry can have large impacts on the produced patterns; thus, domain perturbations should be considered in any sensitivity analyses.

A Dynamical Approach to the α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}–β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document} Displacive Transition of Quartz

The problem of displacive phase transitions (by which crystals pass on heating from a less symmetric to a more symmetric form) is investigated through numerical integration of the Newton equations of motion for a realistic model, in the paradigmatic case of quartz. Usually such transitions are discussed in terms of the positions of the atoms, while the role of normal modes is emphasized here. The key preliminary property established, in agreement with the indications given by Landau in his thermodynamic-like approach, is that four well definite modes are sufficient to describe the transition, the remaining modes just acting as a noise. The main result is then that such four modes constitute a closed Hamiltonian subsystem presenting an effective potential parametrically dependent on specific energy. The effective potential is actually computed, through (appropriately defined) time-averages of the accelerations of the relevant modes, and is found to describe, as energy is varied, a pitchfork bifurcation, once more confirming in dynamical terms the Landau result. The effective potential also allows one to advance a possible explanation of the “soft mode” phenomenon, namely the occuring, in the Raman spectrum, of a peak whose frequency depends on temperature and vanishes at the transition.

Transition to chaotic flow, bifurcation, and entropy generation analysis inside a stratified trapezoidal enclosure for varying aspect ratio

In this numerical study, we investigate the effect of aspect ratio (AR) on the natural convection (NC) flow and entropy generation (Sgen) in a stratified fluid confined inside a trapezoidal enclosure with thermally stratified side walls. The bottom of the enclosure is heated, and top of the enclosure is cooled. We use the Finite Volume Method (FVM) for simulating unsteady flows. We consider a Prandtl number (Pr = 0.71 for air) and explore AR of 0.2, 0.5, and 1.0, covering a wide range of Grashof numbers (Gr) from 10 to 108. The outcomes are presented for Grashof numbers and the effect of the AR on fluid flow, rates of heat transfer (HT), and Sgen inside the enclosure. Critical Grashof numbers are identified, marking the shift in flow behavior from being influenced by baroclinic to Rayleigh–Bénard instability, and from a steady to an unsteady state for various AR. In the context of this transition to chaotic flow, several supercritical bifurcations are observed, including a Pitchfork bifurcation from symmetric to asymmetric states and a Hopf bifurcation from a steady state to an unsteady state. Additionally, we analyze the discrepancies in average entropy generation (Savg) and average Bejan number (Beavg) across the entire enclosure, considering various AR and Gr values. It is observed that for enclosures with Gr ≥ 105, Savg increases as AR decreases, indicating an enhancement in HT rate with decreasing AR. Furthermore, a quantitative relationship between Savg, HT, AR, and Gr is presented. The study concludes that, as AR increases, the ecological coefficient of performance (ECOP) decreases, signifying a reduction in thermodynamics efficiency.

Spectral Signatures of Bifurcations

One way to characterize an orbit of a dynamical system is through its frequency content. Using a “spectral bifurcation diagram”, it has been shown earlier how the frequency content changes when a system undergoes a period-doubling cascade. In this paper, we extend the scope of this technique to obtain newer insights into various bifurcations like pitchfork bifurcation, border-collision bifurcation, period-adding cascade, and various torus bifurcations. We show that applying this method can enrich our understanding of bifurcations by providing vital information about generating or annihilating frequency components in a bifurcation.

Isochronous bifurcations in a two-parameter twist map.

Isochronous islands in phase space emerge in twist Hamiltonian systems as a response to multiple resonant perturbations. According to the Poincaré-Birkhoff theorem, the number of islands depends on the system characteristics and the perturbation. We analyze, for the two-parameter standard map, also called two-harmonic standard map, how the island chains are modified as the perturbation amplitude increases. We identified three routes for the transition from one chain, associated with one harmonic, to the chain associated with the other harmonic, based on a combination of pitchfork and saddle-node bifurcations. These routes can present intermediate island chains configurations. Otherwise, the destruction of the islands always occurs through the pitchfork bifurcation.

Mechanochemical patterning and wave propagation in multicellular tubes

Multicellular tubes are fundamental tissues for transporting and distributing liquids and gases in living organisms. Although the molecular, cellular and mechanical aspects in tube formation have been addressed experimentally, how these factors are coupled to control tube patterning and dynamics at the tissue level remains incompletely understood. Here, we propose a three-dimensional (3D) vertex model that incorporates a mechanochemical feedback loop correlating cell deformation and actomyosin signaling pathway to probe the morphodynamics of multicellular tubes. We show that diverse patterns, including ring, helix, double helix, and labyrinth, are generated in tubes through pitchfork bifurcation, where spatial fluctuations of both biochemical signaling and 3D cell deformation are remarkably involved. The mechanochemical feedback loop enables cell oscillations via Hopf bifurcation, which induces the mechanical and chemical patterns to propagate successively as either traveling or pulse waves while their spatial configurations are maintained, strikingly distinct from the classical Turing instability. Our simulations, together with stability analysis of a minimal model, uncover the essential role of mechanochemical principles in sculpting biological tubes.

The effect of a psychological scare on the dynamics of the tumor-immune interaction with optimal control strategy

Contracting cancer typically induces a state of terror among the individuals who are affected. Exploring how chemotherapy and anxiety work together to affect the speed at which cancer cells multiply and the immune system’s response model is necessary to come up with ways to stop the spread of cancer. This paper proposes a mathematical model to investigate the impact of psychological scare and chemotherapy on the interaction of cancer and immunity. The proposed model is accurately described. The focus of the model’s dynamic analysis is to identify the potential equilibrium locations. According to the analysis, it is possible to establish three equilibrium positions. The stability analysis reveals that all equilibrium points consistently exhibit stability under the defined conditions. The bifurcations occurring at the equilibrium sites are derived. Specifically, we obtained transcritical, pitchfork, and saddle-node bifurcation. Numerical simulations are employed to validate the theoretical study and ascertain the minimum therapy dosage necessary for eradicating cancer in the presence of psychological distress, thereby mitigating harm to patients. Fear could be a significant contributor to the spread of tumors and weakness of immune functionality.

Bifurcation analysis with chaotic attractor for a special case of jerk system

This article focuses on investigating local bifurcations in a special type of chaotic jerk system. It examines the occurrence and non-occurrence of saddle-node, transcritical, zero-Hopf, Hopf, and pitchfork bifurcations at the origin. The parameters that result in a zero-Hopf equilibrium point at the origin are characterized for the proposed system. Additionally, a demonstration is provided to show that the utilization of the first-order averaging theory leads to the emergence of a single periodic solution branching out from the zero-Hopf equilibrium located at the origin. Furthermore, the focus quantities method is applied to explore the periodicity of the cubic part of the system. This method helps determine the number of periodic solutions that can emerge from the Hopf point. Due to the computational load for computing singular quantities, only three singular quantities are found. Under specific conditions, it is shown that three periodic solutions can bifurcate from the origin of the system. Finally, the study also examines the chaotic attractors of the system.

Effects of nonlinearities and geometric imperfections on multistability and deformation localization in wrinkling films on planar substrates

Compressed elastic films on soft substrates release part of their strain energy by wrinkling, which represents a loss of symmetry, characterized by a pitchfork bifurcation. Its development is well understood at the onset of supercritical bifurcation, but not beyond, or in the case of subcritical bifurcation. This is mainly due to nonlinearities and the extreme imperfection sensitivity. In both types of bifurcations, the energy–displacement diagrams that can characterize an energy landscape are non-convex, which is notoriously difficult to determine numerically or experimentally, let alone analytically. To gain an elementary understanding of such potential energy landscapes, we take a thin beam theory suitable for analyzing large displacements under small strains and significantly reduce its complexity by reformulating it in terms of the tangent rotation angle. This enables a comprehensive analytical and numerical analysis of wrinkling elastic films on planar substrates, which are effective stiffening and/or softening due to either geometric or material nonlinearities. We also validate our findings experimentally. We explicitly show how effective stiffening nonlinear behavior (e.g., due to substrate or membrane deformations) leads to a supercritical post-bifurcation response, makes the energy landscape non-convex through energy barriers causing multistability, which is extremely problematic for numerical computation. Moreover, this type of nonlinearity promotes uni-modal, uniformly distributed, periodic deformation patterns. In contrast, nonlinear effective softening behavior leads to subcritical post-bifurcation behavior, similarly divides the energy landscape by energy barriers and conversely promotes localization of deformations. With our theoretical model we can thus explain an experimentally observed phenomenon that in structures with effective softening followed by an effective stiffening behavior, the symmetry is initially broken by localizing the deformation and later restored by forming periodic, distributed deformation patterns as the load is increased. Finally, we show that initial imperfections can significantly alter the local or global energy-minimizing deformation pattern and completely remove some energy barriers. We envision that this knowledge can be extrapolated and exploited to convexify extremely divergent energy landscapes of more sophisticated systems, such as wrinkling compressed films on curved substrates (e.g., on cylinders and spheres) and that it will enable elementary analysis and the development of specialized numerical tools.

Dynamics of a predator–prey system with foraging facilitation and group defense

Foraging facilitation and group defense are widespread phenomena in population ecosystem, which promote and prevent the predation in opposite processes respectively. In this work, a predator–prey system with foraging facilitation among predators and group defense in prey is proposed and investigated. To demonstrate the impact of both the foraging facilitation and the group defense on the system dynamics, the analysis primarily focuses on the qualitative properties of equilibria and possible bifurcations that the system may exhibit. It is shown that there are various non-hyperbolic equilibria for various parameter values such as the saddle–node, the weak focus of multiplicity at most three and the nilpotent cusp of codimension at most two as well as the system exhibits numerous kinds of bifurcation phenomena as the parameter values vary including the saddle–node bifurcation, the transcritical bifurcation, the pitchfork bifurcation, the Hopf bifurcation, the homoclinic bifurcation and the Bogdanov–Takens bifurcation. The results reveal that both foraging facilitation and group defense mechanisms play an important role in enriching the dynamics of predator–prey system and promoting the balance and diversity of ecosystem.

Nonlinear Static and Dynamic Responses of a Floating Rod Pendulum

Abstract A novel nonlinear dynamics model is developed in this paper to describe the static and dynamic nonlinear behaviors of a rod pendulum partially immersed in still water. The pendulum is hinged above the water level and subject to nonlinear gravity, hydrostatic, and hydrodynamic loads, all of which are incorporated into the system dynamics. The nonlinear static behavior and stability of the pendulum have been characterized by analyzing the fixed points. It is found that Pitchfork bifurcation governs the relationship between the rod density (the control parameter) and the static equilibrium angle. The pendulum's nonlinear response to external harmonic torque is obtained using Harmonic Balance Method (HBM). The influence of system parameters, including hinge height, rod diameter, and rod density, on the nonlinear frequency response is examined. Upon altering the system parameters, particularly the rod density, it is found that the system exhibits either a softening or a hardening effect.

Asymptotic stability and bifurcations of a perturbed McMillan map

This paper presents various bifurcations of the McMillan map under perturbations of its coefficients, such as period-doubling, pitchfork, and hysteresis bifurcation. The associated existence regions are located. Using the quasi-Lyapunov function method, the existence of asymptotically stable fixed point is also demonstrated.