Distinct Bifurcations Research Articles

Limit cycle bifurcation from a zero-Hopf equilibrium for a class of 3-dimensional Kolmogorov systems

A zero-Hopf equilibrium point p of a 3-dimensional autonomous differential system in R3 is an equilibrium point such that the eigenvalues of the linear part of the system at p are 0 and ±ωi with ω≠0. A zero-Hopf bifurcation takes place when from a zero-Hopf equilibrium bifurcate some small-amplitude limit cycles moving the parameters of the system.Polynomial Kolmogorov systems are the natural extension of the polynomial Lotka–Volterra systems of degree 2 to higher degree. The Kolmogorov systems have been studied intensively due to their applications for modeling many natural phenomena.In this paper we characterize, up to first order in the averaging theory, the eight distinct zero-Hopf bifurcations which can exhibit the class of 3-dimensional Kolmogorov systems of degree 3, providing an explicit approximation of the bifurcated small-amplitude limit cycles, together with information about their kind of stability.From each one of this eight different zero-Hopf bifurcations emerges one or two limit cycle using the averaging theory of first order. Moreover we provide an explicit example of each one of these eight zero-Hopf bifurcations.

The propulsion direction of nanoparticles trapped in an acoustic field

Solid particles trapped in an acoustic standing wave have been observed to undergo propulsion. This phenomenon has been attributed to the generation of a steady streaming flow, with a reversal in the propulsion direction at a distinct frequency. We explain the mechanism underlying this reversal by considering the canonical problem of a sphere executing oscillatory rotation in an unbounded fluid that undergoes rectilinear oscillation; these two oscillations occur at identical frequency but with an arbitrary phase difference. Two distinct bifurcations in the flow field occur: (1) a stagnation point first forms with increasing frequency, which (2) splits into a saddle node and a vortex centre. Reversal in the propulsion direction is driven by reversal in the flow far from the sphere, which coincides with the second bifurcation. This flow is identified with that of a Stokeslet whose strength is the net force exerted on the particle, which has implications for studying the flow field around particles of non-spherical geometries and for modelling suspensions of particles in acoustic fields.

How neuronal morphology impacts the synchronisation state of neuronal networks.

The biophysical properties of neurons not only affect how information is processed within cells, they can also impact the dynamical states of the network. Specifically, the cellular dynamics of action-potential generation have shown relevance for setting the (de)synchronisation state of the network. The dynamics of tonically spiking neurons typically fall into one of three qualitatively distinct types that arise from distinct mathematical bifurcations of voltage dynamics at the onset of spiking. Accordingly, changes in ion channel composition or even external factors, like temperature, have been demonstrated to switch network behaviour via changes in the spike onset bifurcation and hence its associated dynamical type. A thus far less addressed modulator of neuronal dynamics is cellular morphology. Based on simplified and anatomically realistic mathematical neuron models, we show here that the extent of dendritic arborisation has an influence on the neuronal dynamical spiking type and therefore on the (de)synchronisation state of the network. Specifically, larger dendritic trees prime neuronal dynamics for in-phase-synchronised or splayed-out activity in weakly coupled networks, in contrast to cells with otherwise identical properties yet smaller dendrites. Our biophysical insights hold for generic multicompartmental classes of spiking neuron models (from ball-and-stick-type to anatomically reconstructed models) and establish a connection between neuronal morphology and the susceptibility of neural tissue to synchronisation in health and disease.

Combined influences of environmental enrichment and harvesting mediate rich dynamics in an intraguild predation fishery system

Natural resources and its enrichment activities are often being associated with the paradoxical effects in ecology such as in predator–prey and fishery models. Several studies have shown that an increase in resources can cause the “Paradox of Enrichment” i.e., the destabilization of certain species population as the system is enriched. This situation is rather contradictory to our logical intuition. Parallel with this view, harvesting effort and the intensity of enrichment activities are among crucial factors in shaping the sustainability of fishery activities and natural resources. This is because an extreme harvesting effort could lead to resource depletion and threaten vulnerable marine ecosystems. In this work, we investigate the joint effects of enrichment and harvesting activities in an intraguild predator–prey fishery model that incorporates the Michaelis–Menten type of harvesting of predatory fishes. We study the existence and stability of the steady states in this system analytically and conduct numerical simulations, together with bifurcation analysis to verify our analytical findings. Our co-dimension one bifurcation analysis shows that the system can exhibit local bifurcations such as transcritical, saddle–node, subcritical, and supercritical Hopf bifurcations. Co-dimension two bifurcation analysis further demonstrates the existence of global dynamics through the appearance of Generalized Hopf (GH) and Zero-Hopf (ZH) bifurcations, which occur due to the interactions of distinct local bifurcations in this ecological system. The limit point bifurcation of cycles that emanates from the GH point has a bounded parameter space region in which tristability phenomenon (between stable steady states and stable limit cycles) occurs. In a scenario where the resources are enriched in the presence of harvesting activity, bistability emerges with the long-term outcomes lead to the exclusion of either prey or predator population, depending on the initial conditions of species. Based on these findings, we propose another interpretation to the “Paradox of Enrichment” as postulated in ecological theories: this predator–prey fishery system can be destabilized through an alternative stable states phenomenon, which results in the extinction of certain species as the ecosystem is further enriched.

Dynamics and bifurcations of laminar annular swirling and non-swirling jets

This paper presents bifurcation analyses characterising the nonlinear dynamics of fully developed laminar annular jets with respect to the centrebody diameter${ {d}}$, Reynolds number${ {Re}}$, and swirl ratio${ {S}}$. Similar flows appear in numerous applications and feature a vibrant range of topological and dynamical characteristics associated with phenomena including shear layer separation and vortex breakdown. Our results begin by describing the non-monotonic evolution of the axisymmetric jet's steady topology under varying${ {S}}$. In accord with earlier reports, the jet progresses through a sequence of wake, breakdown and wall jet regimes in a qualitatively similar manner across a wide span of${ {d}}$and${ {Re}}$values. In the wake regime, the non-swirling jet bifurcates to a plane-symmetric, but not axisymmetric, steady flow pattern beyond a${ {d}}$-dependent critical${ {Re}}$value. With further increase in${ {Re}}$, the steady non-swirling jet destabilises subsequently via multiple distinct Hopf bifurcations. Introducing${ {S}}>0$to the jet also induces unsteadiness by twisting the singly azimuthally periodic ($|m|=1$) asymmetric wake structure and causing it to precess periodically in time about the central axis. Intermediate swirl stabilises this unsteady dynamics and restores the jet's axisymmetry. This stabilising effect is then reversed in the breakdown regime at higher${ {S}}$, where a variety of different$|m|=1$and$|m|=2$instabilities bifurcate from the steady flow as${ {S}}$is increased. Several instances of hysteresis and subcritical behaviour are reported and discussed, including one that manifests precessing vortex core oscillations.

Three Crossing Limit Cycles in a 3D-Filippov System Having a T-Singularity

Our starting point is a discontinuous piecewise linear dynamical system in [Formula: see text] with two zones that present a single invisible two-fold straight line and invariant cylinders. We broke the dynamics of this vector field to obtain a new four-parameter piecewise linear vector field with a [Formula: see text]-singularity. For this vector field, we prove that the upper bound of simple crossing limit cycles (simple CLCs) is three, and provide conditions for the existence of 1, 2, or 3 simple CLCs. We also show the occurrence of three distinct bifurcations involving such simple CLCs that are derived from a bifurcation set with two parameters: (i) a Teixeira Singularity bifurcation (TS-bifurcation); (ii) a Fold bifurcation; and (iii) a Cusp bifurcation.

Unified framework for localized patterns in reaction–diffusion systems; the Gray–Scott and Gierer–Meinhardt cases

A recent study of canonical activator-inhibitor Schnakenberg-like models posed on an infinite line is extended to include models, such as Gray-Scott, with bistability of homogeneous equilibria. A homotopy is studied that takes a Schnakenberg-like glycolysis model to the Gray-Scott model. Numerical continuation is used to understand the complete sequence of transitions to two-parameter bifurcation diagrams within the localized pattern parameter regime as the homotopy parameter varies. Several distinct codimension-two bifurcations are discovered including cusp and quadruple zero points for homogeneous steady states, a degenerate heteroclinic connection and a change in connectedness of the homoclinic snaking structure. The analysis is repeated for the Gierer-Meinhardt system, which lies outside the canonical framework. Similar transitions are found under homotopy between bifurcation diagrams for the case where there is a constant feed in the active field, to it being in the inactive field. Wider implications of the results are discussed for other pattern-formation systems arising as models of natural phenomena. This article is part of the theme issue 'Recent progress and open frontiers in Turing's theory of morphogenesis'.

A mathematical model for the dynamics of happiness.

Positive psychology recognizes happiness as a construct comprising hedonic and eudaimonic well-being dimensions. Integrating these components and a set of theory-led assumptions, we propose a mathematical model, given by a system of nonlinear ordinary differential equations, to describe the dynamics of a person's happiness over time. The mathematical model offers insights into the role of emotions for happiness and why we struggle to attain sustainable happiness and tread the hedonic treadmill oscillating around a relative stable level of well-being. The model also indicates that lasting happiness may be achievable by developing constant eudaimonic emotions or human altruistic qualities that overcome the limits of the homeostatic hedonic system; in mathematical terms, this process is expressed as distinct dynamical bifurcations. This mathematical description is consistent with the idea that eudaimonic well-being is beyond the boundaries of hedonic homeostasis.

Local dispersal, trophic interactions and handling times mediate contrasting effects in prey-predator dynamics

Trophic interactions between prey and predator species are an ubiquitous force in nature and it is reckoned to be one of the most important factors that can shape species geographical range limits. To better understand how local dispersal and predator handling times affect community dynamics, we perform a bifurcation analysis of a partial differential equations model consisting of environmental gradients, trophic (generalist, specialist predator and prey species) interactions and spatial diffusion components. Our co-dimension one bifurcation results depict the existence of numerous threshold values corresponding to supercritical and subcritical Hopf bifurcations, saddle-node bifurcation of cycles and transcritical bifurcation, as the magnitude of handling times and the generalist predation-induced mortality force change in this ecological system with local dispersal. It is discovered that the interaction between these distinct local bifurcations leads to higher codimension (two) bifurcation such as a generalised Hopf bifurcation, which acts as an organising centre of the dynamics. Ecologically, we also observe that: (i) without dispersal, higher levels of generalist predation (which acts as a mortality agent) require shorter specialist predator handling times to promote the maintenance of biodiversity through coexistence steady state of prey-predator species and oscillatory dynamics; however, lower forces of generalist predation result in longer specialist predator handling times and this situation induces destabilisation of species biodiversity; (ii) local dispersal stabilises the trophic dynamics and mediates more species diversity outcomes; (iii) dispersal interacts with lower magnitudes of generalist predator-induced mortality to generate bistability phenomenon between stable limit cycles and coexistence steady state; (iv) interestingly, higher intensities of mortality factor combined with dispersal may destroy some interior attractors, and exclude bistability-oscillatory dynamics in this trophic system; consequently, this situation leads to the extinction of specialist predator species and jeopardises species biodiversity. Taken together, our findings illustrate the influential roles of local dispersal, trophic interactions and handling times in determining community dynamics and these insights may have important consequences in conservation management and biological control strategies.

Patterns, localized structures and fronts in a reduced model of clonal plant growth

A simplified model of clonal plant growth is formulated, motivated by observations of spatial structures in Posidonia oceanica meadows in the Mediterranean Sea. Two levels of approximation are considered for the scale-dependent feedback terms. Both take into account mortality and clonal, or vegetative, growth as well as competition and facilitation, but the first version is nonlocal in space while the second is local. Study of the two versions of the model in the one-dimensional case reveals that both cases exhibit qualitatively similar behavior (but quantitative differences) and describe the competition between three spatially extended states, the bare soil state, the populated state, and a pattern state, and the associated spatially localized structures. The latter are of two types, holes in the populated state and vegetation patches on bare ground, and are organized within distinct snaking bifurcations diagrams. Fronts between the three extended states are studied and a transition between pushed and pulled fronts identified. Numerical simulations in one spatial dimension are used to determine front speeds and confront the predictions from the marginal stability condition for pulled fronts.

Kozai-Lidov mechanism inside retrograde mean motion resonances

As the discoveries of more minor bodies in retrograde resonances with giant planets, such as 2015 BZ509 and 2006 RJ2, our curiosity about the Kozai-Lidov dynamics inside the retrograde resonance has been sparked. In this study, we focus on the 3D retrograde resonance problem and investigate how the resonant dynamics of a minor body impacts on its own Kozai-Lidov cycle. Firstly we deduce the action-angle variables and canonical transformations that deal with the retrograde orbit specifically. After obtaining the dominant Hamiltonian of this problem, we then carry out the numerical averaging process in closed form to generate phase-space portraits on a $e-\omega$ space. The retrograde 1:1 resonance is particularly scrutinized in detail, and numerical results from a CRTBP model shows a great agreement with the our semi-analytical portraits. On this basis, we inspect two real minor bodies currently trapped in retrograde 1:1 mean motion resonance. It is shown that they have different Kozai-Lidov states, which can be used to analyze the stability of their unique resonances. In the end, we further inspect the Kozai-Lidov dynamics inside the 2:1 and 2:5 retrograde resonance, and find distinct dynamical bifurcations of equilibrium points on phase-space portraits.

Identification of Bifurcations from Observations of Noisy Biological Oscillators

Hair bundles are biological oscillators that actively transduce mechanical stimuli into electrical signals in the auditory, vestibular, and lateral-line systems of vertebrates. A bundle’s function can be explained in part by its operation near a particular type of bifurcation, a qualitative change in behavior. By operating near different varieties of bifurcation, the bundle responds best to disparate classes of stimuli. We show how to determine the identity of and proximity to distinct bifurcations despite the presence of substantial environmental noise. Using an improved mechanical-load clamp to coerce a hair bundle to traverse different bifurcations, we find that a bundle operates within at least two functional regimes. When coupled to a high-stiffness load, a bundle functions near a supercritical Hopf bifurcation, in which case it responds best to sinusoidal stimuli such as those detected by an auditory organ. When the load stiffness is low, a bundle instead resides close to a subcritical Hopf bifurcation and achieves a graded frequency response—a continuous change in the rate, but not the amplitude, of spiking in response to changes in the offset force—a behavior that is useful in a vestibular organ. The mechanical load in vivo might therefore control a hair bundle’s responsiveness for effective operation in a particular receptor organ. Our results provide direct experimental evidence for the existence of distinct bifurcations associated with a noisy biological oscillator, and demonstrate a general strategy for bifurcation analysis based on observations of any noisy system.

Grazing-sliding bifurcations, border collision maps and the curse of dimensionality for piecewise smooth bifurcation theory

We show that maps describing border collision bifurcations (continuous but non-differentiable discrete time maps) are subject to a curse of dimensionality: it is impossible to reduce the study of the general case to low dimensions, since in every dimension the bifurcation can produce fundamentally different attractors (contrary to the case of local bifurcations in smooth systems). In particular we show that n-dimensional border collision bifurcations can have invariant sets of dimension k for integer k from 0 to n. We also show that the border collision normal form is related to grazing-sliding bifurcations of switching dynamical systems. This implies that the dynamics of these two apparently distinct bifurcations (one for discrete time dynamics, the other for continuous time dynamics) are closely related and hence that a similar curse of dimensionality holds in grazing-sliding bifurcations.

Hopf Bifurcations and Limit Cycles in Evolutionary Network Dynamics

The replicator-mutator equations from evolutionary dynamics serve as a model for the evolution of language, behavioral dynamics in social networks, and decision-making dynamics in networked multiagent systems. Analysis of the stable equilibria of these dynamics has been a focus in the literature, where symmetry in fitness functions is typically assumed. We explore asymmetry in fitness and show that the replicator-mutator equations exhibit Hopf bifurcations and limit cycles. We prove conditions for the existence of stable limit cycles arising from multiple distinct Hopf bifurcations of the dynamics in the case of circulant fitness matrices. In the noncirculant case we illustrate how stable limit cycles of the dynamics are coupled to embedded directed cycles in the payoff graph. The limit cycles correspond to oscillations of grammar dominance in language evolution and to oscillations in behavioral preferences in social networks; for decision-making systems, the limit cycles correspond to sustained oscillati...

Competition between transport phenomena in a reaction–diffusion–convection system

A reaction–diffusion–convection (RDC) model is introduced as a convenient framework for studying instability scenarios by which chemical oscillators are driven to chaos. Distinct bifurcations in the oscillating patterns are found as diffusion coefficients or Grashof numbers are varied. Singularly there emerge peculiar bifurcation paths in which quasi-periodicity transmutes into a period-doubling sequence to chemical chaos. The opposite influence exhibited by the two parameters in these transitions clearly indicate that diffusion and natural convection are in ‘competition’ for the stability of ordered dynamics: the former has a stabilizing effect while the latter exerts a destabilizing role.

Interactive buckling in sandwich beam-columns

Recent work on localized buckling with different core bending theories in sandwich struts is adapted for a sandwich beam-column that combines the effects of bending and compression. The two distinct bifurcations present in the pure compression case are replaced by a single pitchfork bifurcation point that combines local and global buckling behaviour. Several models with a various number of degrees of freedom (DOFs) are developed to account for the explicit effect of end moments. All models are formulated using total potential energy and variational principles, which are used to derive the governing equations that are solved using the numerical continuation package Auto97. Results are validated with a finite element model formulated in the commercial package Abaqus. Comparisons with the interactive buckling profiles and the mechanical response in the nonlinear range are very favourable and the models with more DOFs are determined to be superior.

Ruelle–Takens–Newhouse scenario in reaction-diffusion-convection system

Direct numerical simulations of the transition process from periodic to chaotic dynamics are presented for two variable Oregonator-diffusion model coupled with convection. Numerical solutions to the corresponding reaction-diffusion-convection system of equations show that natural convection can change in a qualitative way, the evolution of concentration distribution, as compared with convectionless conditions. The numerical experiments reveal distinct bifurcations as the Grashof number is increased. A transition to chaos similar to Ruelle-Takens-Newhouse scenario is observed. Numerical results are in agreement with the experiments.

Global optimal control and system-dependent solutions in the hardening Helmholtz–Duffing oscillator

A method for controlling nonlinear dynamics and chaos based on avoiding homo/heteroclinic bifurcations is applied to the hardening Helmholtz–Duffing oscillator, which is an archetype of a class of asymmetric oscillators having distinct homoclinic bifurcations occurring for different values of the parameters. The Melnikov's method is applied to analytically detect these bifurcations, and, in the spirit of the control method developed by the authors, the results are used to select the optimal shape of the excitation permitting the maximum shift of the undesired bifurcations in parameter space. The two main novelties with respect to previous authors' works consist in the occurrence (i) of a more involved control scenario, and (ii) of system-dependent optimal solutions. In particular, the possibility of having global control “with” or “without symmetrization”, related to the optimization of the critical amplitudes versus the critical gains, is illustrated. In the former case, two sub-cases are found, namely, “pursued” and “achieved” symmetrization, which take care of the possibly simultaneous occurrence of right and left homoclinic bifurcations under optimal excitation.

ON THE BEHAVIOR OF A SELF-EXCITING FARADAY DISK HOMOPOLAR DYNAMO WITH A VARIABLE NONLINEAR SERIES MOTOR

In this paper we seek to bridge the gap between the study of a self-exciting Faraday disk homopolar dynamo with a linear series motor [Hide et al., 1996] and the case when the torque acting on the armature of the motor is proportional to the square of the current flowing through the dynamo [Hide, 1998]. We also focus on the issue of when the nonlinear quenching of oscillatory solutions can occur. The present study is a special case of the more general problem when azimuthal eddy currents are permitted to flow [Moroz & Hide, 2000] and shares with that problem the existence of multiple steady states and Hopf bifurcations. This results in distinct double-zero bifurcations for the trivial and the nontrivial equilibrium states as well as other codimension-two bifurcations, which leads to the suppression of oscillatory solutions.

Catastrophic bifurcation from riddled to fractal basins.

Most existing works on riddling assume that the underlying dynamical system possesses an invariant subspace that usually results from a symmetry. In realistic applications of chaotic systems, however, there exists no perfect symmetry. The aim of this paper is to examine the consequences of symmetry-breaking on riddling. In particular, we consider smooth deterministic perturbations that destroy the existence of invariant subspace, and identify, as a symmetry-breaking parameter is increased from zero, two distinct bifurcations. In the first case, the chaotic attractor in the invariant subspace is transversely stable so that the basin is riddled. We find that a bifurcation from riddled to fractal basins can occur in the sense that an arbitrarily small amount of symmetry breaking can replace the riddled basin by fractal basins. We call this a catastrophe of riddling. In the second case, where the chaotic attractor in the invariant subspace is transversely unstable so that there is no riddling in the unperturbed system, the presence of a symmetry breaking, no matter how small, can immediately create fractal basins in the vicinity of the original invariant subspace. This is a smooth-fractal basin boundary metamorphosis. We analyze the dynamical mechanisms for both catastrophes of riddling and basin boundary metamorphoses, derive scaling laws to characterize the fractal basins induced by symmetry breaking, and provide numerical confirmations. The main implication of our results is that while riddling is robust against perturbations that preserve the system symmetry, riddled basins of chaotic attractors in the invariant subspace, on which most existing works are focused, are structurally unstable against symmetry-breaking perturbations.