Doubling Bifurcation Research Articles

Bifurcation Theory of a Racetrack Economy in a Spatial Economy Model

Racetrack economy is a conventional spatial platform for economic agglomeration in spatial economy models. Studies of this economy up to now have been conducted mostly on $2^k$ cities, for which agglomerations proceed via so-called spatial period doubling bifurcation cascade. This paper aims at the elucidation of agglomeration mechanisms of the racetrack economy in a general setting of an arbitrary number of cities. First, an attention was paid to the existence of invariant solutions that retain their spatial distributions when the transport cost parameter is changed. A complete list of possible invariant solutions, which are inherent for replicator dynamics and are dependent on the number of cities, is presented. Next, group-theoretic bifurcation theory is used to describe bifurcation from the uniform state, thereby presenting an insightful information on spatial agglomerations. Among a plethora of theoretically possible invariant solutions, those which actually become stable for spatial economy models are obtained numerically. Asymptotic agglomeration behavior when the number of cities become very large is studied.

Polynomial maps with hidden complex dynamics

The dynamics of a class of one-dimensional polynomial maps are studied, and interesting dynamics are observed under certain conditions: the existence of periodic points with even periods except for one fixed point; the coexistence of two attractors, an attracting fixed point and a hidden attractor; the existence of a double period-doubling bifurcation, which is different from the classical period-doubling bifurcation of the Logistic map; the existence of Li-Yorke chaos. Furthermore, based on this one-dimensional map, the corresponding generalized Henon map is investigated, and some interesting dynamics are found for certain parameter values: the coexistence of an attracting fixed point and a hidden attractor; the existence of Smale horseshoe for a subshift of finite type and also Li-Yorke chaos.

The dynamics of ensemble of neuron-like elements with excitatory couplings

We investigated the phenomenological model of ensemble of two FitzHugh–Nagumo neuron-like elements with symmetric excitatory couplings. The main advantage of proposed model is the new approach to model the coupling which is implemented by smooth function that approximates rectangular function and reflects main important properties of biological synaptic coupling. The proposed coupling depends on three parameters that define: a) the beginning of activation of an element α, b) the duration of the activation δ and c) the strength of the coupling g. We observed a rich diversity of different types of neuron-like activity, including regular in-phase, anti-phase and sequential spiking. In the phase space of the system, these regular regimes correspond to specific asymptotically stable periodic motions (limit cycles). We also observed the canard in-phase solutions and the chaotic anti-phase activity, which corresponds to a strange attractor that appears via the cascade of period doubling bifurcations of limit cycles.In addition, we investigated an interesting phenomenon when two different chaotic attractive regimes corresponding for two different types of chaotic anti-phase activity merge in a single strange attractor. As a result, a new type of chaotic anti-phase regime appears by explosion from the collision of these two strange attractors.We also provided the detailed study of bifurcations which lead to the transitions between all these regimes. We detected on the (α, δ) parameter plane regions that correspond to the above-mentioned regimes. We also showed numerically the existence of bistability regions where various non-trivial regimes coexist. For example, in some regions, one can observe either anti-phase or in-phase oscillations depending on initial conditions. We also specified regions corresponding to coexisting various types of sequential activity.

Three-dimensional steady and oscillatory flow in a double bifurcation airway model

Using Magnetic Resonance Velocimetry, we investigate the steady expiratory and the oscillatory flow in a planar double bifurcation model with geometric proportions relevant to the respiratory human airways for a range of Reynolds (Re) and Womersley (Wo) numbers.

Complex dynamics analysis for a remanufacturing duopoly model with nonlinear cost

In this paper, a duopoly remanufacturing game of competition on output based on boundedly rational expectation is formulated. It involves an original equipment manufacturer (OEM) who makes new products and a third-party independent remanufacture (IR) who uses returned cores to produce differentiated products. Both parties adopt nonlinear cost functions. The study shows that consumer’s willingness-to-pay (WTP) has distinct effects on the OEM’s performance and the stability of Nash equilibrium. We present results on existence, stability and local bifurcations of the equilibrium points. Numerical simulations demonstrate that the system with varying model parameters may be driven to chaos and the loss of stability may be caused by period doubling bifurcations. It is also shown that the state variables feedback and parameter variation method can be used to keep the system from instability and chaos.

Double Hopf bifurcation of differential equation with linearly state-dependent delays via MMS

In this paper the dynamics of differential equation with two linearly state-dependent delays is considered, with particular attention focused on non-resonant double Hopf bifurcation. Firstly, we identify the sufficient and necessary conditions of the double Hopf bifurcation by formal linearization, linear stability analysis and Hopf bifurcation theorem. Secondly, in the first time the method of multiple scales (MMS) is employed to classify the dynamics in the neighborhoods of two kinds of non-resonant double Hopf bifurcation points, i.e. Cases III and Ib, in the bifurcation parameters plane. Finally, numerical simulation is executed qualitatively to verify the previously analytical results and demonstrate the rich phenomena, including the stable equilibrium, stable periodic solutions, bistability and stable quasi-periodic solution and so on. It also implies that MMS is simple, effective and correct in higher co-dimensional bifurcation analysis of the state-dependent DDEs. Besides, the other complicated dynamics, such as the switch between torus and phase-locked solution, period-doubling and the route of the break of torus to chaos are also found in this paper.

Homoclinic-doubling and homoclinic-gluing bifurcations in the Takens-Bogdanov normal form with symmetry.

In this paper, we investigate the dynamics of a fourth-order normal form near a double Takens-Bogdanov bifurcation. The reduced system of this normal form possesses eight pairs of homoclinic orbits for certain parameter values. The nonlinear time transformation method is applied to obtain an analytical approximation of the homoclinic orbit in the perturbed system and to construct the homoclinic bifurcation curve as well. Using numerical continuation, period-doubling and homoclinic-doubling cascades emanating from a codimension-2 bifurcation point are found. A codimension-2 homoclinic-gluing bifurcation point at which several homoclinic orbits concerning the origin glue together to form a new homoclinic orbit is also obtained. It is shown that in the vicinity of these bifurcation points, the system may exhibit chaos and chaotic attractors.

Complex spatiotemporal behavior and coherent excitations in critically-coupled chains of neural circuits.

We investigate a critically-coupled chain of nonlinear oscillators, whose dynamics displays complex spatiotemporal patterns of activity, including regimes in which glider-like coherent excitations move about and interact. The units in the network are identical simple neural circuits whose dynamics is given by the Wilson-Cowan model and are arranged in space along a one-dimensional lattice with nearest neighbor interactions. The interactions follow an alternating sign rule, and hence the "synaptic matrix" embodying them is tridiagonal antisymmetric and has purely imaginary (critical) eigenvalues. The model illustrates the interplay of two properties: circuits with a complex internal dynamics, such as multiple stable periodic solutions and period doubling bifurcations, and coupling with a "critical" synaptic matrix, i.e., having purely imaginary eigenvalues. In order to identify the dynamical underpinnings of these behaviors, we explored a discrete-time coupled-map lattice inspired by our system: the dynamics of the units is dictated by a chaotic map of the interval, and the interactions are given by allowing the critical coupling to act for a finite period , thus given by a unitary matrix . It is now explicit that such critical couplings are volume-preserving in the sense of Liouville's theorem. We show that this map is also capable of producing a variety of complex spatiotemporal patterns including gliders, like our original chain of neural circuits. Our results suggest that if the units in isolation are capable of featuring multiple dynamical states, then local critical couplings lead to a wide variety of emergent spatiotemporal phenomena.

Period doubling and route to chaos in reduced graphene oxide, an experimental evidence

We use Nonlinear Schrödinger Equation (NLSE) to investigate chaotic dynamics inside a graphene-based optical system. Numerical results show that with increasing the control parameters, the character of Modulation instability (MI) is changed from convective to absolute. In consequence, a hyper-chaotic map is more expressive to explain the dynamical state of the system. It is also indicated that a reverse transition from the chaotic regime to the stable state can be resulted if the optical input intensity is further increased. The optical input intensity and feedback strength are distinguished as the two control parameters of the system. We then use Reduced Graphene Oxide (RGO) dispersion with large nonlinear refractive index to verify if a transition to instability and chaotic behavior is experimentally realized for the visible input light wave. The observation begins with saturation in the nonlinear response followed by an Optical Bistability (OB) phenomena. Afterward, a quasi-periodic fluctuations appear expressing the absolute MI. In agreement with our simulation result based on the period doubling bifurcation diagram, this quasi-periodic state is recognized as a route to the chaos. We infer that the graphene Quantum Dots (QDs) which are formed as a result of reduction process are responsible for energy transfer through the optical near-field interactions and thus, strengthen the required feedback.

Pitchfork-bifurcation-delay-induced bursting patterns with complex structures in a parametrically driven Jerk circuit system**Project supported by the National Key Basic Research Program of China (Grant No.2015CB057400) and the National Natural Science Foundation of China (Grant No.11672201).

This paper aims to present the novel routes to periodic and chaotic bursting oscillations, i.e. the different routes via delayed pitchfork bifurcations, a cascade of inverse period doubling bifurcations, Hopf bifurcations and homoclinic orbits, based on the parametrically driven Jerk circuit system. Firstly, by calculating the corresponding characteristic polynomial, we obtain the stabilities of different attractors and the critical values related to different bifurcations. Moreover, the transition mechanisms among different stable attractors have been revealed, including homoclinic connection and chaotic attractors. Secondly, based on the analysis of stabilities, bifurcations and transitions among different stable attractors, we investigate the mechanisms of different bursting oscillations. A distinct delayed supercritical pitchfork bifurcation is observed when the slow-varying parameter passes through the supercritical pitchfork bifurcation point periodically. We see that the delayed behavior may terminate at different parameter areas, which leads to different types of bursting oscillations. In particular, two novel chaotic bursting patterns, the ‘delayed sup-pitchfork/a cascade of inverse flip/supHopf’ chaotic bursting and the ‘delayed sup-pitchfork/a cascade of inverse flip/homoclinic connection/a cascade of inverse flip/supHopf’ chaotic bursting, are revealed. Our research enriches the routes to bursting oscillations and deepens the understanding of bursting phenomena. Finally, the accuracy of our study is verified by the numerical simulations.

Hyperchaos in constrained Hamiltonian system and its control

This paper first formulates a Hamiltonian system with hyperchaotic phenomena and investigates the equilibrium point and double Hopf bifurcation of the system. We obtain the result that the Hamiltonian system has hyperchaotic behaviors when any system parameter varies. The influences of holonomic constraint and nonholonomic constraint on the equilibrium points, invariance and the hyperchaotic state of the Hamiltonian system are then studied. Finally, we achieve the hyperchaotic control of the Hamiltonian system by introducing the constraint method. The studies indicate that the constraint can not only change the Hamiltonian system from hyperchaotic state to periodic state or chaotic state, but also make the Hamiltonian system become globally asymptotically stable. Numerical simulations, including Lyapunov exponents, bifurcation diagrams, Poincare maps and phase portraits for systems, exhibit the complex dynamical behaviors.

The study of non-smooth dynamics of a class of multi-degree-of-freedom vibro-impact system with clearance

The period doubling bifurcation and the Neimark-Sacker bifurcation of the multi-degree-of-freedom vibro-impact system with clearance are studied in this paper. The periodic solution of vibro-impact system is solved and a six-dimensional Poincaré map is established, and the doubling bifurcation and the Neimark-Sacker bifurcation are analyzed. The study of bifurcation and chaos of the system provides reliable basis for the design and fault diagnosis and provides theoretical guidance and technical support for the actual design in the safe and stable operation of some rotating machinery.

Period doubling bifurcation and chaos in oscillations of two interacting cavitation bubbles under ultrasonic

Two interacting bubbles oscillator under ultrasonic is a highly nonlinear system. The nature of the cavitation will be understood more deeply in the study of two interacting bubbles’ internal mechanism. In this paper, the model to describe the oscillation motion of the two interacting bubbles in the sound field is the Keller-Miksis equation. The dynamics of two interacting bubbles has been studied by adopting the methods of time domain wave, phase trajectory, Poincaré section, power spectrum and bifurcation diagram. We also studied the nonlinear oscillations of bubbles with different initial radii and characteristics of bifurcation structure of normalized radius as ultrasonic pressure amplitude changed. Results show that two bubbles oscillation traverses the similar mode of period doubling bifurcation to cascade chaos. Furthermore, the motion of two interacting bubbles shows discrepant oscillation characteristics under the action of same sound field. In addition, the periodic window and chaotic region of bubble’s oscillation occur alternately with the increase of pressure amplitude.

Climate change: Back to development

The Paris Agreement has created a double bifurcation. First, from top-down approach (with an emission limit imposed from above) to a bottom-up approach based on national emissions reduction pledges. And second, from a mitigation-centered policy to a more balanced mitigation and adaptation efforts. The following work proposes, however, that further steps must be taken to bring the theme of development back to the center of the fight against climate change.

Double Hopf Bifurcation of a Hematopoietic System with State Feedback Control

The dynamics of a system composed of hematopoietic stem cells and its relationship with neutrophils is ubiquitous due to periodic oscillating behavior induce cyclical neutropenia. Underlying the methodology of state feedback control with two time delays, double Hopf bifurcation occurs as varying time delay to reach its threshold value. By applying center manifold theory, the analytical analysis of system exposed the different dynamical feature in the classified regimes near double Hopf point. The novel dynamics as periodical solution and quasi-periodical attractor coexistence phenomena are explored and verified by numerical simulation.

A unique chaotic snap system with a smoothly adjustable symmetry and nonlinearity: Chaos, offset-boosting, antimonotonicity, and coexisting multiple attractors

This work proposes and systematically investigates the dynamics of a novel snap system with a single parameterized nonlinearity in the form φk(z)=0.5(exp(kz)−exp(−z)). The form of nonlinearity is physically interesting in the sense that the corresponding circuit realization involves only off-the shelf electronic components such as resistors, semiconductor diodes and operational amplifiers. Parameter k (i.e. a control resistor) serves to smoothly adjust the nonlinearity, and hence the symmetry of the system. In particular, for k=1, the nonlinearity is a hyperbolic sine, and thus the system is point symmetry about the origin. For k ≠ 1, the system is non-symmetric. The fundamental dynamics of the system are investigated in terms of equilibria and stability, phase space trajectory plots, bifurcations diagrams, and graphs of Lyapunov exponents. When monitoring the system parameters, some striking phenomena are found including period doubling bifurcation, reverse bifurcations, merging crises, coexisting bifurcations, hysteresis and offset boosting. Several windows in the parameters space are depicted in which the novel snap system displays a plethora of coexisting attractors (i.e. two, three, four, five or six different attractors) depending solely on the choice of the initial conditions. The magnetization of the state space due to the presence of multiple competing solutions is illustrated by means of basins of attraction. Laboratory experimental results confirm the theoretical predictions.

Complex dynamics of a 4D Hopfield neural networks (HNNs) with a nonlinear synaptic weight: Coexistence of multiple attractors and remerging Feigenbaum trees

This contribution investigates the nonlinear dynamics of a model of a 4D Hopfield neural networks (HNNs) with a nonlinear synaptic weight. The investigations show that the proposed HNNs model possesses three equilibrium points (the origin and two nonzero equilibrium points) which are always unstable for the set of synaptic weights matrix used to analyze the equilibria stability. Numerical simulations, carried out in terms of bifurcation diagrams, Lyapunov exponents graph, phase portraits and frequency spectra, are used to highlight the rich and complex phenomena exhibited by the model. These rich nonlinear dynamic behaviors include period doubling bifurcation, chaos, periodic window, antimonotonicity (i.e. concurrent creation and annihilation of periodic orbits) and coexistence of asymmetric self-excited attractors (e.g. coexistence of two and three disconnected periodic and chaotic attractors). Finally, PSpice simulations are used to confirm the results of the theoretical analysis.

Dynamic analysis of a novel jerk system with composite tanh-cubic nonlinearity: chaos, multi-scroll, and multiple coexisting attractors

It is well known that the dynamics of very simple physical systems can be quite complex if a sufficient amount of nonlinearity is present. In this contribution, a novel jerk system with smooth composite tanh-cubic nonlinearity is proposed and investigated. Interestingly, the new nonlinearity takes advantage of the classical smooth cubic polynomial in the sense that it induces more complex and interesting dynamics (e.g. five equilibria instead of three in the case of a (traditional) cubic nonlinearity, multi-scroll, and multistability). The fundamental properties of the model are discussed including equilibria and stability, phase portraits, Poincare sections, bifurcation diagrams and Lyapunov exponent’s plots. Period doubling bifurcation, antimonotonicity, chaos, hysteresis, and coexisting bifurcations are reported. In particular, a rare phenomenon is found in which two different pairs of coexisting limit cycles born from the Hopf bifurcation follow each a different sequence of period-doubling bifurcations, then merge to form a three-scroll chaotic attractor as a parameter is smoothly changed. As another major result of this work, several windows in the parameter space are depicted in which the novel jerk system develops the striking behaviour of multiple coexisting attractors (i.e. coexistence of three, four, six, or eight disjoint periodic and chaotic attractors for the same parameters set) due to the combined effects of three families of parallel bifurcation branches and hysteresis. An electronic analogue of the new jerk system is designed and simulated in PSPICE. As far as the authors’ knowledge goes, no example of such a simple and ‘elegant’ 3D autonomous system with this combination of features is reported in the relevant literature.

A plethora of behaviors in a memristor based Hopfield neural networks (HNNs)

This work investigates the nonlinear dynamics of a model of memristor based Hopfield Neural Networks (HNNs). The investigations show that the memristive HNNs possesses three equilibrium points among which the origin and two nonzero equilibrium points. It is found that the origin and the nonzero equilibrium points are always unstable for the set of synaptic weights matrix used for the analysis. Numerical simulations, carried out in terms of bifurcation diagrams, graphs of Lyapunov exponents, phase portraits and frequency spectra, are used to highlight the complex dynamical behaviors exhibited by the model. The results indicate that the introduced memristive HNNs exhibits rich nonlinear dynamical behaviors including period doubling bifurcation, symmetry restoring crisis, chaos, periodic window, antimonotonicity (i.e. concurrent creation and annihilation of periodic orbits) and coexisting self-excited attractors (e.g. coexistence of two, four and six disconnected periodic and chaotic attractors) not reported in the relevant literature. Finally, PSpice simulation investigations support the results of theoretical analyses.

Weakly resonant double Hopf bifurcation in coupled nonlinear systems with delayed freedback and application of homotopy analysis method

In this paper, we study the dynamical behaviors of coupled nonlinear systems with delay coupling by the multiple scales method and the homotopy analysis method. Firstly, we analyze the distribution...