Cascade Of Bifurcations Research Articles

Mixed-mode oscillations and chaos in a complex chemical reaction network involving heterogeneous catalysis.

The nonlinear dynamical behavior in a complex isothermal reaction network involving heterogeneous catalysis is studied. The method first determines the multiple steady states in the reaction network. This is followed by an analysis of bifurcation continuations to identify several kinds of bifurcations, including limit point, Bogdanov-Takens, generalized Hopf, period doubling, and generalized period doubling. Numerical simulations are performed around the period doubling and generalized period doubling bifurcations. Rich nonlinear behaviors are observed, including simple sustained oscillations, mixed-mode oscillations, non-mixed-mode chaotic oscillations, and mixed-mode chaotic oscillations. Concentration-time plots, 2D phase portraits, Poincaré maps, maximum Lyapunov exponents, frequency spectra, and cascade of bifurcations are reported. Period-doubling and period-adding routes leading to chaos are observed. Maximum Lyapunov exponents are positive for all the chaotic cases, but they are also positive for some non-chaotic orbits. This result diminishes the reliability of using maximum Lyapunov exponents as a tool for determining chaos in the network under study.

Numerical investigation of tip bifurcation on aerodynamic characteristics and flow field structure in a compressor cascade

In this study, the tip bifurcation of the compressor cascade is proposed, referring to the biological characteristics of bird wing bifurcation. The influence of tip bifurcation with different structural parameters on the aerodynamic performance and flow field structure of compressor cascade at different incidence angles is studied by numerical simulation. The research shows that the tip bifurcation can make the starting position of the corner separation move backward, inhibit the separation along the pitch and blade height direction, and reduce the flow loss of the cascade. The separation control effect of the corner region first increases and then decreases as the bifurcation height increases, and gradually diminishes as the arrangement position approaches the trailing edge. When the tip bifurcation is set at the starting position of the corner separation and its height is equal to 4 %H, the flow improvement effect in the passage is the most obvious, and the cascade flow loss is reduced by 14.4 %. At the same time, the tip bifurcation cascade has effective positive incidence angle characteristics, but when the incidence angle is less than the minimum loss incidence angle, the tip bifurcation increases the cascade loss.

Nonlinear dynamics of a laser diode subjected to both optical injection and optical feedback.

We analyse theoretically the nonlinear dynamics of a single-mode laser diode subjected to both optical injection and optical feedback. Detailed mappings of the laser dynamics reveal that, due to optical feedback (OF), the locking boundaries resulting from optical injection (OI) shift towards larger negative detunings and higher injection rates and display a periodic pattern of the injection locking boundaries. We demonstrate how feedback induces a cascade of quasiperiodic bifurcations associated with abrupt dynamic changes, hence altering the route to locking. A close inspection of the laser optical spectra for increasing feedback rate reveals the complex interplay between undamped relaxation oscillations and external cavity frequencies.

Evolving division of labor in a response threshold model

The response threshold model explains the emergence of division of labor (i.e., task specialization) in an unstructured population by assuming that the individuals have different propensities to work on different tasks. The incentive to attend to a particular task increases when the task is left unattended and decreases when individuals work on it. Here we derive mean-field equations for the stimulus dynamics and show that they exhibit complex attractors through period-doubling bifurcation cascades when the noise disrupting the thresholds is small. In addition, we show how the fixed threshold can be set to ensure specialization in both the transient and equilibrium regimes of the stimulus dynamics. However, a complete explanation of the emergence of division of labor requires that we address the question of where the threshold variation comes from, starting from a homogeneous population. We then study a structured population scenario, where the population is divided into a large number of independent groups of equal size, and the fitness of a group is proportional to the weighted mean work performed on the tasks during a fixed period of time. Using a winner-take-all strategy to model group competition and assuming an initial homogeneous metapopulation, we find that a substantial fraction of workers specialize in each task, without the need to penalize task switching.

Heterogeneous multistability and antimonotonicity for a new 3D system with a triple well nonlinearity: theoretical study, control and microcontroller implementation

We propose a new 3D autonomous multistable jerk-like system with a nonlinear term consisting of a six-order triple well function. The presence of six equilibrium points with symmetrical locations along the x-axis represents one of the main distinguishing properties of the new system. Strikingly, the stability analysis of equilibria reveals a cascade of Hopf bifurcations at three specific values of a single control parameter, which results in several forms of complexity. Accordingly, various forms of coexisting attractors such as stable fixed points, limit cycles of diverse periodicities, and chaotic attractors are depicted for some special parameter values. Moreover, It is found that the new jerk-like system with six order triple well polynomial function exhibit extremely complex nonlinear behaviors such as anti-monotone bifurcations, hysteresis and parallel bifurcation branches. These latter aspects explain the presence of multiple (i.e. up to four) coexisting asymmetric attractors for some special rank of parameters. In the presence of multiple competing dynamics, we resort to basins of attraction in order to highlight the how the state space is magnetized. The combination of dynamic features discussed in the new jerk-like system with triple well polynomials nonlinearity introduced in this article is unique and rarely reported. An electronic version of the new system with triple well polynomial nonlinearity is implemented in PSpice. Moreover, a hardware digital implementation of the system is also carried out using an Arduino microcontroller. A very good agreement is captured between PSpice simulation results, the laboratory measurements and the theoretical predictions.

Chaos detection in predator-prey dynamics with delayed interactions and Ivlev-type functional response

<p>Regarding delay-induced predator-prey systems, extensive research has focused on the phenomenon of delayed destabilization. However, the question of whether delays contribute to stabilizing or destabilizing the system remains a subtle one. In this paper, the predator-prey interaction with discrete delay involving Ivlev-type functional response is studied by theoretical analysis and numerical simulations. The positivity and boundedness of the solution for the delayed model have been discussed. When time delay is accounted as a bifurcation parameter, stability analysis for the coexistence equilibrium is given in theoretical aspect. Supercritical Hopf bifurcation is detected by numerical simulation. Interestingly, by choosing suitable groups of parameter values, the chaotic solutions appear via a cascade of period-doubling bifurcations, which is also detected. The theoretical analysis and numerical conclusions demonstrate that the delay mechanism plays a crucial role in the exploration of chaotic solutions.</p>

Bursting activity in the reduced mean-field model of neuron-glial interaction

The study of neural activity synchronization in the brain is a central focus of modern neurobiology and neurodynamics. In a healthy brain, cognitive functions necessitate the accurate integration of neural activity at specific spatiotemporal scales. Some neurological and psychiatric disorders exhibit observable variations in patterns of synchronous population activity. One of the most significant and compelling synchronous population activity patterns is the bursting activity that participates in various informational and physiological processes, including epilepsy. Several mathematical models have been proposed to explain the mechanisms of formation of the bursting activity. The most intriguing and scientifically sound models enable incorporation of astrocytic modulation of neural activity [1]. In this paper, we present a novel phenomenological model for accurately replicating the bursting neural population activity. Our model builds upon the Tsodyks-Markram [2] model and incorporates vital aspects of neuron-glial interaction through a tripartite synapse. The presented model is a simplified version of the previously proposed model [3]. During astrocyte activity (which lasts seconds), we can set the probability of neurotransmitter release, u, which is determined in fractions of a second. This enables the proposed model to be written in the following manner: r=–E+α ln(1+exp((Ju(Y)xE+I0)/α)), =(1–x)/τD–u(Y)xE, =–y/τY+βσy(X). Here, E(t) represents the average neuronal activity of the excitatory population, while x(t) denotes the quantity of available neurotransmitter. Additionally, y(t) describes the concentration of gliotransmitters, which are released as a result of biochemical reactions during neuron-astrocyte interactions. The function describes the alteration in the likelihood of neurotransmitter release when gliotransmitters are present u(y)=u0+(∆u0)/(1+exp(–50(y–ythr)). Here u0 represents the probability of neurotransmitter release without astrocytic influence; ∆u0 indicates the change in the probability of neurotransmitter release caused by the interaction of the gliotransmitter with the presynaptic terminal, and ythr represents the threshold value that determines the change in the probability of neurotransmitter release due to the effect of a gliotransmitter. The function describes the influence of neurotransmitter on the concentration of the gliotransmitter σy(x)=1/(1+exp(–20(x–xthr)), where xthr is the astrocyte activation threshold. Note that the proposed model does not account for synaptic depression’s mechanism. Therefore, various dynamical regimes’ formation in the model is solely controlled by astrocytic dynamics. In the study, control parameters I0 and u0 were selected while the remaining parameters were held constant. Specifically, τ=0.013, τD=0.08, α=1.58, J=3.07. Changes in the concentration of neurotransmitters and gliotransmitters were characterized by parameters ∆u0=0.305, τν=3.3, β=0.3, xxthr=0.75, ythr=0.4. The research was conducted using computer modeling and numerical methods in nonlinear dynamics. The proposed model exhibits a diverse range of population activity patterns, such as spiking and bursting regular and chaotic activity. Chaotic dynamics zones are identified in the control parameter plane. An account is given of the mechanisms behind the emergence of these activity patterns through bifurcation analysis. It is shown that the occurrence of chaotic activity in the system can be associated both with the cascade of period doubling bifurcations and with the further development of chaos, as a result of which a homoclinic attractor appears in the system according to the Shilnikov scenario. It was also shown that multistability is observed in the system in a certain range of parameters. Note that the emergence of multistability and bursting activity in the model is independent of the intricacy of neuron and glial cell dynamics. These dynamics are instead influenced by the feedback loop between the presynaptic neuron and the glial cell. The effects of neuron-like dynamics and neural-glial interaction demonstrated are generally applicable, without specifying particular characteristics of neural-glial interaction, network architecture, or single-cell neuron dynamics. It should be noted that our proposed model solely concerns the potentiation of synaptic transmission by astrocytes. In summary, the proposed phenomenological model for population activity can replicate various patterns of neuron activity found in a wide range of dynamic memory and information processing studies. This research can potentially lead to the development of new, effective treatment methods for neurological diseases that involve neuron-glial interaction. Another potential application for these results is in the development of an optimized live chip that has preset functions. This requires a deeper understanding of rhythmogenesis in neural networks and brain function.

Mathematical Modeling of the Evolutionary Dynamics of a Planktonic Community Using a Discrete-Time Model

This study proposes a discrete-time eco-genetic model of a planktonic community that includes zooplankton and two competing phytoplankton haplotypes with and without a toxicity trait. The Holling type II response function describes predator consumption. We use the Ricker model to consider density limitation and regulation. The model is analytically and numerically studied. The loss of stability of fixed points occurs via the Neimark–Sacker scenario and a cascade of period-doubling bifurcations. The model reveals bistability and multistability. Therefore, the initial conditions can determine which of the coexisting dynamic modes will be attracted. If the competition of haplotypes is weaker than their self-regulation, then the variation in the current densities of community components can shift the observed dynamics, while the evolution direction remains unchanged. The ratio of haplotype fitnesses and predator pressure generally determines the asymptotic genetic composition of phytoplankton. If competition of haplotypes is higher than their self-regulation, then the bistability of monomorphic fixed points occurs when the displacement of one haplotype by another depends on initial conditions. The presence of predators can maintain the genetic polymorphism of the prey. This system shows dynamic modes similar to experimental dynamics: oscillation with delay, long-period antiphase fluctuations, and cryptic cycles emerging due to rapid evolution.

Effects of Competition Intensities and R&D Spillovers on a Cournot Duopoly Game of Digital Economies

In this paper, we introduce a Cournot duopoly game that can characterize fierce competition in digital economies and employ it to examine the effects of research and development (R&D) spillovers while considering various competition intensities. We obtain the analytical solution of the Nash equilibrium and the expression of commodity price, firm production, and variable profit under some key competition intensities. Furthermore, we analyze the local stability of the Nash equilibrium and derive that the equilibrium may lose its stability only through a 1:4 resonance bifurcation. Numerical simulations are conducted, through which we find that the Nash equilibrium transitions to complex dynamics through a cascade of period-doubling bifurcations. Phase portraits are also provided to illustrate more details of the dynamics, which confirm the previous theoretical finding that the Nash equilibrium loses its stability through a 1:4 resonance bifurcation.

Arnold Tongue-Like Structures and Coexisting Attractors in the Memristive Muthuswamy–Chua–Ginoux Circuit Model

The three-dimensional Muthuswamy–Chua–Ginoux (MCG) circuit model is a generalization of the paradigmatic canonical Muthuswamy–Chua circuit, where a physical memristor assumes the role of a thermistor, and it is connected in series with a linear passive capacitor, a linear passive inductor, and a nonlinear resistor. The physical memristor presents an electrical resistance which is a function of temperature. Nowadays, the MCG circuit model has gained considerable attention due to the lack of extensive numerical explorations and their distinct dynamical properties, exemplified by phenomena such as the transition from torus breakdown to chaos, giving rise to a double spiral attractor associated to independent period-doubling cascades. In this contribution, the complex dynamics of the MCG circuit model is studied in terms of the Lyapunov exponents spectra, Kaplan–Yorke (KY) dimension, and the number of local maxima (LM) computed in one period of oscillation, as two parameters are simultaneously varied. Using the Lyapunov spectra to distinguish different dynamical regimes, KY dimension to estimate the attractors’ dimension, and the number of LM to characterize different periodic attractors, we construct high-resolution two-dimensional stability diagrams considering specific ranges of the parameter pairs [Formula: see text]. These parameters are associated with the inverse of the capacitance in the passive capacitor, and the heat capacitance and dissipation constant of the thermistor, respectively. Unexpectedly, we identify sequences of infinite self-organized generic stable periodic structures (SPSs) and Arnold tongues-like structures (ATSs) merged into chaotic dynamics domains, and the coexistence of different attracting sets (attractors) for the same parameter combinations and different initial conditions (multistability). We explore the multistable dynamics using the stability analysis and computation of Lyapunov coefficients, the inspection of the coexisting attractors, bifurcations diagrams, and basins of attraction. The periods of the ATSs and a particular sequence of shrimp-shaped SPSs obey specific generating and recurrence rules responsible for the bifurcation cascades. As the MCG circuit model has the crucial properties presented by the usual Muthuswamy–Chua circuit model, specific properties explored in our study should be helpful in real problems involving circuits with the presence of physical memristor playing the role of thermistors.

Emergence and evolution of unusual inhomogeneous limit cycles displacing hyperchaos in three quorum-sensing coupled identical ring oscillators

We demonstrate that strongly asymmetric limit cycles can be observed in the system of three identical ring oscillators (3-gene networks known as Repressilators) globally coupled by signal molecule diffusion added to the model in a way like the known bacterial “quorum-sensing” mechanism. These cycles are stable over a wide interval of the coupling strengths where they expel the dominant hyperchaotic regime existing in three Repressilators in very large areas of parameters. The bifurcations of the inhomogeneous limit cycle, with a high-amplitude orbit for one oscillator and two low-amplitude identical orbits for the other two, are traced. Bifurcation analysis reveals an unusual cascade of bifurcations ended in the appearance of a new limit cycle with splitted (slightly nonidentical) low-amplitude orbits. Both cycles lose stability giving birth inhomogeneous chaos in the small parameter interval. Hyperchaos dominates in the parameter plane around the “island” with inhomogeneous limit cycles, and this accounts for very long hyper chaotic transients when the system is returning to stable asymmetric cycles after their perturbations. In turn, it is the cycles that contribute the asymmetric and often rather long pieces in hyperchaotic trajectories. The presented cycles differ from the known asymmetric attractors: inhomogeneous limit cycles born from “oscillation death” and the cycles observing in “smallest chimeras”.

Feigenbaum scenario without parameters.

Typically, the period-doubling bifurcations exhibited by nonlinear dissipative systems are observed when varying systems' parameters. In contrast, the period-doubling bifurcations considered in the current research are induced by changing the initial conditions, whereas parameter values are fixed. Thus, the studied bifurcations can be classified as the period-doubling bifurcations without parameters. Moreover, we show a cascade of the period-doubling bifurcations without parameters, resulting in a transition to deterministic chaos. The explored effects are demonstrated by means of numerical modeling on an example of a modified Anishchenko-Astakhov self-oscillator where the ability to exhibit bifurcations without parameters is associated with the properties of a memristor. Finally, we compare the dynamics of the ideal-memristor-based oscillator with the behavior of a model taking into account the memristor forgetting effect.

Complex Dynamics Modes in a Simple Model of Prey-Predator Community: Bistability and Multistability

The paper proposes and studies a two-component discrete-time model of the prey-predator community considering zooplankton and fish interactions and their development features. Discrete-time systems of equations allow us to take into account naturally the rhythm of many processes occurring in marine and freshwater communities, which are subject to cyclical fluctuations due to the daily and seasonal cycle. We describe the dynamics of fish and zooplankton populations constituting the community by Ricker’s model, which is well-studied and widely used in population modeling. To consider the species interaction, we use the Holling-II type response function taking into account predator saturation. We carried out the study of the proposed model. The system is shown to have from one to three non-trivial equilibria, which gives the existence of the complete community. In addition to the saddle-node bifurcation, which generates bistability of stationary dynamics, a nontrivial equilibrium loses its stability according to the Neimark-Sacker scenario with an increase in the reproductive potential of both predator and prey species, as a result of which the community exhibits long-period oscillations similar to those observed in experiments. With the higher bifurcation parameter, the reverse Neimark-Sacker bifurcation is shown to occur followed by the closed invariant curve collapses, and dynamics of the population stabilizes, later losing stability through a cascade of period-doubling bifurcations. Multistability complicates the birth and disappearance of the invariant curve in the phase space scenario by the emergence of another irregular dynamics in the system with the single unstable nontrivial fixed point. At fixed values of the model parameters and different initial conditions, the system considered is shown to demonstrate various quasi-periodic oscillations. Despite extreme simplicity, the proposed discrete-time model of community dynamics demonstrates a wide variety and variability of dynamic modes. It shows that the influence of environmental conditions can change the type and nature of the observed dynamics.

Quasi-periodic structure in chaotic bursting attractor for a controlled jerk oscillator

The main purpose of the paper is to show that parts of the trajectory for chaotic bursting oscillations may oscillate according to different limit cycles, which implies orderly movement exists in chaos. For a typical jerk oscillator with a cubic nonlinearity, when a slow-varying controlling term is introduced, bursting oscillations may occur, appearing in the combination of large-amplitude and small-amplitude oscillations. Two coexisting stable equilibrium points may evolve to chaos via different cascades of period-doubling bifurcation, respectively, which means different types of stable attractors may coexist with the variation of slow-varying parameter, while several periodic windows exist in the chaotic region. It is found that the spiking oscillations of the full system may change according to different cycles not only for 2-D torus bursting oscillations but also for the chaotic bursting attractor. Furthermore, whether the influence of the short periodic window of the fast subsystem on the bursting oscillations appears or not depends on the size of the slow scale, since the inertia of the movement may cause the delay effect of the bifurcations at both two edges of the window. The findings suggest that coupling of multiple scales may be one of key factors for the diversity of real world.

Bifurcation cascade, self-similarity, and duality in the three-rotor problem.

The three-rotor system concerns equally massive point particles moving on a circle subject to attractive cosine potentials of strength g. The quantum theory models chains of coupled Josephson junctions. Classically, it displays order-chaos-order behavior with increasing energy E along with a seemingly globally chaotic phase for 5.33g≲E≲5.6g. It is also known to admit pendulum and isosceles breather families of periodic orbits at all energies. While pendula display a doubly infinite sequence of stability transitions accumulating at their libration to rotation threshold at E=4g, breathers undergo only one stability transition. Here, we show that these stability transitions are associated with forward and reverse fork-like isochronous and period-doubling bifurcations. The new family of periodic orbits born at each of these bifurcations is found using an efficient search algorithm starting from a transverse perturbation to the parent orbit. The graphs of stability indices of various classes of orbits born at pendulum bifurcations meet at E=4g forming "fans." The transitions in the librational and rotational phases are related by an asymptotic duality between bifurcation energies and shapes of newly born periodic orbits. The latter are captured by solutions to a Lamé equation. We also find and numerically validate values of scaling constants for self-similarity in (a) stability indices of librational and rotational pendula and (b) shapes of newly born orbits as E→4g. Finally, we argue that none of the infinitely many families of periodic orbits we have found is stable for 5.33g≲E≲5.6g, providing further evidence for global chaos in this energy band.

Minimal Chaotic Networks of Linear Neuron-Like Elements with Single Rectification: Three Prototypes

Chaotic oscillations induced by single rectification in networks of linear neuron-like elements are examined on three prototype models: one nonautonomous system and two autonomous systems. The first is a system of coupled neurons with periodic input; the second is a system of three coupled neurons with six couplings; the third is a ring of four unidirectionally coupled neurons with one reverse coupling. In each system, the output function of one neuron is ramp and that of the others is linear. Each system is piecewise linear and the phase space is separated into two domains by a single border. Steady states, periodic solutions and homoclinic orbits are derived rigorously and their stability is evaluated with the eigenvalues of the Jacobian matrices. The bifurcation analysis of the three systems shows that chaotic attractors could be generated through cascades of period-doubling bifurcations of periodic solutions.

Universal Bifurcation Chaos Theory and Its New Applications

In this work, an analytical and numerical analysis of the transition to chaos in five nonlinear systems of ordinary and partial differential equations, which are models of autocatalytic chemical processes and interacting populations, is carried out. It is shown analytically and numerically that in all considered systems of equations, further complication of the dynamics of solutions and the transition to chemical and biological turbulence is carried out in full accordance with the universal Feigenbaum-Sharkovsky-Magnitskii bifurcation theory through subharmonic and homoclinic cascades of bifurcations of stable limit cycles. In this case, irregular (chaotic) attractors in all cases are exclusively singular attractors in the sense of the FShM theory. The obtained results once again indicate the wide applicability of the universal bifurcation FShM theory for describing laminar–turbulent transitions to chaotic dynamics in complex nonlinear systems of differential equations and that chaos in the system can be confirmed only by detection of some main cycles or tori in accordance with the universal bifurcation diagram presented in the article.

Chaotic Bloch oscillations in dissipative optical systems driven by a periodic train of coherent pulses

We study the response of an optical system with the Kerr nonlinearity demonstrating Bloch oscillations to a periodic train of coherent pulses. It has been found out that the intensity of the field excited in the system by pulses resonantly depends on the train period. It is demonstrated numerically and analytically that the response of the system is stronger when the period of the driving pulses is commensurate with the period of the Bloch oscillations. Moreover, large enough pulses are capable to induce the instabilities which eventually lead to onset of chaotic Bloch oscillations of the wave-function envelope bouncing both in time and space. The analysis reveals that these instabilities are associated with period-doubling bifurcations. A cascade of such bifurcations with increase of the pulse amplitude triggers the chaotic behavior.

Scenarios for the creation of hyperchaotic attractors in 3D maps

We study bifurcation mechanisms for the appearance of hyperchaotic attractors in three-dimensional diffeomorphisms, i.e. such attractors whose orbits have two positive Lyapunov exponents in numerical experiments. In particular, periodic orbits belonging to the attractor should have two-dimensional unstable invariant manifolds. We discuss several bifurcation scenarios which create such periodic orbits inside the attractor. This includes cascades of supercritical period-doubling bifurcations of saddle periodic orbits and supercritical Neimark–Sacker bifurcations of stable periodic orbits, as well as various combinations of these cascades. These scenarios are illustrated by an example of the three-dimensional Mirá map.

Period-doubling bifurcations and islets of stability in two-degree-of-freedom Hamiltonian systems.

In this paper, we show that the destruction of the main Kolmogorov-Arnold-Moser (KAM) islands in two-degree-of-freedom Hamiltonian systems occurs through a cascade of period-doubling bifurcations. We calculate the corresponding Feigenbaum constant and the accumulation point of the period-doubling sequence. By means of a systematic grid search on exit basin diagrams, we find the existence of numerous very small KAM islands ("islets") for values below and above the aforementioned accumulation point. We study the bifurcations involving the formation of islets and we classify them in three different types. Finally, we show that the same types of islets appear in generic two-degree-of-freedom Hamiltonian systems and in area-preserving maps.