Period-adding Bifurcation Research Articles

On Forced Oscillations in a Relay System with Hysteresis

This paper discusses the phenomenon associated with the forced synchronization (“entrainment of a self-sustained oscillator by an external force”) in a relay system with hysteresis, which manifests itself in the occurrence of periodic motions close to the rhythmic activity of neurons, when packets of fast oscillations are interspersed with intervals of the slow dynamics. To study this phenomenon, we introduce a circle mapping, which, depending on the parameters, can be a circle diffeomorphism or discontinuous map (“gap map”). In both cases, this mapping demonstrates the so-called period-adding bifurcation structure. It is demonstrated that packets number of fast oscillations in the period of periodic motion is determined by the rotation number, and the length of the intervals between the packets may be found of the boundaries of the absorbing interval. The change in the number of pulses in the packet occurs through the border-collision bifurcation.

Complex firing activities and bifurcations in memristor-coupled Hindmarsh–Rose neuron

Due to the unique synaptic plasticity and memory effect, a memristor can not only mimic biological synapses but also characterize the influence of external electromagnetic radiation. In this paper, a ReLU-type non-ideal memristor with a simple structure is first coupled to a classical three-dimensional Hindmarsh–Rose neuron to describe the electromagnetic induction effect, which can show period-doubling, period-adding, and saddle-node bifurcations by varying the coupling strength of the memristor. Furthermore, complex discharge behaviors of the system, including bursting discharge and spiking discharge, are exhibited, and some coexisting discharge modes associated with initial values are also presented. Finally, an analog circuit scheme consuming fewer circuit components is designed, and it was found that the experimental results are consistent with the numerical results.

Regular and chaotic dynamics in a 2D discontinuous financial market model with heterogeneous traders

We develop a financial market model where three types of traders operate simultaneously: fundamentalists and chartists of two types, namely, trend followers and contrarians. The dynamics of this model is described by a two-dimensional discontinuous map defined by two linear functions, where one acts in the partition between two (parallel) discontinuity lines and the other one acts outside this partition. Our analysis shows that despite the linearity of the map components, its dynamics can be quite complex, with various, possibly coexisting attracting cycles and chaotic attractors. As a first step towards the understanding how the overall bifurcation structure observed in the parameter space of the map is organized, we obtain analytically the boundaries of periodicity regions related to the simplest attracting period-n cycles, n≥3, with one point in the middle partition and n−1 points outside it. These boundaries can be related to border-collision bifurcations (when a point of the cycle collides with a discontinuity line) as well as to degenerate bifurcations (associated with eigenvalues on the unit circle). We show also some elements of period-adding and period-incrementing bifurcation structures for which the cycles mentioned above are basic. From an economic point of view, our study confirms that a fairly simple financial market model with heterogeneous agents is able to produce complicated boom-bust dynamics typical of real financial markets.

Experimental investigation of the period-adding bifurcation route to chaos in plasma.

Since the characteristic timescales of the various transport processes inside the discharge plasma span several orders of magnitude, it can be regarded as a typical fast-slow system. Interestingly, in this work, a special kind of complex oscillatory dynamics composed of a series of large-amplitude relaxation oscillations and small-amplitude near-harmonic oscillations, namely, mixed-mode oscillations (MMOs), was observed. By using the ballast resistance as the control parameter, a period-adding bifurcation sequence of the MMOs, i.e., from L^{s} to L^{s+1}, was obtained in a low-pressure DC glow discharge system. Meanwhile, a series of intermittently chaotic regions caused by inverse saddle-node bifurcation was embedded between the two adjacent periodic windows. The formation mechanism of MMOs was analyzed, and the results indicated that the competition between electron production and electron loss plays an important role. Meanwhile, the nonlinear time series analysis technique was used to study the dynamic behavior quantitatively. The attractor in the reconstructed phase space indicated the existence of the homoclinic orbits of type Γ^{-}. In addition, by calculating the largest Lyapunov exponent (LLE), the chaotic nature of these states was confirmed and quantitatively characterized. With the decrease in the ballast resistance, the return map of the chaotic state gradually changed from the nearly one-dimensional single-peak structure to the multibranch structure, which indicates that the dissipation of the system decreased. By further calculating the correlation dimension, it was shown that the complexity of the strange attractors increased for higher-order chaotic states.

A 2D piecewise-linear discontinuous map arising in stock market modeling: Two overlapping period-adding bifurcation structures

We consider a 2D piecewise-linear discontinuous map defined on three partitions that drives the dynamics of a stock market model. This model is a modification of our previous model associated with a map defined on two partitions. In the present paper, we add more realistic assumptions with respect to the behavior of sentiment traders. Sentiment traders optimistically buy (pessimistically sell) a certain amount of stocks when the stock market is sufficiently rising (falling); otherwise they are inactive. As a result, the action of the price adjustment is represented by a map defined by three different functions, on three different partitions. This leads, in particular, to families of attracting cycles which are new with respect to those associated with a map defined on two partitions. We illustrate how to detect analytically the periodicity regions of these cycles considering the simplest cases of rotation number 1/n,n≥3, and obtaining in explicit form the bifurcation boundaries of the corresponding regions. We show that in the parameter space, these regions form two different overlapping period-adding structures that issue from the center bifurcation line. In particular, each point of this line, associated with a rational rotation number, is an issue point for two different periodicity regions related to attracting cycles with the same rotation number but with different symbolic sequences. Since these regions overlap with each other and with the domain of a locally stable fixed point, a characteristic feature of the map is multistability, which we describe by considering the corresponding basins of attraction. Our results contribute to the development of the bifurcation theory for discontinuous maps, as well as to the understanding of the excessively volatile boom-bust nature of stock markets.

Field coupling-induced synchronization of a functional neuron with magnetic field perception

It is difficult to explore the induced mechanism of neuronal firing activity and the cooperation between neurons experimentally, so some neuronal circuits are established to train mode transformation and selection. Furthermore, some functional device, for example, piezoelectric ceramic, thermistor, phototube are inset in neuronal circuits and the functional neurons are developed to perceive some specific physical signal. In this paper, a linear resistance and an ideal Josephson junction are parallelly connected to FitzHugh-Nagumo (FHN) neuronal circuit, and then a functional neuron is established to percept the induction currents induced by the external magnetic field. Some basic dynamics are analyzed in terms of two-parameter bifurcation, one-parameter bifurcation and interspike interval(ISI for short) bifurcation, it is indicated that the functional neuron is sensitive to the electromagnetic stimulation and has rich multimodal transformations. The period-adding bifurcations accompanied by multi-period and chaos between two period cascades occupy the entire bifurcation interval when the external magnetic field is given in steady state, while the irregular modal transformations between period and chaos can be observed when the functional neuron encounters a periodic external magnetic field. Furthermore, two functional neurons are bridged with a capacitor, so field coupling is induced, then the effects of coupling strengths on complete synchronization are investigated by calculating the synchronization error function and the bifurcation diagram. In addition, the local dynamics of the network node play an important role in collective behavior and synchronous transition, so two capacitor-coupled functional neurons is presented as the network nodes, and a chain neural network is constructed to explore the effects of external magnetic field and coupling strength on network synchronous behavior. Obviously, the orderliness of the neural network can enhance or destroy under different modulation of external magnetic stimulation and coupling strength. It can give insights to investigate synchronization on neural networks with field coupling and useful guidance for implementing artificial synapse for signal processing.

Nested mixed-mode oscillations, Part III: Comparison of bifurcation structures between a driven Bonhoeffer–van der Pol oscillator and Nagumo–Sato piecewise-linear discontinuous one-dimensional map

In our previous studies (Inaba and Kousaka, (2020); Inaba and Tsubone, (2020)), we discovered bifurcation structures represented by nested mixed-mode oscillations (MMOs) generated by a driven Bonhoeffer–van der Pol (BVP) oscillator. BVP oscillators are equivalent to FitzHugh–Nagumo models and have been a subject of intense research for the last six decades. In this study, we consider the case in which the diode included in a driven BVP oscillator is assumed to operate as an ideal switch. In this case, Poincaré return maps can be rigorously constructed one-dimensionally, which consist of two downward convex branches. We also consider the Poincaré return map that is approximated as a two-segment piecewise-linear discontinuous one-dimensional map. Such a piecewise-linear map was proposed by Nagumo and Sato and generates nested period-adding bifurcations. We show that un-nested, singly, and doubly nested MMO-incrementing bifurcations generated by the driven BVP oscillator coincide with one of the possible un-nested, singly, and doubly nested period-adding bifurcations, respectively, generated with the Nagumo–Sato map.

ReLU-type Hopfield neural network with analog hardware implementation

Rectified Linear Unit (ReLU) is an extremely simple activation function with two-segment linearity and is usually used in convolutional neural network. To simplify the analog hardware implementation, this article presents a bi-neuron Hopfield neural network (HNN) using ReLU function instead of hyperbolic tangent (Tanh) function as the activation function. With the HNN model, the boundedness properties can be proved by virtue of the Lyapunov method and the stability distributions of time-varying equilibrium point are clarified according to four independent affine regions. Using numerical measures, complex dynamical behaviors are uncovered therein, including period-adding bifurcation, spiking/bursting pattern, fast-slow effect, and coexisting multistable patterns. In addition, a multiplierless electronic circuit is developed for implementing the HNN model and its experimental measurements well validate the numerical ones. The results manifest that the ReLU-type HNN model can display intricate and rich dynamics, and its analog hardware implementation is simpler than other HNN models.

Hidden Dynamics, Multistability and Synchronization of a Memristive Hindmarsh–Rose Model

Reliable neuron models play an essential role in identifying the electrical activities, global bifurcation patterns, and dynamic mechanisms of neurons in electromagnetic environments. Considering that memristive autapse can characterize the self-induced effect of neurons, a five-dimensional Hindmarsh–Rose (HR) neuron model involving electric and magnetic fields is established. The detailed existence and stability analyses for equilibrium points are performed, and the complex time-varying stability, saddle-node bifurcation, and Hopf bifurcation behaviors are demonstrated. Interestingly, the bistable structures consisting of quiescent state and periodic bursting modes near the subcritical Hopf bifurcation and counterintuitive dynamic phenomena can be induced via appropriately adjusting the memristive current. Accordingly, the mechanism of positive feedback autaptic current decreases its firing frequency, while negative feedback autaptic current promotes its excitability and is revealed by the fast–slow dynamic analysis. Generally, the system possesses period-adding bifurcation patterns and comb-shaped chaotic structures as demonstrated by the numerical results. Importantly, it can be confirmed that the electrical activities and multistability of the system can be accurately predicted by analyzing the global dynamic behaviors of the Hamilton energy. Furthermore, it is verified that the unidirectional coupling controller involving energy is far more efficient and consumes lower energy than electrical synaptic coupling in achieving complete synchronization with mismatched parameters. These results provide potential guidance and help for further research in computational neuroscience and the design and control of intelligent sensors.

Three-dimensional memristive Morris-Lecar model with magnetic induction effects and its FPGA implementation.

To characterize the magnetic induction flow induced by neuron membrane potential, a three-dimensional (3D) memristive Morris-Lecar (ML) neuron model is proposed in this paper. It is achieved using a memristor induction current to replace the slow modulation current in the existing 3D ML neuron model with fast-slow structure. The magnetic induction effects on firing activities are explained by the spiking/bursting firings with period-adding bifurcation and periodic/chaotic spiking-bursting patterns, and the bifurcation mechanisms of the bursting patterns are elaborated using the fast-slow analysis method to create two bifurcation sets. In particular, the 3D memristive ML model can also exhibit the homogeneous coexisting bursting patterns when switching the memristor initial states, which are effectively illustrated by the theoretical analysis and numerical simulations. Finally, a digitally FPGA-based hardware platform is developed for the 3D memristive ML model and the experimentally measured results well verify the numerical ones.

On Bifurcations Along the Spiral Organization of the Periodicity in a Hopfield Neural Network

In this work we report numerical results involving a certain Hopfield-type three-neurons network, with the hyperbolic tangent as the activation function. Specifically, we investigate a place of a two-dimensional parameter-space of this system where typical periodic structures, the so-called shrimps, are embedded in a chaotic region. We show that these structures are organized themselves as a spiral that coil up toward a focal point, while undergo period-adding bifurcations. We also indicate the locations along this spiral in the parameter-space, where such bifurcations happen.

Bifurcation analysis of mixed-mode oscillations and Farey trees in an extended Bonhoeffer–van der Pol oscillator

The two-variable Bonhoeffer–van der Pol oscillator, which is equivalent to the FitzHugh–Nagumo model, can be represented by a natural circuit, i.e., a circuit consisting only of simple two-terminal elements. An extended Bonhoeffer–van der Pol (BVP) oscillator is a circuit extended to a three-variable system from a two-variable BVP oscillator; this is obtained by adding an inductor–resistor branch to the original BVP oscillator. The extended BVP oscillator is known to generate mixed-mode oscillations. In this study, we classify the bifurcations of the simple sequences and their daughters, which constitute asymmetric Farey trees. Because the extended BVP oscillator is an extremely simple circuit, the parents–daughter processes can be quite precisely explained. We confirm that simple sequences 1s (s≥0) are born via saddle–node bifurcations and can be basic parents, which correspond to each stable fixed point of a Poincaré return map, where 1s represents one large excursion followed by a number s of small peaks. The two basic parents 1s and 1s+1 generate daughters [1s+1,1s×n] sequentially for the successive values of n via mixed-mode oscillation-incrementing bifurcations. The terms [1s+1,1s×n] indicate that 1s+1 is followed by 1sn-times. The daughters are born in a similar fashion to period-adding bifurcations generated by the circle map, and the parents–daughter processes satisfy Farey arithmetic and are terminated by a saddle–node bifurcation through which 1s appears. We create multiple two-parameter bifurcation diagrams and confirm that these phenomena are stably observed in these diagrams over broad ranges, i.e., complex bifurcations, such as codimension-two bifurcations or cusps, do not appear in the diagrams. Furthermore, the theoretical results are confirmed using laboratory measurements and experiments.

Firing patterns and bifurcation analysis of Purkinje cell dendrite model

This paper mainly studied firing patterns and related bifurcations in the Purkinje cell dendrite model. Based on the methods of equivalent potentials and time scale analysis, the initial six-dimensional (6D) dendrite model is reduced to a 3D form to facilitate the calculation. We numerically show that the dendrite model could exhibit period-adding bifurcation and four bursting patterns for several vital parameters. Then the bifurcation mechanisms and transition of these four bursting patterns are discussed by phase plane analysis, and two-parameter bifurcation analysis of the fast subsystem, respectively. Moreover, we computed the first Lyapunov coefficient to determine the stability of Hopf bifurcation. Ultimately, we analyzed the codimension-two bifurcation of the whole system and gave a detailed theoretical derivation of the Bogdanov–Takens bifurcation.

Bifurcation Analysis of an Ecological System with State-Dependent Feedback Control and Periodic Forcing

By assuming a periodic variation in the intrinsic growth rate of the prey, a nonlinear ecological system with periodic forcing and state-dependent feedback control is proposed. The main purpose of the present paper is to study the dynamical behavior generated by periodic forcing and nonlinear impulse perturbations and their effects on pest control. To do this, we first investigate the existence and stability of the boundary periodic solution, and then we employ the numerical bifurcation techniques, mainly including one-dimensional and two-dimensional parameter bifurcation analyses, to reveal that the system exhibits rich and complex dynamic behaviors. Especially, period-adding bifurcation with chaos is found in the two-parameter bifurcation plane. Moreover, we find the periodic structure similar to Arnold tongues, and they are arranged according to the sequence of a Farey tree. In addition, one-dimensional bifurcation diagrams reveal the existence of order-[Formula: see text] periodic, and chaotic solutions, multiple coexisting attractors, period-doubling bifurcations, period-halving bifurcations, and so on. Finally, the effects of the initial population density of pests and natural enemies on the pulse frequency and the biological significance related to the numerical results are studied and discussed.

Nested sequences of period-adding stability phases in a CO2 laser map proxy

• Nested sequences of period-adding bifurcations found in Henon map. • Period-adding sequences exist in the Hamiltonian limits of the map. • Adding cascade crisscross the whole control parameter space. We report the discovery of wide nested sequences of period-adding complexification routes spawning the whole control parameter space of the Hénon map, a discrete-time proxy of single-mode loss-modulated CO 2 lasers. For a realistic map, the new adding routes found reproduce analogous phenomena previously observed in continuous-time differential equations describing, for instance, oscillatory phases in an enzyme reaction model. In contrast to differential equations, the map is analytically tractable to a large extent, and provides a prototypic framework to investigate intricate features of generic dissipative flows.

Period-doubling route to mixed-mode chaos.

Mixed-mode oscillations (MMOs) are a complex dynamical behavior in which each cycle of oscillation consists of one or more large amplitude spikes followed by one or more small amplitude peaks. MMOs typically undergo period-adding bifurcations under parameter variation. We demonstrate here, in a set of three identical, linearly coupled van der Pol oscillators, a scenario in which MMOs exhibit a period-doubling sequence to chaos that preserves the MMO structure, as well as period-adding bifurcations. We characterize the chaotic nature of the MMOs and attribute their existence to a master-slave-like forcing of the inner oscillator by the outer two with a sufficient phase difference between them. Simulations of a single nonautonomous oscillator forced by two sine functions support this interpretation and suggest that the MMO period-doubling scenario may be more general.

Dynamic response of the e-HR neuron model under electromagnetic induction

Due to the fluctuation of membrane potential of neuron, there are complex time-varying electromagnetic fields in nervous systems, and the exciting electromagnetic field will further regulate the discharge activities of neurons. In this paper, the coupling of magnetic flux variables to the membrane potential is realised by using a magnetron memristor, and then a 5D extended Hindmarsh–Rose (e-HR) neuron model is established. With the help of Matcont software, the distribution and bifurcation properties of the equilibrium point in the e-HR model is analysed. It is found that there are subcritical Hopf bifurcation, coexisting oscillation modes and hidden limit cycle attractors with period 1 and period 2. In addition, by applying the washout controller, the subcritical Hopf bifurcation point can be transformed into the supercritical Hopf bifurcation point. Thus, the hidden oscillation behaviour of the model can be effectively eliminated. In order to analyse the influence of various parameters on the bifurcation behaviour, numerical simulation of two-parameter bifurcation, single-parameter bifurcation, maximum Lyapunov exponential and time response are given. It is found that the e-HR neuron has a complex bifurcation structure, i.e., the bifurcation structure with period-doubling bifurcations, inverse period-doubling bifurcations, period-adding bifurcations with and without chaos. At the same time, the study also finds that the coexistence behaviour of the periodic cluster discharge and the mixed-mode oscillations (MMOs) can be observed from the bifurcation structure with unique ‘periodic dislocation layer’ on the two-parameter plane. Interestingly, the bursting mode of the system is converted into MMOs when the system parameters are randomly perturbed. The results of this study provide useful insights into the complex discharge patterns and hidden discharge behaviours of neurons under electromagnetic induction.

Discrete Modeling and Period-Adding Bifurcation of DC–DC Converter Feeding Constant Power Load

In this work, an including the switched evolution process discrete model of a DC–DC converter feeding a switched constant power load is proposed based on the Poincare mapping rule and matrix exponent method. Numerical simulations of the bifurcation scheme based on the proposed discrete model reveal the period-adding bifurcation structures of the system as multiple devils’ staircases. Further, the calculation of the stable regions of the multiple-period orbits of the nested period-adding bifurcation structures are illustrated. The experimental results verify the validity of the discrete model and analytical accuracy of the period-adding bifurcation characteristics of the system. The proposed model solves the difficulties associated with discrete modeling for power electronic nonlinear switched systems. The introduction of the switched evolution process in the proposed discrete model provides a specific mathematical model for studying the influence of the switched evolution process on the dynamics of power electronic nonlinear switched systems. This study is expected to be a foundation for the theory and further research on such power electronic nonlinear switched systems.

Chaos in the peroxidase-oxidase oscillator.

The peroxidase-oxidase (PO) reaction involves the oxidation of reduced nicotinamide adenine dinucleotide by molecular oxygen. When both reactants are supplied continuously to a reaction mixture containing the enzyme and a phenolic compound, the reaction will exhibit oscillatory behavior. In fact, the reaction exhibits a zoo of dynamical behaviors ranging from simple periodic oscillations to period-doubled and mixed mode oscillations to quasiperiodicity and chaos. The routes to chaos involve period-doubling, period-adding, and torus bifurcations. The dynamic behaviors in the experimental system can be simulated by detailed semiquantitative models. Previous models of the reaction have omitted the phenolic compound from the reaction scheme. In the current paper, we present new experimental results with the oscillating PO reaction that add to our understanding of its rich dynamics, and we describe a new variant of a previous model, which includes the chemistry of the phenol in the reaction mechanism. This new model can simulate most of the experimental behaviors of the experimental system including the new observations presented here. For example, the model reproduces the two main routes to chaos observed in experiments: (i) a period-doubling scenario, which takes place at low pH, and a period-adding scenario involving mixed mode oscillations (MMOs), which occurs at high pH. Our simulations suggest alternative explanations for the pH-sensitivity of the dynamics. We show that the MMO domains are separated by narrow parameter regions of chaotic behavior or quasiperiodicity. These regions start as tongues of secondary quasiperiodicity and develop into strange attractors through torus breakdown.

Firing patterns and bifurcation analysis of neurons under electromagnetic induction

<p style='text-indent:20px;'>Based on the three-dimensional endocrine neuron model, a four-dimensional endocrine neuron model was constructed by introducing the magnetic flux variable and induced current according to the law of electromagnetic induction. Firstly, the codimension-one bifurcation and Interspike Intervals (ISIs) analysis were applied to study the bifurcation structure with respect to external stimuli and parameter <inline-formula><tex-math id="M1">$ k_0 $</tex-math></inline-formula>, and two dynamical behaviors were found: period-adding and period-doubling bifurcation leading to chaos. Besides, Hopf bifurcation was specially discussed corresponding to the transformation of the state. Secondly, the different firing patterns such as regular bursting, subthreshold oscillations, fast spiking, mixed-mode oscillations (MMOs) etc. can be observed by changing the external stimuli and the induced current. The neuron model presented more firing activities under strong coupling strength. Finally, the codimension-two bifurcation analysis shown more details of bifurcation. At the same time, the Bogdanov-Takens bifurcation point was also analyzed and three bifurcation curves were derived.</p>