International Journal of Bifurcation and Chaos

The stochastically forced three-dimensional Hindmarsh–Rose model of neural activity is considered. We study the effect of random disturbances in parametric zones where the deterministic model exhibits mono- and bistable dynamic regimes with period-adding bifurcations of oscillatory modes. It is shown that in both cases the phenomenon of noise-induced bursting is observed. In the monostable zone, where the only attractor of the system is a stable equilibrium, this effect is connected with a stochastic generation of large-amplitude oscillations due to the high excitability of the model. In a parametric zone of coexisting stable equilibria and limit cycles, bursts appear due to noise-induced transitions between the attractors. For a quantitative analysis of the noise-induced bursting and corresponding stochastic bifurcations, an approach based on the stochastic sensitivity function (SSF) technique is applied. Our estimations of the strength of noise that generates such qualitative changes in stochastic dynamics are in a good agreement with the direct numerical simulation. A relationship of the noise-induced generation of bursts with transitions from order to chaos is discussed.

The paper completes the cycle of the research devoted to the development of the experimental bifurcation analysis (not computer simulations) in order to answer the following questions: whether qualitative changes occur in the dynamics of local climate systems in a centennial timescale?; how to analyze such qualitative changes with daily resolution for local and regional space-scales?; how to establish one-to-one daily correspondence between the dynamics evolution and economic consequences for productions? To answer the questions, the unconventional conceptual model to describe the local climate dynamics was proposed and verified in the previous parts. That model (HDS-model) originates from the hysteresis regulator with double synchronization and has a variable structure due to competition between the amplitude quantization and the time quantization. The main advantage of the HDS-model is connected with the possibility to describe “internally” (on the basis of the self-regulation) the specific causal effects observed in the dynamics of local climate systems instead of “external” description of three states of the hysteresis behavior of climate systems (upper, lower and transient states). As a result, the evolution of the local climate dynamics is based on the bifurcation diagrams built by processing the data of meteorological observations, where the strange effects of the essential interannual daily variability of annual temperature variation are taken into account and explained. It opens the novel possibilities to analyze the local climate dynamics taking into account the observed resultant of all internal and external influences on each local climate system. In particular, the paper presents the viewpoint on how to estimate economic damages caused by climate-related hazards through the bifurcation analysis. That viewpoint includes the following ideas: practically each local climate system is characterized by its own time pattern of the natural qualitative changes in temperature dynamics over a century, so, any unified time window to determine the local climatic norms seems to be questionable; the temperature limits determined for climate-related technological hazards should be reasoned by the conditions of artificial human activity, but not by the climatic norms; the damages caused by such hazards can be approximately estimated in relation to the average annual profit of each production. Now, it becomes possible to estimate the minimal and maximal numbers of the specified hazards per year in order, first of all, to avoid unforeseen latent damages. Also, it becomes possible to make some useful relative estimation concerning damage and profit. We believe that the results presented in the cycle illustrate great practical competence of the current advances in the experimental bifurcation analysis. In particular, the developed QHS-analysis provides the novel prospects towards both how to adapt production to climatic changes and how to compensate negative technological impacts on environment.

Using a systematic computer search, four simple chaotic flows with cubic nonlinearities were found that have the unusual feature of having a curve of equilibria. Such systems belong to a newly introduced category of chaotic systems with hidden attractors that are important and potentially problematic in engineering applications.

We devote to studying the problem for the existence of homoclinic and heteroclinic orbits of Unified Lorenz-Type System (ULTS). Other than the known results that the ULTS has two homoclinic orbits to $E_0=(0,0,0)$ for $b=-2a_1$, $d=-a_1$, $a_1^2+a_{2}c>0$, $e<0$ and two heteroclinic orbits to $E_{1,2}=(\pm \sqrt{-\frac{2(a_1^2+a_{2}c)}{e}},\mp \frac{a_1}{a_2}\sqrt{-\frac{2(a_1^2+a_{2}c)}{e}},-\frac{a_1^2+a_{2}c}{a_{2}e})$ for $b=-2a_1$, $d=-a_1$, $a_1^2+a_{2}c<0$, $e>0$ on its invariant algebraic surface $Q(x,y,z)=z-\frac{x^2}{2a_2}=0$, formulated in the literature by Yang and Chen [2014], we seize two new heteroclinic orbits of this Unified Lorenz-Type System. Namely, we rigorously prove that this system has another two heteroclinic orbits to $E_0$ and $E_{\pm }=(\pm \sqrt{\frac{b(a_{2}c-a_{1}d)}{a_{1}e}},\mp \frac{a_1}{a_2}\sqrt{\frac{b(a_{2}c-a_{1}d)}{a_{1}e}},\frac{a_{1}d-a_{2}c}{a_{2}e})$ while no homoclinic orbit when $a_1<0$, $e<0$, $a_1+d<0$, $a_2\ne 0$, $a_{2}c-a_{1}d>0$, $b+2a_1\ge 0$.

We study the $(1+1)$-dimensional generalized cubic-quintic nonlinear Schrödinger equation (GCQ–NLSE) with two nonlinear terms that is in cubic-quintic formulation and the general format of polytropic approximation. We find the analytical solution of the GCQ–NLSE through the $F$-expansion method without utilizing any integrability condition. We apply the analytical results to the concrete system that supports dark soliton and reach the analytical formula for the sound speed based on the results from the theoretical treatment of GCQ–NLSE. We show that our derived results match pretty well with measured sound velocity values in the off-center regions of the BCS–BEC crossover regime under the same parametric setting, demonstrating the applicability of our theoretical treatment.

An adaptive training procedure is developed for a network of electronic neurons, which controls a mobile robot driving around in an unknown environment while avoiding obstacles. The neuronal network controls the angular velocity of the wheels of the robot based on the sensor readings. The nodes in the neuronal network controller are clusters of neurons rather than single neurons. The adaptive training procedure ensures that the input–output behavior of the clusters is identical, even though the constituting neurons are nonidentical and have, in isolation, nonidentical responses to the same input. In particular, we let the neurons interact via a diffusive coupling, and the proposed training procedure modifies the diffusion interaction weights such that the neurons behave synchronously with a predefined response. The working principle of the training procedure is experimentally validated and results of an experiment with a mobile robot that is completely autonomously driving in an unknown environment with obstacles are presented.

We propose a kind of delayed diffusive intraguild predation model with Holling II functional response in this paper. By analyzing the eigenvalue spectrum, it is found that the stability or instability of equilibria can be induced by delay. By utilizing the local bifurcation theory of partial functional differential equations, Hopf bifurcation of the proposed system with time delay as bifurcation parameter is investigated. It reveals that the time delay has a destabilizing effect in the intraguild predation model dynamics and a phenomenon of Hopf bifurcation occurs when the delay increases through a certain threshold. Then we give the explicit formulas to determine the direction, stability of Hopf bifurcation by utilizing the normal form method and center manifold reduction for PFDEs. Numerical simulations are performed to illustrate our theoretical results and show that delay and diffusion can induce the system into chaos and even trigger the emergence of different types of spatial patterns, including spiral wave pattern and chaotic wave pattern, which are induced by Hopf instability.

In this paper, we investigate the global dynamics of a memristor oscillator

This paper discusses the fast-scale instability in the one-cycle controlled H-bridge inverter which is widely used in practical applications. Simulations show that the fast-scale instability which occurs around the peak value of the inductive current is a type of local instability. The discrete-time model is derived to analyze the fast-scale instability. According to the Jacobian matrix derived from the discrete-time model, the theoretical analysis shows that the fast-scale instability manifests itself as a period-doubling bifurcation occurs, which reveals the intrinsic mechanism of the instability of one-cycle controlled H-bridge inverter. Furthermore, with the approximation of the matrix exponential in the discrete-time model and based on the Jacobian matrix, the stability boundary is obtained by the analytical expression which is helpful to select the proper values of parameters for avoiding the fast-scale instability. The experiments are carried out to validate the results of the simulation and theoretical analysis. It is shown that the analysis method is able to facilitate the design of the inverter.

In recent years, memristor has become one of the research focuses. Considerable open literature has shown that many devices and systems possess memristive properties. Thus, research into supplying power source to these special devices and making them work steadily has been a meaningful study. This paper investigates the slow-scale instability whose oscillating frequency is between the line frequency and the switch frequency that occurs in a voltage controlled H-bridge inverter with memristive load. The mechanism of this medium-frequency oscillation is identified via harmonic balance method and Floquet theory. The results show that it is due to the occurrence of Hopf bifurcation. Finally, experimental results are presented to verify the theoretical analysis.

In this paper, one-cycle compensation (OCC) method is proposed to realize adaptive control of fast-scale bifurcation in the peak current controlled buck-boost inverter because the proposed control method can adjust the slope of the integrator’s output voltage automatically through extracting a sinusoidal signal from the absolute value of the reference voltage. In order to reveal their underlying mechanisms of fast-scale bifurcations, a modified averaged model which can capture the sample-and-hold effect is derived in detail to describe the fast-scale dynamics of the buck-boost inverter. Based on the proposed model, a theoretical analysis is performed to identify both the fast-scale period-doubling bifurcation and the fast-scale Hopf one by judging in what way the poles loci move. It has been shown that the OCC method can be used not only to discover the unknown dynamical behaviors (i.e. fast-scale Hopf bifurcation), but also to enlarge the stable region in peak current controlled buck-boost inverter. In addition, the critical bifurcation angles and the parameter behavior boundary are given to verify the effectiveness of the adaptive bifurcation control method. Finally, PSpice circuit experiments are performed to verify the above theoretical and numerical results.

The speech-evoked auditory brainstem response (sABR) shows how complex sounds such as speech and music are processed in the auditory system. Speech-ABR could be used to evaluate particular impairments and improvements in auditory processing system. Many researchers used linear approaches for characterizing different components of sABR signal, whereas nonlinear techniques are not applied so commonly. The primary aim of the present study is to examine the underlying dynamics of normal sABR signals. The secondary goal is to evaluate whether some chaotic features exist in this signal. We have presented a methodology for determining various components of sABR signals, by performing Ensemble Empirical Mode Decomposition (EEMD) to get the intrinsic mode functions (IMFs). Then, composite multiscale entropy (CMSE), the largest Lyapunov exponent (LLE) and deterministic nonlinear prediction are computed for each extracted IMF. EEMD decomposes sABR signal into five modes and a residue. The CMSE results of sABR signals obtained from 40 healthy people showed that 1st, and 2nd IMFs were similar to the white noise, IMF-3 with synthetic chaotic time series and 4th, and 5th IMFs with sine waveform. LLE analysis showed positive values for 3rd IMFs. Moreover, 1st, and 2nd IMFs showed overlaps with surrogate data and 3rd, 4th and 5th IMFs showed no overlap with corresponding surrogate data. Results showed the presence of noisy, chaotic and deterministic components in the signal which respectively corresponded to 1st, and 2nd IMFs, IMF-3, and 4th and 5th IMFs. While these findings provide supportive evidence of the chaos conjecture for the 3rd IMF, they do not confirm any such claims. However, they provide a first step towards an understanding of nonlinear behavior of auditory system dynamics in brainstem level.

A seasonally forced predator–prey system with generalized Holling type IV functional response is considered in this paper. The influence of seasonal forcing on the system is investigated via numerical bifurcation analysis. Bifurcation diagrams for periodic solutions of periods one and two, containing bifurcation curves of codimension one and bifurcation points of codimension two, are obtained by means of a continuation technique, corresponding to different bifurcation cases of the unforced system illustrated in five bifurcation diagrams. The seasonally forced model exhibits more complex dynamics than the unforced one, such as stable and unstable periodic solutions of various periods, stable and unstable quasiperiodic solutions, and chaotic motions through torus destruction or cascade of period doublings. Finally, some phase portraits and corresponding Poincaré map portraits are given to illustrate these different types of solutions.

This paper deals with the limit cycle bifurcations for piecewise smooth Hamiltonian systems. By using the first order Melnikov function of piecewise near-Hamiltonian systems given in [Liu & Han, 2010], we give a lower bound and an upper bound of the number of limit cycles that bifurcate from the period annulus between the center and the generalized eye-figure loop up to the first order of Melnikov function.

This paper is concerned with how the symmetry and singularity of a system of differential equations affect generic dynamics and bifurcations. By computation of Hopf-pitchfork point in a two-parameter nonlinear problem satisfying a $D_3$-symmetry condition, the mode interactions in two-parameter bifurcations with a single zero and two pairs of imaginary roots are considered. The codimension two normal form with equivariant Hopf-pitchfork bifurcations are given. Through analyzing the unfolding structure, local classification in the neighborhood of equivariant Hopf-pitchfork bifurcations point for the $D_3$-symmetric is undertaken. A rich variety of dynamical and bifurcation behaviors is pointed out. Beyond a stable fixed point or a pair of stable fixed points, some interesting phenomena are also found, such as the coexistence of two periodic solutions which are verified both theoretically and numerically.

This paper studies a new class of three-dimensional maps in a Jerk-like structure with a special concern of their hidden chaotic dynamics. Our investigation focuses on the hidden chaotic attractors in three typical scenarios of fixed points, namely no fixed point, single fixed point, and two fixed points. A systematic computer search is performed to explore possible hidden chaotic attractors, and a number of examples of the proposed maps are used for demonstration. Numerical results show that the routes to hidden chaotic attractors are complex, and the basins of attraction for the hidden chaotic attractors could be tiny, so that using the standard computational procedure for localization is impossible.

In this paper, a shallow water wave model is used to introduce the concepts of peakon, periodic peakon and compacton. Traveling wave solutions of the shallow water equation are presented. The corresponding traveling wave system is a singular planar dynamical system with one singular straight line. By using the method of dynamical systems, bifurcation diagrams and explicit exact parametric representations of the solutions are given, including solitary wave solution, periodic wave solution, peakon solution, periodic peakon solution and compacton solution under different parameter conditions.

The dynamical properties of a kind of Lorenz-type systems with time-varying parameters are studied. The time-varying ultimate bounds are estimated, and a simple method is provided to generate strange attractors, including both the strange nonchaotic attractors (SNAs) and chaotic attractors. The approach is: (i) take an autonomous system with an ultimate bound from the Lorenz family; (ii) add at least two time-varying parameters with incommensurate frequencies satisfying certain conditions; (iii) choose the proper initial position and time by numerical simulations. Three interesting examples are given to illustrate this method with computer simulations. The first one is derived from the classical Lorenz model, and generates an SNA. The second one generates an SNA or a Lorenz-like attractor. The third one exhibits the coexistence of two SNAs and a Lorenz-like attractor. The nonautonomous Lorenz-type systems present more realistic models, which provide further understanding and applications of the numerical analysis in weather and climate predication, synchronization, and other fields.

Based on our previous works on liquid crystal (LC) microlenses driven electrically, we present a new type of dual-mode liquid crystal microlens arrays (DLCMAs) for chaotic encryption applications. Currently, the DLCMAs developed by us consist of a top electrode couple constructed by two layers of controlling electrode and a bottom planar electrode. Aluminium and Indium-Tin Oxide (ITO) materials are respectively deposited over both sides of a glass substrate for shaping the top electrode couple, which is used to act as a key mode-control-part in the DLCMAs. Another ITO layer is deposited over the surface of another glass substrate for shaping the bottom public electrode. Both glass substrates with fabricated electrode structures are coupled into a microcavity fully filled by a layer of nematic liquid crystal materials. The DLCMAs proposed in this paper present excellent beam divergence and light convergence performances through loading relatively low driving voltage signals. The common optical properties of the devices, leading to a type of optical modulator of chaotic beams or light intensity adjustment devices for chaotic light coupling between functioned components, are demonstrated experimentally.

Based on the polarization switching mechanism in an optically injected vertical-cavity surface-emitting laser (VCSEL), and relying on the new electro-optic modulation theory, we propose a novel approach to implement optoelectronic logic gates. The two linear-polarization (LP) lights from the output of the laser are considered as two logic outputs. Under the electro-optic modulation, one of the logic outputs is the NOT operation with the other one. With the same logic input signal, we perform various digital signal processing (AND, OR, XNOR, NAND, NOR and XOR) in the optical domain, controlling the logic operations of the applied electric field between the two logic input signals. On this basis, the logic operation of half-adder is further implemented.