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The present manuscript relies on the companion paper entitled “Memristor Circuits: Flux–Charge Analysis Method,” which has introduced a comprehensive analysis method to study the nonlinear dynamics of memristor circuits in the flux–charge $(\varphi ,q)$ –domain. The Flux–Charge Analysis Method is based on Kirchhoff Flux and Charge Laws and constitutive relations of circuit elements in terms of incremental fluxes and incremental charges. The straightforward application of the method has previously provided a full portrait of the nonlinear dynamics and bifurcations of the simplest memristor circuit composed by a capacitor and a flux-controlled memristor. This paper aims to show that the method is effective to analyze nonlinear dynamics and bifurcations in memristor circuits with more complex dynamics including Hopf bifurcations (originating persistent oscillations) and period–doubling cascades (leading to chaotic behavior). One key feature of the method is that it makes clear how initial conditions give rise to bifurcations for an otherwise fixed set of circuit parameters. To the best of the authors’ knowledge, these represent the first results that relate such bifurcations, which are referred to in the paper as Bifurcations without Parameters, with physical circuit variables as the initial conditions of dynamic circuit elements. [ABSTRACT FROM AUTHOR]

This paper is concerned with a method for design and analysis of specific neuronal networks, called central pattern generators (CPGs), which produce primary rhythmic patterns in animals. In particular, the paper is focused on synthetic CPGs made up of few basic elements and governing quadrupeds’ gaits and gait transitions, under the control of an external drive. The method combines the principles of bifurcation theory, geometric properties of symmetry, and numerical analysis based on the recently proposed toolbox CEPAGE. The method is applied to two CPGs, one bio-inspired and one purely synthetic. In both the cases, the method provides a way to obtain a desired sequence of gaits by continuously changing a bifurcation parameter related to the external drive. [ABSTRACT FROM AUTHOR]

This paper aims to introduce a technique that utilizes both dynamical mechanisms and machine learning to reduce dimensionality in high-dimensional complex systems. Specifically, the method employs Hopf bifurcation theory to establish a model paradigm and use machine learning to train location parameters. The effectiveness of the proposed method is evaluated by testing the Van Der Pol equation and it is found that it possesses good predictive ability. In addition, simulation experiments are conducted using a hunting motion model, which is a well-known practice in high-speed rail, demonstrating positive results. To ensure the robustness of the proposed method, we tested it on noisy data. We introduced simulated Gaussian noise into the original dataset at different signal-to-noise ratios (SNRs) of 10 db, 20 db, 30 db, and 40 db. All data and codes used in this paper have been uploaded to GitHub. [ABSTRACT FROM AUTHOR]

Three types of bifurcations (instabilities) in the DC-DC converter under valley current mode control (VCMC) are analyzed, which is an extension of a recent paper on the peak current mode control (PCMC). The three bifurcations are border-collision, period-doubling, and saddle-node bifurcations. The corresponding critical (instability) conditions are derived. The derived conditions divide the parameter space into different operating modes. Contrary to the general belief that the critical conditions for VCMC and PCMC are symmetric with respect to the duty cycle D=1/2, most critical conditions are actually asymmetric. The dynamics symmetry is converter, bifurcation, loading, and parameter dependent, verified by bifurcation diagrams and time-domain simulations. The VCMC boost converter has rich dynamics and it is extensively analyzed. [ABSTRACT FROM AUTHOR]

This paper develops and validates a comprehensive and universally applicable computational concept for solving nonlinear differential equations (NDEs) through a neurocomputing concept based on cellular neural networks (CNNs). High-precision, stability, convergence, and lowest-possible memory requirements are ensured by the CNN processor architecture. A significant challenge solved in this paper is that all these cited computing features are ensured in all system-states (regular or chaotic ones) and in all bifurcation conditions that may be experienced by NDEs.One particular quintessence of this paper is to develop and demonstrate a solver concept that shows and ensures that CNN processors (realized either in hardware or in software) are universal solvers of NDE models. The solving logic or algorithm of given NDEs (possible examples are: Duffing, Mathieu, Van der Pol, Jerk, Chua, Rössler, Lorenz, Burgers, and the transport equations) through a CNN processor system is provided by a set of templates that are computed by our comprehensive templates calculation technique that we call nonlinear adaptive optimization. This paper is therefore a significant contribution and represents a cutting-edge real-time computational engineering approach, especially while considering the various scientific and engineering applications of this ultrafast, energy-and-memory-efficient, and high-precise NDE solver concept. For illustration purposes, three NDE models are demonstratively solved, and related CNN templates are derived and used: the periodically excited Duffing equation, the Mathieu equation, and the transport equation. [ABSTRACT FROM PUBLISHER]

Purpose The purpose of the paper is to summarize recent achievements and suggest further research directions in numerical studies of round free jets with particular attention on the influence of the inlet parameters (mean velocity, turbulence intensity, length and time scales) on the jet dynamics.Design/methodology/approach The large eddy simulation (LES) and direct numerical simulation (DNS) are regarded as accurate tools which can support expensive and requiring sophisticated measurements techniques experimental studies. In the paper, the authors present challenges and recent findings related to the LES and DNS of jet type flows in isothermal, heated, excited and reactive conditions.Findings LES of the isothermal jet allowed to identify the new jet instability mechanism leading to the self-sustained oscillations and to determine conditions required to trigger this phenomenon. Numerical simulation on the low-density round jet captured the phenomenon of absolute instability with a very good agreement with the experimental findings. LES/DNS of excited jet exhibited bifurcating and blooming jet and showed that the jet can be directly controlled by excitation frequency what is crucial issue also for flame shape control.Originality/value The paper shows complexity of seemingly simple jet type flow and proves that despite a huge interest in these flows and relatively deep knowledge on the jet dynamics there are still some open issues requiring further studies. [ABSTRACT FROM AUTHOR]

• A new car-following model is proposed to describe the dynamic behavior of autonomous vehicle. • The bifurcation characteristic of traffic flow composed of autonomous vehicle is analyzed. • A controller that considers multi-step memory and multi-step prediction effect is designed. • A hybrid control strategy is proposed, which combines multi-step prediction and memory mechanism with PID. This paper is committed to capturing the dynamic behaviors of homogenous flow of autonomous vehicles (AVs), and exploring the control strategies to improve traffic conditions, which can alleviate traffic congestion and improve traffic efficiency. Firstly, a car-following model of AVs considering real-time driving state is established. Secondly, based on bifurcation theory and stability theory, bifurcation analysis is carried out and the relationship between bifurcation and stability is revealed. In order to suppress the bifurcation and improve the stability, a controller considering multi-step prediction and memory mechanism (MPM) is designed, and the root trajectories for eigenvalues and stable time length of the model controlled by MPM controller are calculated. In response to the limitations of the MPM controller, a hybrid controller including the MPM controller and PID controller is further proposed, and it is found that the model controlled by hybrid controller has greater range of stable bifurcation parameter and stable time length, which means better ability of bifurcation suppression. Finally, the capabilities of the controller proposed in this paper are effectively demonstrated by numerical experiments in MATLAB and simulation experiments in the ROS-Gazebo environment. [ABSTRACT FROM AUTHOR]

The use of resonant injection-locked frequency dividers in frequency synthesizers has increased in recent years due to their lower power consumption compared to conventional digital prescalers. Numerous circuit ideas have been proposed, but there are few and sometimes contradictory estimates of the locking ranges (LRs) in these dividers. Despite several attempts, there is still no accurate analytical method to predict the LR. In this paper, we present a nonlinear approach based on the application of bifurcation theory to predict where the LR is wider rather than to say precisely how wide it is. The results presented have been derived by numerical bifurcation analysis using AUTO and have been verified by circuit simulations and experiments. In Part II, we will show how the insights developed in this paper lead to a rigorous design methodology. [ABSTRACT FROM AUTHOR]

S-type locally-active memristor (LAM) has a great potential for brain- inspired neuromorphic computing, where the S-type LAM-based oscillator is a fundamental building block. Concerning the S-type LAM, this paper constructs a material-independent model in simple mathematical expression, which can be relatively easily analyzed. By biasing the memristor into the locally- active region, and connecting it with a capacitor, a second-order oscillator can be built. The small-signal equivalent circuit of the memristor and its frequency response are applied to determine the period oscillation frequency range and compensation capacitance. Hopf bifurcation theory is used to analyze oscillation mechanism of the second-order circuit and appropriate capacitance. By adding an extra inductor into the second-order oscillator, a novel third-order chaotic circuit is developed, where a saddle-focus is derived to create chaos. Its dynamic characteristics are investigated via Lyapunov exponents, bifurcation diagrams, dynamic route map, and so on. The local activities of the single memristor, second-order oscillator, and third-order chaotic circuit are verified through the mathematical analysis. Finally, physical circuit realizations of the S- type LAM-based oscillators, including the memristor emulator, are presented. Both simulation and experimental results demonstrate the practicability of the proposed mathematical model and the validity of the theoretical analysis. [ABSTRACT FROM AUTHOR]

This paper studies the impacts of stochastic load fluctuations, namely the fluctuation intensity and the load power variation speed, on power system dynamic voltage stability. Additionally, the trade-off relationship between the two parameters is revealed, which provides important insights regarding the potential of using energy storage to maintain voltage stability under high uncertainty. To this end, Stochastic Differential-Algebraic Equations (SDAEs) are used to model the stochastic load variation; bifurcation analysis is carried out to explain the influence of stochasticity. Numerical study and Monte Carlo simulations on the IEEE 14-bus system demonstrate that a larger fluctuation intensity or a slower load power variation speed may lead to a smaller voltage stability margin. To the best of authors’ knowledge, this work uncovers the impacts of the time evolution property of the driving parameters, i.e., the load power variation speed and its trade off effect with the fluctuation intensity on the size of the dynamic voltage stability margin. [ABSTRACT FROM AUTHOR]

This paper presents a COordinate Rotation DIgital Computer (CORDIC) based Adaptive Exponential Integrate and Fire (AdEx) neuron for efficient large scale biological neural network implementation. The accuracy of the modified model is investigated by both calculating various errors and bifurcation analysis; both show that the proposed model follows the same signaling, dynamical behavior, and bifurcation pattern as the original model. Network behavior of the original and proposed models are also observed to be very much alike in following the same activity patterns. The model is hardware synthesized and implemented on FPGA as a proof of concept. Measurement results show that the proposed model can reproduce neuronal behaviors similar to the original model. Hardware device utilization and speed also confirm the efficiency of the realized hardware compared with previous works. [ABSTRACT FROM PUBLISHER]

An intuitive approach to analyze the behavior of an LC Injection-Locked Frequency Divider was presented in Part I of this work; that paper provided insight into the locking behavior in the valid design area of the circuit. In this paper, we present a rigorous design methodology which provides a closed form equation showing where the locking range is wider. Theoretical predictions of the locked regions are verified by simulations of the circuit in Spectre RF using 0.35-\mu \m CMOS technology models. [ABSTRACT FROM PUBLISHER]

The problem of stabilization of unknown steady states of chaotic systems by means of linear controllers is studied. A complete solution of the problem is given for the case of state feedback stabilization. For the case of output feedback stabilization, some partial results are given. The paper shows that the control theoretic concepts of controllability, stabilizability and root-locus are instrumental in the solution of the problem. All results are illustrated by means of the stabilization of unknown steady states of a controlled Lorenz system. [ABSTRACT FROM AUTHOR]

This paper describes a method for analyzing the bifurcation phenomena in switched dynamical systems whose switching borders are varying periodically with time. The type of systems under study coven most of power electronics circuits where two or more dynamical systems are cyclically switched according to the interaction of the state variables and some periodically moving borders. In particular, the complex bifurcation behavior of a voltage feedback buck converter is studied in detail. The analytical method developed in this paper allows bifurcation scenarios to be clearly revealed in any chosen parameter space. [ABSTRACT FROM AUTHOR]

In this paper, a discrete-time seasonally forced SIR epidemic model is investigated for different types of bifurcations. Although, many researchers already suggested numerically that this model can exhibit chaotic dynamics but not much focus is given to the bifurcation theory of the model. We prove analytically and numerically the existence of different types of bifurcations in the model. First, the one and two parameters bifurcations of this model are investigated by computing their critical normal form coefficients. Secondly, the flip, Neimark–Sacker, and strong resonances bifurcations are detected for this model. The critical coefficients identify the scenario associated with each bifurcation. The complete complex dynamical behavior of the model is investigated. Some graphical representations of the model are presented to verify the obtained results. [ABSTRACT FROM AUTHOR]

Many dynamical systems, including power systems, recover from perturbations more slowly as they approach critical transitions—a phenomenon known as critical slowing down. If the system is stochastically forced, autocorrelation and variance in time-series data from the system often increase before the transition, potentially providing an early warning of coming danger. In some cases, these statistical patterns are sufficiently strong, and occur sufficiently far from the transition, that they can be used to predict the distance between the current operating state and the critical point. In other cases CSD comes too late to be a good indicator. In order to better understand the extent to which CSD can be used as an indicator of proximity to bifurcation in power systems, this paper derives autocorrelation functions for three small power system models, using the stochastic differential algebraic equations (SDAE) associated with each. The analytical results, along with numerical results from a larger system, show that, although CSD does occur in power systems, its signs sometimes appear only when the system is very close to transition. On the other hand, the variance in voltage magnitudes consistently shows up as a good early warning of voltage collapse. [ABSTRACT FROM AUTHOR]

Three-phase power-factor-correction (PFC) power supplies are commonly used to convert ac power from a three-phase grid to a regulated dc voltage with unity input power factor. The output voltage regulation is normally achieved by an outer voltage feedback loop and a sinusoidal pulse-width-modulated (SPWM) inner current loop. However, the non-ideal power grid may drive the converter to enter a low-frequency instability region. In this paper, a low-frequency instability phenomenon in three-phase PFC power supplies is reported. The converter can also be regarded as exhibiting a Hopf-type bifurcation in which the dc output voltage oscillates at 150 Hz, and a large amount of harmonics appear on the line current. We develop an averaged model of the grid-converter system to predict the low-frequency instability. The effects of circuit and grid parameters can be studied with this model. Additionally, as a result of the emergence of low-frequency oscillation, the stability boundary leading to a catastrophic bifurcation changes dramatically. Simulation results also provide design-oriented boundaries of operation. This phenomenon is confirmed by experimental measurement. [ABSTRACT FROM PUBLISHER]

As the backbone of social development, college students' level of physical exercise has always been the focus of research by experts and scholars. Most of the research methods are on the strength of literature data, questionnaire survey, mathematical statistics and comparative analysis. Based on the classification of college students and the influence and flow law of inter-class population, this paper establishes a differential equation system. By analyzing the existence and stability of the equilibrium of this system and the possible fold or backward bifurcations at the equilibrium, the quantitative analysis of college students' physical exercise trends on campus is carried out. This paper aims to improve the participation of college students in physical exercise by maximizing the number of students in the third categories. The results of theoretical proof, sensitivity analysis and numerical simulation show that in the initial stage, promoting peer-to-peer communication is the most effective measure. Secondly, when the effect of peer-to-peer interaction reaches saturation point, the way to improve physical education can achieve significant results. To fundamentally improve the enthusiasm of college students to participate in sports activities, we should start from the level of consciousness and enhance students' awareness of physical exercise from an early age. [ABSTRACT FROM AUTHOR]

This paper finds and investigates the application of fixed-event-time based discretized sliding mode (DSM) controller in dc-dc buck converter, to achieve fast transient response and high robustness under wide parameters variation. We show that how these can be achieved by integrating the concept of Utkin's equivalent control law and discontinuous border-collision bifurcation (DBCB) theory developed for 2-D discontinuous piecewise smooth (PWS) maps. Moreover, based on derived 2-D discontinuous maps of DSM-controlled converter, we investigate its inherent steady-state dynamical properties or bifurcation behaviors under different parameters variation. Numerically as well as experimentally obtained bifurcation diagrams are then presented to show the domains of existence of different oscillatory modes and their sequence of occurrence. Such phenomena are not only useful to study the robustness of the system but may also facilitates to design the input filter with fast transient response. The performance of DSM controller is experimentally verified and compared with hysteresis and classical peak current-mode controller. [ABSTRACT FROM PUBLISHER]

Standalone photovoltaic-battery hybrid power systems are attractive renewable power generation systems, a popular form of which is based on a photovoltaic system and a battery connected to an output dc bus via a buck converter and a bidirectional buck/boost converter, respectively. Due to variation of the available sunlight intensity, the battery voltage and load condition, the system's structures and operating modes are switched from time to time. The dynamic behavior is thus quite complex, and the design for stable operation of the system requires consideration of the stability conditions for all possible operating modes. This paper studies the dynamic behavior of this system and reveals the smooth and non-smooth bifurcation phenomena. Under certain conditions, when the system switches its operating mode, a non-smooth bifurcation, manifested as a jump from stable to unstable behavior, has been observed and verified with full-circuit simulations. Moreover, a detailed analysis based on averaged modes is performed to identify the two types of bifurcation and evaluate the stability boundaries of the system. The results provide useful insights and information about the behavior of the system and the interacting effects of control parameters on the design of the control loops. [ABSTRACT FROM PUBLISHER]

In this paper, we present a formal analysis and description of the steady-state behavior of an electrostatic vibration energy harvester operating in constant-charge mode and using different types of electromechanical transducers. The method predicts parameter values required to start oscillations, allows a study of the dynamics of the transient process, and provides a rigorous description of the system, necessary for further investigation of the related nonlinear phenomena and for the optimisation of converted power. We show how the system can be presented as a nonlinear oscillator and be analysed by the multiple scales method, a type of perturbation technique. We analyse two the most common cases of the transducer geometry and find the amplitude and the phase of steady-state oscillations as functions of parameters. The analytical predictions are shown to be in good agreement with the results obtained by behavioral modeling. [ABSTRACT FROM PUBLISHER]

Total ampere-turns of the primary and secondary windings of a transformer are taken as one of variables to analyze the bifurcation behaviors and the border collision phenomenon on voltage-mode-controlled flyback converter. Six operation modes which are separated by the two borderlines of capacitor voltage and the three borderlines of total ampere-turns are obtained. A class of explicit piecewise smooth discrete-time maps with six operation modes is derived to describe the dynamics of the system. The numerical calculations are carried out, the bifurcation diagrams and the maximal Lyapunov exponent spectrums with the control period as a parameter are obtained. Then, the paper particularly analyzes the stability and existence conditions of main periodic orbits. The structure of bifurcation diagrams and the characteristics of operation modes at the border collision points are studied too. Finally, PSIM simulation and experimental results testify the validity of theoretical analysis. [ABSTRACT FROM PUBLISHER]

Purpose – An image contrast enhancement is one of the most important low-level image pre-processing tasks required by the vision-based advanced driver assistance systems (ADAS). This paper seeks to address this important issue keeping the real time constraints in focus, which is especially vital for the ADAS. Design/methodology/approach – The approach is based on a paradigm of nonlinear-coupled oscillators in image processing. Each layer of the colored images is treated as an independent grayscale image and is processed separately by the paradigm. The pixels with the lowest and the highest gray levels are chosen and their difference is enhanced to span all the gray levels in an image over the entire gray level range, i.e. [0 1]. This operation enhances the contrast in each layer and the enhanced layers are finally combined to produce a color image of a much improved quality. Findings – The approach performs robust contrast enhancement as compared to other approaches available in the relevant literature. Generally, other approaches do need a new setting of parameters for every new image to perform its task, i.e. contrast enhancement. These approaches are not useful for real-time applications such as ADAS. Whereas, the proposed approach presented in this paper performs contrast enhancement for different images under the same setting of parameters, hence giving rise to the robustness in the system. The unique setting of parameters is derived through a bifurcation analysis explained in the paper. Originality/value – The proposed approach is novel in different aspects. First, the proposed paradigm comprises of coupled differential equations, and therefore, offers a continuous model as opposed to other approaches in the relevant literature. This continuity in the model is an inherent feature of the proposed approach, which could be useful in realizing real-time image processing with an analog implemented circuit of the approach. Furthermore, a novel framework combining coupled oscillatory paradigm and cellular neural network is also possible to achieve ultra-fast solution in image contrast enhancement. [ABSTRACT FROM AUTHOR]

This paper proposes a small-signal model for variable-switching-frequency hysteretic converters operating in continuous conduction mode (CCM) and discontinuous conduction mode (DCM). The model is capable of predicting flip-bifurcation phenomena when the hysteretic converter operates below the CCM/DCM boundary. By developing a discrete-time state-space model of the converter, the z-domain model is derived using the advanced z-transform. Such a tool is suitable to transform in the z-domain signals varying within a switching cycle. A comparison with exact simulations is provided to validate the proposed model. The propagation delay of the feedback control circuit is included due to its significant influence on the low-frequency open-loop gain. Experimental results run on a test chip including a 10-MHz TSMC 0.18\ \mu\m-process hysteretic-mode buck converter are also reported along with the model results. [ABSTRACT FROM PUBLISHER]

In this paper, the slow-scale instability in a boost PFC converter under average current mode control is studied via the harmonic balance method and Floquet theory. Systematic and general numerical algorithm of the technique is developed. The objective is to identify instability of the system. If instability exists, the type of bifurcation and the boundaries of slow-scale instability in the parameter space of the system can be found for facilitating the design of such converters. Firstly, based on the fact that the current compensator is designed very well and what we considered is the slow-scale instability, the simplified model of a boost PFC converter under average current mode control is derived. And then, the solution of the simplified model is calculated by using the harmonic balance method. Subsequently, both the stability of the circuit system and the type of bifurcation are identified via Floquet theory. Meanwhile, the critical conditions of the stable operation in the parameter space of the system are given accurately. Finally, experimental results are presented for the verification of the analytical results. [ABSTRACT FROM AUTHOR]

A single-inductor two-input two-output power electronic dc-dc converter can be used to regulate two generally nonsymmetric positive and negative outputs by means of a pulsewidth modulation with a double voltage feedback. This paper studies the dynamic behavior of this system. First, the operation modes and the steady-state properties of the converter are addressed, and, then, a stability analysis that includes both the power stage and control parameters is carried out. Different bifurcations are determined from the averaged model and from the discrete-time model. The Routh-Hurwitz criterion is used to obtain the stability regions of the averaged (slow-scale) dynamics in the design parameter space, and a discrete-time approach is used to obtain more accurate results and to detect possible (fast-scale) subharmonic oscillations. Experimental measurements were taken from a system prototype to confirm the analytical results and numerical simulations. Some possible nonsmooth bifurcations due to the change in the switching patterns are also illustrated. [ABSTRACT FROM AUTHOR]

This paper investigates the coexisting fast-scale and slow-scale bifurcations in simple dc/dc converters under peak current-mode control operating in continuous conduction mode. Our focus is the boost converter as it is a representative form of dc/dc converter requiring current-mode control. Effects of varying the input voltage and some chosen parameters on the qualitative behavior of the system are studied in detail. Analysis based on a nonlinear simplified discrete-time model, which takes into account the effects of parasitics, is performed to investigate the coexistence of fast-scale and slow-scale bifurcations, and to identify the different types of bifurcation. Boundaries of stable region, slow-scale bifurcation region, fast-scale bifurcation region, coexisting fast and slow-scale bifurcation region are identified. Experimental measurements of the boost converter are provided for verification of the analytical results. [ABSTRACT FROM AUTHOR]

The dynamics of a zero-average dynamic strategy controlled dc-dc Buck converter, modelled by a set of differential equations with discontinuous right-hand side is studied. Period-doubling and corner-collision bifurcations are found to occur close to each other under small parameter variations. Closer examination of the parameter space leads to the discovery of a novel bifurcation. This type of bifurcation has not been reported so far in the literature and it corresponds to a corner-collision bifurcation of a nonhyperbolic cycle. The bifurcation boundaries are computed analytically in this paper and the system dynamics are unfolded close to the novel bifurcation point. [ABSTRACT FROM AUTHOR]

To study the stability of a nominal cyclic steady state in power electronic converters, it is necessary to obtain a linearization around the periodic orbit. In many past studies, this was achieved by explicitly deriving the Poincaré map that describes the evolution of the state from one clock instant to the next and then locally linearizing the map at the fixed point. However, in many converters, the map cannot be derived in closed form, and therefore this approach cannot directly be applied. Alternatively, the orbital stability can be worked out by studying the evolution of perturbations about a nominal periodic orbit, and some studies along this line have also been reported. In this paper, we show that Filippov's method—which has commonly been applied to mechanical switching systems—can be used fruitfully in power electronic circuits to achieve the same end by describing the behavior of the system during the switchings. By combining this and the Floquet theory, it is possible to describe the stability of power electronic converters. We demonstrate the method using the example of a voltage-mode-controlled buck converter operating in continuous conduction mode. We find that the stability of a converter is strongly dependent upon the so-called saltation matrix—the state transition matrix relating the state just after the switching to that just before. We show that the Filippov approach, especially the structure of the saltation matrix, offers some additional insights on issues related to the stability of the orbit, like the recent observation that coupling with spurious signals coming from the environment causes intermittent subharmonic windows. Based on this approach, we also propose a new controller that can significantly extend the parameter range for nominal period-1 operation. [ABSTRACT FROM AUTHOR]

This paper introduces a novel fourth-order double-torus chaotic circuit. Based on this basic circuit, a systematic theoretical design approach is proposed for generating 1-D n torus, 2-D n x m-torus, 3-D n x rn x 1 torus, and 4-D n x in x I x p torus chaotic attractors. This is the first autonomous circuit reported in the literature for generating multidirectional multi-torus (MDMT) chaotic attractors. The dynamical behav- iors of these MDMT chaotic systems are further investigated, including equilibrium points, bifurcations, Lyapunov exponents, and Poincaré maps. Theoretical analysis shows that the MDMT chaotic attractors can be generated by switching and displacing a basic linear circuit. Finally, a block circuit diagram is designed for hardware implementation of the MDMT chaotic attractors. This is also the first time in the literature to experimentally verify a maximal 1-D 20-torus, a maximal 2-D 5 x 5 torus, and a maximal 4-D 5 x 5 x 3 x 3 torus chaotic attractors. [ABSTRACT FROM AUTHOR]

This paper reports slow-scale instability in a single-stage power-factor-correction (PFC) power supply, which is a popular design solution for low power applications. The circuit employs a cascade configuration of a boost converter and a forward converter, which share an active switch and operate in discontinuous-conduction mode (DCM), to provide input PFC and tight output regulation. Main results are given by "exact" cycle-by-cycle circuit simulations. The effect of the slow-scale instability on the attainable power factor is illustrated in terms of total harmonic distortion which can be found by taking the fast Fourier transform of the input current. The slow-scale instability usually manifests itself as local oscillations within a line cycle. Based on the critical condition of DCM for the buck converter, the underlying mechanism of such instability is further investigated. It has been found that border collision is the underlying cause of the phenomenon. Moreover, it has been shown that the border collision observed here is effectively a nonsmooth Neimark-Sacker bifurcation. Finally, experimental results are presented for verification purposes. [ABSTRACT FROM AUTHOR]

Considering a pulsewidth modulated dc/dc converter as an example, this paper describes a border-collision bifurcation that can lead to the appearance of quasi-periodicity in piecewise-smooth dynamical systems. We demonstrate how a two-dimensional torus can arise from a periodic orbit through a bifurcation in which two complex-conjugate Poincaré characteristic multipliers jump abruptly from the inside to the outside of the unit circle. The torus may be ergodic or resonant. However, in both cases the diameter of the torus develops approximately linearly with the distance to the bifurcation point as opposed to the characteristic parabolic form of the well-known Neimark-Sacker bifurcation. The paper also considers the birth of a torus via a subcritical Neimark-Sacker bifurcation in the piecewise-smooth system. Particular emphasis is given to the development of resonance zones via border-collision bifurcations. [ABSTRACT FROM AUTHOR]

A systematic analysis of singular bifurcations in semistate or differential-algebraic models of electrical circuits is presented in this paper. The singularity-induced bifurcation (SIB) theorem describes the divergence of one eigenvalue through infinity when an operating point or equilibrium locus of a parameterized differential-algebraic model crosses a singular manifold. The present paper extends this result to cover situations in which several eigenvalues diverge; we prove a multiple SIB theorem which states that a minimal rank (resp. index) change makes it possible to compute the number of diverging eigenvalues in terms of an index (resp. rank) change in the matrix pencil characterizing the linearized problem. The scope of the work comprises quasi-linear ordinary differential equations, semiexplicit index-i differential-algebraic equation (DAEs), and Hessenberg index-2 DAEs, describing different electrical configurations. The electrical features from which singularities and, specifically, singular bifurcations stem are extensively discussed. Examples displaying simple, double, and triple SIB points illustrate different ways in which the spectrum may diverge. [ABSTRACT FROM AUTHOR]

In this paper, we present an efficient algorithm to compute singular points and singularity-induced bifurcation points of differential-algebraic equations for a multimachine power-system model Power systems are often modeled as a set of differential-algebraic equations (DAE) whose algebraic part brings singularity issues into dynamic stability assessment of power systems. Roughly speaking, the singular points are points that satisfy the algebraic equations, but at which the vector field is not defined. In terms of power-system dynamics, around singular points, the generator angles (the natural states variables) are not defined as a graph of the load bus variables (the algebraic variables). Thus, the causal requirement of the DAE model breaks down and it cannot predict system behavior. Singular points constitute important organizing elements of power-system DAE models. This paper proposes an iterative method to compute singular points at any given parameter value. With a lemma presented in this paper, we are also able to locate singularity induced bifurcation points upon identifying the singular points. The proposed method is implemented into voltage stability toolbox and simulations results are presented for a 5-bus and IEEE 118-bus systems. [ABSTRACT FROM AUTHOR]

Large disturbances in power systems often initiate complex interactions between continuous dynamics and discrete events. The paper develops a hybrid automaton that describes such behavior. Hybrid systems can be modeled in a systematic way by a set of differential algebraic equations, modified to incorporate impulse action and constraint switching. The paper outlines a number of inverse problems including parameter uncertainty, parameter estimation, grazing bifurcations, boundary value problems and dynamic embedded optimization.

Control of a water hyacinth-fish ecological system is required for a healthy and sustainable environment. This paper aims to investigate the global dynamics of a water hyacinth fish ecological system under ratio-dependent state impulsive control. First, we study the positivity and boundedness of the solution of the controlled system. By studying the local stability of the equilibrium, we find that the system has two situations. One is that there are two equilibria, namely a saddle point and a boundary equilibrium. In the second case, there are four equilibria, namely, two saddle points, a boundary equilibrium, and a focus point. For the first case, when we select an appropriate ratio-dependent control threshold, the trajectory will globally converge to the boundary equilibrium. For the second case, when the control line is located below the focus point, by using Poincare mapping method, flip bifurcation theory, and vector field analysis techniques, we find that the solution of the controlled system either globally converges to the boundary equilibrium, order-1 periodic solution, or order-2 periodic solution under certain conditions. When the control line is located above the focus point, the solution of the controlled system either globally converges to the focus point, order-1 or order-2 periodic solution. Finally, we use examples to verify the correctness and validity of the theoretical results. [ABSTRACT FROM AUTHOR]

A standard approach in analyzing dynamical systems consists in identifying and understanding the eigenvalues bifurcations when crossing the imaginary axis. Efficient methods for crossing imaginary roots identification exist. However, to the best of the author's knowledge, the multiplicity of such roots was not deeply investigated. In recent papers by the authors [1], [2], it is emphasized that the multiplicity of the zero spectral value can exceed the number of the coupled scalar delay-differential equations and a constructive approach Vandermonde-based allowing to an adaptive bound for such a multiplicity is provided. Namely, it is shown that the zero spectral value multiplicity depends on the system structure (number of delays and number of non zero coefficients of the associated quasipolynomial) rather than the degree of the associated quasipolynomial [3]. This technical note extends the constructive approach in investigating the multiplicity of crossing imaginary roots $j\omega$ where $\omega\neq 0$ and establishes a link with a new class of functional confluent Vandermonde matrices. A symbolic algorithm for computing the LU-factorization for such matrices is provided. As a byproduct of the proposed approach, a bound sharper than the Polya-Szegö generic bound arising from the principle argument is established. [ABSTRACT FROM AUTHOR]

This work reveals and analyzes an unfamiliar bifurcation in weakly-coupled inductive wireless power transfer systems (IWPT) employing diode rectifier. This paper firstly provides a local analysis calculating the required RF source power, based on the rectifier ac-dc conversion gain and the two-port analysis of the coupled coils. The local analysis is supported by measured results on silicon. Successively, we demonstrate that if the secondary coil and the diode rectifier have a high quality factor, the rectifier diode varactor can create a turning-point bifurcation in the solution curve of the output dc voltage versus the RF source power. With the bifurcation, the local analysis becomes invalid and a higher source power is required. Bifurcation analysis based on large-signal capacitor modeling and conversion matrix analysis are provided, substantiating the results. The introduced bifurcation analysis is verified by three IWPT examples on PCB. A proposed design with an auxiliary nonlinear varactor is proposed to mitigate the bifurcation with improved performance. [ABSTRACT FROM AUTHOR]

This paper focuses on the species interaction within a tri-trophic food chain model involving species' mutual interference, characterized by Crowley–Martin functional response with strong Allee effect in prey species. First, the evaluation of the solutions' positivity and boundedness is conducted. Subsequently, the local stability of equilibrium points is analysed, and the Lyapunov function is employed to assess the global stability of interior equilibrium. We have also established adequate criteria for detecting local bifurcations. Sotomayor's theorem infers the presence of saddle–node and transcritical bifurcations. Utilizing the centre manifold theory and the Hopf bifurcation theorem, the emergence and stability of Hopf bifurcation are investigated. Following this, numerical simulations are performed so as to validate the theoretical findings and analyse the impacts of Allee effect in greater details. The analysis reveals that the system undergoes a range of local bifurcations in co-dimension one, including saddle–node, subcritical and supercritical Hopf bifurcations. Furthermore, the existence of global dynamics can be observed by the appearance of Generalized-Hopf bifurcation in the co-dimension two bifurcation diagram. The findings suggest that strong Allee effect could trigger instability and induce a bi-stability phenomenon in the system's dynamical behaviours. • A tri-trophic food chain model with a strong Allee effect is considered. • The occurrence of transcritical and saddle–node bifurcations is derived using Sotomayor's theorem. • The analysis of intricate dynamical features is conducted by means of co-dimension one and two bifurcations. • The model exhibits diverse behaviours and bi-stability phenomena at various Allee thresholds. • Prey with strong Allee effect have a higher chance of survival. [ABSTRACT FROM AUTHOR]

Reaction–diffusion equations serve as fundamental tools in describing pattern formation in biology. In these models, nonuniform steady states often represent stationary spatial patterns. Notably, these steady states are not unique, and unveiling them mathematically presents challenges. In this paper, we introduce a framework based on bifurcation theory to address pattern formation problems, specifically examining whether nonuniform steady states can bifurcate from trivial ones. Furthermore, we employ linear stability analysis to investigate the stability of the trivial steady-state solutions. We apply the method to two classic reaction–diffusion models: the Schnakenberg model and the Gray–Scott model. For both models, our approach effectively reveals many nonuniform steady states and assesses the stability of the trivial solution. Numerical computations are also presented to validate the solution structures for these models. • Nonuniform steady states correspond to stationary spatial patterns. • Most nonuniform steady states bifurcate from trivial steady-state solutions. • Bifurcation theory can be used to find biological patterns. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the various responses in cancer immune therapy through bifurcation theory. Our results characterize the influence of key parameters on different treatment responses and illustrate the response regions through bifurcation diagrams using parameters tuned by therapy. In particular, we examine a periodic outcome with a delayed therapy response followed by a cancer relapse. The repetitive pattern in this periodic solution is formed by two quasi-steady cancer states connected by two fast transitions. Our results suggest that the transient behavior of the delayed response, which is the fast transition after the cancer progression state, is induced by a periodic solution exhibiting an exploding period near a saddle–node bifurcation. Another transient behavior, which shows as a fast relapse after a long period of dormancy state, is caused by the slow change near a saddle-type equilibrium. Our findings have the potential to guide and inform future therapeutic research. • Various responses in cancer immune therapy are studied through bifurcation theory. • Key parameters on different treatment responses are characterized. • Different response regions are illustrated through bifurcation diagrams. • Transient dynamics in delayed responses are explained mathematically. • Our findings have the potential to guide and inform future therapeutic research. [ABSTRACT FROM AUTHOR]

Highlights • The analysis of multiple spatial limit sets and chaos existence in MIMO cascade nonlinear systems consisting of large number of cascades (subsystems) using theorems is done in this paper. • It is shown that based on bifurcation diagrams of these systems the appearance of multiple spatial limit sets and chaos can be analyzed, also. • It should be noticed that in the case of MIMO system with large number of cascades monitoring dynamics of the whole system, using bifurcation diagram, is based on monitoring of all outputs of the last cascade x (k + 1) = [ x 1 (k + 1) , ... , x n (k + 1) ] as a function of control parameter l n. • Bifurcation diagram simulated in this way completelly shows the dynamics of the whole system. This is confirmed comparing segments of bifurcation diagram with phase portraits. Abstract The appearance of multiple spatial limit sets and chaos in MIMO cascade nonlinear systems is analyzed in this paper. The necessary and sufficient conditions for the existence of stable multiple spatial limit sets and the rules for the existence of chaos are given, too. The validity of these conditions is confirmed with bifurcation diagrams and spatial phase portraits of MIMO2 and MIMO4 cascade nonlinear systems. [ABSTRACT FROM AUTHOR]

Highlights • Turing instability condition for a Gierer–Meinhardt system of discrete time and space form are obtained. • Coupled map lattice is applied to study the spatiotemporal dynamics. • Hopf, flip bifurcation to chaos, together with Turing bifurcation are coupled to account for the complex patterns. • Striking patterns such as circle, spiral, mosaic and chaos are shown by numerical simulation. Abstract This paper is concerned with the spatiotemporal behaviors of a Gierer–Meinhardt system in discrete time and space form. Through the linear stability analysis, the parametric conditions are gained to ensure the stability of the homogeneous steady state of the system. Based on the bifurcation theory, as well as center manifold theorem, we derive the critical parameter values of the flip, Neimark–Sacker and Turing bifurcation respectively. Besides, the specific parameter expression to form patterns are also determined. In order to identify chaos among regular behaviors, we calculate the Maximum Lyapunov exponents. The results obtained in this paper are illustrated by numerical simulations. From the simulations, we can see some complex dynamics, such as period doubling cascade, invariant cycles, periodic windows, chaotic behaviors, and some striking Turing patterns, e.g. circle, mosaic, spiral, spatiotemporal chaotic patterns and so on, which can be produced by flip-Turing instability, Neimark–Sacker–Turing instability and chaos. [ABSTRACT FROM AUTHOR]

Stochastic bifurcation theory plays an important role in stochastic dynamical systems, thus, in this paper, we mainly consider the stochastic bifurcation of a stochastic logistic model with distributed delay in the weak kernel case, where the birth rate of species is disturbed by white noise. In order to study the bifurcation of stochastic logistic system, we use the intrinsic growth rate of species as a bifurcation parameter. Firstly, we study the stochastic D-bifurcation and stochastic P-bifurcation for stochastic logistic model with distributed delay. Furthermore, by deriving the corresponding Fokker–Planck equation, we obtain the exact expression of the joint density function of the stochastic system near the positive equilibrium point. Finally, we give some conclusions. [ABSTRACT FROM AUTHOR]

Purpose When a reflected shock interacts with the boundary layer in a shock tube, the shock bifurcation occurs near the walls. Although the study of the shock bifurcation has been carried out by many researchers for several decades, little attention has been devoted to investigate the instability pattern of the bifurcation. This research work aims to successfully capture the asymmetry of the whole flow field, and attempt to achieve the instability mechanism of the shock bifurcation by a direct numerical simulation of the reflected shock wave/boundary layer interaction at Ma = 1.9. In addition, the reason for the formation of the bifurcated structure is also explored.Design/methodology/approach The spatial and temporal evolution of the shock bifurcation is obtained by solving the two-dimensional compressible Navier–Stokes equations using a seventh-order accurate weighted essentially non-oscillatory (WENO) scheme and a three-step Runge–Kutta time advancing approach.Findings The results show that the formation of shock bifurcation is mainly because of the shock/gradient field interaction, and the height of the bifurcated foot increases with the growth of the shock intensity and the gradient field. The unsteady asymmetry of the upper and bottom shock bifurcated structures is because of the vortex shedding with high frequency in the rear recirculation zone, which leads to the fluctuation of the recirculation area. The vortex shedding process behind the bifurcated structure closely resembles the Karman vortex street formed by the flow around the cylinder. The dimensionless vortex shedding frequency varies between 0.01 and 0.02. In comparison to the scenario at Ma = 1.9, the occurring time of instability is delayed and the upper and bottom bifurcated feet intersect in a relatively short time at Ma = 3.5. The region behind the bifurcated shock is a transitional flow field containing obvious cell structures and “isolated islands.”Originality/value This paper discovers an unsteady flow pattern of the shock bifurcation, and the mechanism of this instability in the reflected shock/boundary layer interaction is revealed in detail. [ABSTRACT FROM AUTHOR]

This paper analyses a dynamic model of Cournot–Stackelberg which involves a feedback regulation system based on the costs of production. The manufacturer makes decisions based on costs of production and wholesale prices as the leader of the market, while the retailers make decisions according to order amount. The decision-making process of the system is divided into two stages in order to carry out the research more clearly and the doubling period bifurcation will occur with speed of adjusting decision-making increase. It is proved that the stability of the model can be considered separately into two decision-making processes, the cross-effects of the two decision-making stages are analyzed. The manufacturer's decision-making has a decisive influence on the stability of the whole system when the manufacturer is a marketing leader and the parameters also have a significant effect on the stability of the system. Finally, the paper proposes a chaos control method based on adaptive method, which is inspired by decision-making methods. The result of the paper has great significance to the Cournot–Stackelberg game with variable costs and master-slave status. [ABSTRACT FROM AUTHOR]

In this paper we study the phenomenon of localization of deformation in fault gouges during seismic slip. This process is of key importance to understand frictional heating and energy budget during an earthquake. A infinite layer of fault gouge is modeled as a Cosserat continuum taking into account Thermo-Hydro-Mechanical (THM) couplings. The theoretical aspects of the problem are presented in the companion paper (Rattez et al., 2017a), together with a linear stability analysis to determine the conditions of localization and estimate the shear band thickness. In this Part II of the study, we investigate the post-bifurcation evolution of the system by integrating numerically the full system of non-linear equations using the method of Finite Elements. The problem is formulated in the framework of Cosserat theory. It enables to introduce information about the microstructure of the material in the constitutive equations and to regularize the mathematical problem in the post-localization regime. We emphasize the influence of the size of the microstructure and of the softening law on the material response and the strain localization process. The weakening effect of pore fluid thermal pressurization induced by shear heating is examined and quantified. It enhances the weakening process and contributes to the narrowing of shear band thickness. Moreover, due to THM couplings an apparent rate-dependency is observed, even for rate-independent material behavior. Finally, comparisons show that when the perturbed field of shear deformation dominates, the estimation of the shear band thickness obtained from linear stability analysis differs from the one obtained from the finite element computations, demonstrating the importance of post-localization numerical simulations. [ABSTRACT FROM AUTHOR]

This paper investigates the orbital stability and instability of solitary waves for the generalized Boussinesq equation with two nonlinear terms. Firstly, according to the theory of Grillakis–Shatah–Strauss orbital stability, we present the general results to judge orbital stability of the solitary waves. Further, we deduce the explicit expression of discrimination d ′′( c ) to judge the stability of the two solitary waves, and give the stable wave speed interval. Moreover, we analyze the influence of the interaction between two nonlinear terms on the stable wave speed interval, and give the maximal stable range for the wave speed. Finally, some conclusions are given in this paper. [ABSTRACT FROM AUTHOR]

We study the dynamics of a model of interdependent consumer behavior defined by a family of two-dimensional noninvertible maps. This family belongs to a class of coupled logistic maps with different nonlinearity parameters and coupling terms that depend on one variable only. In our companion paper we considered the case of independent consumers as well as the case of uni -directionally connected consumers. The present paper aims at describing the dynamics in the case of a bi -directional connection. In particular, we investigate the bifurcation structure of the parameter plane associated with the strength of coupling between the consumers, focusing on the mechanisms of qualitative transformations of coexisting attractors and their basins of attraction. [ABSTRACT FROM AUTHOR]

Persistence of solitary wave solutions of the regularized long wave equation with small perturbations are investigated by the geometric singular perturbation theory and bifurcation theory. Two different kinds of the perturbations are considered in this paper: one is the Kuramoto–Sivashinsky perturbation, the other is the Marangoni effects. Indeed, the solitary wave persists under small perturbations. Furthermore, the different perturbations do affect the proper wave speed c ensuring the persistence of the solitary waves. Finally, numerical simulations are utilized to confirm the theoretical results. [ABSTRACT FROM AUTHOR]