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In this paper, the qualitative behaviors of an important four-dimensional Lorenz system with wild pseudohyperbolic attractor that proposed in (Gonchenko et al 2021 Nonlinearity 34 2018–47) are considered. Here, we prove that the four-dimensional Lorenz system with varying parameters is global bounded according to Lyapunov's direct method. Furthermore, we provide a collection of global absorbing sets, where in addition we obtain the rate of the trajectories going from the exterior to the global absorbing set. In particular, we solve the critical case k → 0 + that cannot be resolved by using the previous methods. The fundamental qualitative behaviors are analyzed theoretically and numerically. We present bifurcation diagrams to further explore the complicated dynamical behaviors of this system. The period-doubling bifurcation phenomenon is found. To illustrate the efficiency of our method, we present numerical simulations to show the validity of our research results. Finally, we present some applications of our research results in this paper. [ABSTRACT FROM AUTHOR]

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

This study proposes a novel fractional discrete-time macroeconomic system with incommensurate order. The dynamical behavior of the proposed macroeconomic model is investigated analytically and numerically. In particular, the zero equilibrium point stability is investigated to demonstrate that the discrete macroeconomic system exhibits chaotic behavior. Through using bifurcation diagrams, phase attractors, the maximum Lyapunov exponent and the 0–1 test, we verified that chaos exists in the new model with incommensurate fractional orders. Additionally, a complexity analysis is carried out utilizing the approximation entropy (ApEn) and C 0 complexity to prove that chaos exists. Finally, the main findings of this study are presented using numerical simulations. [ABSTRACT FROM AUTHOR]

In their seminal paper (2008 Chaos 18 037113), Ott and Antonsen showed that large groups of phase oscillators driven by a certain type of common force display low dimensional long-term dynamics, which is described by a small number of ordinary differential equations. This fact was later used as a simplifying reduction technique in many studies of synchronisation phenomena occurring in networks of coupled oscillators and in neural networks. Most of these studies focused mainly on partially synchronised states corresponding to the equilibrium-type dynamics in the so called Ott–Antonsen manifold. Going beyond this paradigm, here we propose a universal approach for the efficient analysis of partially synchronised states with non-equilibrium periodic collective dynamics. Our method is based on the observation that the Poincaré map of the complex Riccati equation, which describes the dynamics in the Ott–Antonsen manifold, coincides with the well-known Möbius transformation. To illustrate the possibilities of our method, we use it to calculate a complete bifurcation diagram of travelling chimera states in a ring network of phase oscillators with asymmetric nonlocal coupling. [ABSTRACT FROM AUTHOR]

Memristor chaotic systems have aroused great attention in recent years with their potentials expected in engineering applications. In this paper, a five-dimension (5D) double-memristor hyperchaotic system (DMHS) is modeled by introducing two active magnetron memristor models into the Kolmogorov-type formula. The boundness condition of the proposed hyperchaotic system is proved. Coexisting bifurcation diagram and numerical verification explain the bistability. The rich dynamics of the system are demonstrated by the dynamic evolution map and the basin. The simulation results reveal the existence of transient hyperchaos and hidden extreme multistability in the presented DMHS. The NIST tests show that the generated signal sequence is highly random, which is feasible for encryption purposes. Furthermore, the system is implemented based on a FPGA experimental platform, which benefits the further applications of the proposed hyperchaos. [ABSTRACT FROM AUTHOR]

We present a class of two-dimensional memristive maps with a cosine memristor. The memristive maps do not have any fixed points, so they belong to the category of nonlinear maps with hidden attractors. The rich dynamical behaviors of these maps are studied and investigated using different numerical tools, including phase portrait, basins of attraction, bifurcation diagram, and Lyapunov exponents. The two-parameter bifurcation analysis of the memristive map is carried out to reveal the bifurcation mechanism of its dynamical behaviors. Based on our extensive simulation studies, the proposed memristive maps can produce hidden periodic, chaotic, and hyper-chaotic attractors, exhibiting extremely hidden multi-stability, namely the coexistence of infinite hidden attractors, which was rarely observed in memristive maps. Potentially, this work can be used for some real applications in secure communication, such as data and image encryptions. [ABSTRACT FROM AUTHOR]

We investigate the classical equilibrium properties and metamorphosis of the ground state of interacting dipoles with fixed locations on a helix. The dipoles are shown to align themselves along separate intertwined dipole chains forming single, double, and higher-order helical chains. The number of dipole chains, and their properties such as chirality and length scale on which the chains wind around each other, can be tuned by the geometrical parameters. We demonstrate that all possible configurations form a self-similar bifurcation diagram which can be linked to the Sternâ€"Brocot tree and the underlying Farey sequence. We describe the mechanism responsible for this behavior and subsequently discuss corresponding implications and possible applications. [ABSTRACT FROM AUTHOR]

We consider a Hamiltonian system of coupled Van der Polâ€"Duffing (VdPD) oscillators with balanced loss and gain. The system is analyzed perturbatively by using renormalization group (RG) techniques as well as multiple scale analysis (MSA). Both the methods produce identical results in the leading order of the perturbation. The RG flow equation is exactly solvable and the slow variation of amplitudes and phases in time can be computed analytically. The system is analyzed numerically and shown to admit periodic solutions in regions of parameter-space, confirming the results of the linear stability analysis and perturbation methods. The complex dynamical behavior of the system is studied in detail by using time-series, PoincarĂ©-sections, power-spectra, auto-correlation function and bifurcation diagrams. The Lyapunov exponents are computed numerically. The numerical analysis reveals chaotic behaviour in the system beyond a critical value of the parameter that couples the two VdPD oscillators through linear coupling, thereby providing yet another example of Hamiltonian chaos in a system with balanced loss and gain. Further, we modify the nonlinear terms of the model to make it a non-Hamiltonian system of coupled VdPD oscillators with balanced loss and gain. The non-Hamiltonian system is analyzed perturbatively as well as numerically and shown to posses regular periodic as well as chaotic solutions. It is seen that the P T -symmetry is not an essential requirement for the existence of regular periodic solutions in both the Hamiltonian as well as non-Hamiltonian systems. [ABSTRACT FROM AUTHOR]

A novel memristor-based multi-scroll hyperchaotic system is proposed. Based on a voltage-controlled memristor and a modulating sine nonlinear function, a novel method is proposed to generate the multi-scroll hyperchaotic attractors. Firstly, a multi-scroll chaotic system is constructed from a three-dimensional chaotic system by designing a modulating sine nonlinear function. Then, a voltage-controlled memristor is introduced into the above-designed multi-scroll chaotic system. Thus, a memristor-based multi-scroll hyperchaotic system is generated, and this hyperchaotic system can produce various coexisting hyperchaotic attractors with different topological structures. Moreover, different number of scrolls and different topological attractors can be obtained by varying the initial conditions of this system without changing the system parameters. The Lyapunov exponents, bifurcation diagrams and basins of attraction are given to analyze the dynamical characteristics of the multi-scroll hyperchaotic system. Besides, the field programmable gate array (FPGA) based digital implementation of the memristor-based multi-scroll hyperchaotic system is carried out. The experimental results of the FPGA-based digital circuit are displayed on the oscilloscope. [ABSTRACT FROM AUTHOR]

Synchronization refers to the adjustment of rhythms of two nonidentical oscillators due to interaction. In this study, we report for the first time on the ability of computational fluid dynamics (CFD) to simulate the synchronization of self-sustained acoustic oscillations in two coupled thermoacoustic (TA) oscillators that interfere with each other through acoustic radiation. A bifurcation diagram containing both asynchronous and synchronous states is first mapped out by changing the end-to-end distance and the resonator length difference. It is found that the two TA oscillators are analogous to mutually coupled van der Pol oscillators with reactive coupling. Then, the dynamic characteristics of beating and periodic oscillations in asynchronous and synchronous states are analysed separately to facilitate the comprehension of the synchronization process. This research demonstrates that the CFD methodology provides a valuable numerical tool for studying the synchronization of thermally induced acoustic oscillations in TA systems. [ABSTRACT FROM AUTHOR]

One-dimensional maps with discontinuities are known to exhibit bifurcations somewhat different to those of continuous maps. Freed from the constraints of continuity, and hence from the balance of stability that is maintained through fold, flip, and other standard bifurcations, the attractors of discontinuous maps can appear as if from nowhere, and change period or stability almost arbitrarily. But in fact this is misleading, and if one includes states inside the discontinuity in the map, highly unstable 'hidden orbits' are created that have iterates on the discontinuity. These populate the bifurcation diagrams of discontinuous maps with just the necessary unstable branches to make them resemble those of continuous maps, namely fold, flip, and other familiar bifurcations. Here we analyse such bifurcations in detail, focussing first on folds and flips, then on bifurcations characterised by creating infinities of orbits, chaotic repellers, and infinite accumulations of sub-bifurcations. We show the role that hidden orbits play, and how they capture the topological structures of continuous maps with steep branches. This suggests both that a more universal dynamical systems theory marrying continuous and discontinuous systems is possible, and shows how discontinuities can be used to approximate steep jumps in continuous systems without losing any of their topological structure. [ABSTRACT FROM AUTHOR]

Multiple steady solutions and hysteresis phenomenon in the square cavity flows driven by the surface with antisymmetric velocity profile are investigated by numerical simulation and bifurcation analysis. A high order spectral element method with the matrix-free pseudo-arclength technique is used for the steady-state solution and numerical continuation. The complex flow patterns beyond the symmetry-breaking at Re ≃ 320 are presented by a bifurcation diagram for Re < 2500. The results of stable symmetric and asymmetric solutions are consistent with those reported in literature, and a new unstable asymmetric branch is obtained besides the stable branches. A novel hysteresis phenomenon is observed in the range of 2208 < Re < 2262, where two pairs of stable and two pairs of unstable asymmetric steady solutions beyond the stable symmetric state coexist. The vortices near the sidewall appear when the Reynolds number increases, which correspond to the bifurcation of topology structure, but not the bifurcation of Navier–Stokes equations. The hysteresis is proposed to be the result of the combined mechanisms of the competition and coalescence of secondary vortices. [ABSTRACT FROM AUTHOR]

Three-dimensional flow past surface mounted elliptic cylinder has been investigated numerically using OpenFOAM. Axis ratio (AR) of the elliptic cylinder is varied from 0.5 to 2. Effects of shear intensity (K) of the nonuniform incoming flow and Reynolds number (Re) varying from 150 to 250 on wake dynamics have been examined using Hilbert Huang transformation and Recurrence analysis. Qualitative and quantitative analyses of wake unsteadiness are illustrated using Hilbert spectra and degree of stationarity. Various recurrence quantification estimates have been reported to reveal insight about unsteady wake transition with the change in Re, AR and K. Four types of distinguishable wake flows, viz., steady, periodic, quasi-periodic and chaotic have been characterized with the help of vorticity contours, frequency spectra and bifurcation diagram. In aperiodic flow, wake oscillation contains multiple laminar states or slow changing states which has been disclosed using laminarity and trapping time. Nonstationarity and synchronous wake behavior along the height of the cylinder have been examined using cross-recurrence analysis and quantified in terms of recurrence rate, determinism and average diagonal length. In addition, the effect of change in K for each value of Re and AR has been studied on the mean drag coefficient of the cylinder. [ABSTRACT FROM AUTHOR]

The ability to control, manipulate, and read out nanomechanical resonators is of great significance for many applications. In this work, we start by constructing a nonlinear dynamic model that is deduced from the fundamental beam-photon–electron interaction and energy band theories, with the aim of describing a complicated cavity-free optomechanical coupling process. Based on the model established, we first reveal the manipulation of a resonator's response, including softening and hardening effects due to laser injection. By driving the laser parametrically, we comprehensively investigate the control of the resonator's dynamics, in particular, in the nonlinear regime. It is found that both the laser power and frequency can be used to directly manipulate the NEMS resonator's dynamics, e.g., by amplitude amplification, periodicity changes, and periodic–chaotic state conversion. We then provide bifurcation diagrams, which evidence a deterministic evolution of dynamics. Finally, we perform a special study of the control of chaotic states of the nanomechanical resonator using laser parametric driving. The maximal Lyapunov exponents together with time series calculation show that the chaotic states can be controlled at a few specific frequency points of the injecting laser. This work not only provides guidance for using lasers to control nanoscale resonators, but also sheds light on the exploration of novel applications based on nonlinear NEMS resonators. [ABSTRACT FROM AUTHOR]

In the paper we deal with a quintic Liénard system of the form with a -symmetry, where and a3b2 ≠ 0. A complete study of this system with b2 > 0, called the saddle case, is finished, showing that the system exhibits at most two limit cycles, and the necessary and sufficient conditions are obtained on the existence of two limit cycles and a two-saddle heteroclinic loop. We also present a global bifurcation diagram and the corresponding phase portraits of this system, including Hopf bifurcation, Bautin bifurcation, two-saddle heteroclinic loop bifurcation and double limit cycle bifurcation. [ABSTRACT FROM AUTHOR]

Tools for analyzing dynamics in a class of 2 degrees-of-freedom Hamiltonian impact systems with underlying separable integrable structure are derived. Integrable, near-integrable and far-from integrable cases are considered. In particular, a generalization of the energy momentum bifurcation diagram, Fomenko graphs and the hierarchy of bifurcations framework to this class is constructed. The projection of Liouville leaves of the smooth integrable dynamics to the configuration space allows to extend these tools to impact surfaces which produce far from integrable dynamics. It is suggested that such representations classify dynamically different regions in phase space. For the integrable and near integrable cases these provide global information on the dynamics whereas for the far from integrable regimes (caused by finite deformations of the impact surface), these provide information on the singular set and on the non-impact orbits. The results are presented and demonstrated for the Duffing-center system with impacts from a slanted wall. [ABSTRACT FROM AUTHOR]

For a continuous map on the unit interval or circle, we define the bifurcation set to be the collection of those interval holes whose surviving set is sensitive to arbitrarily small changes of (some of) their endpoints. By assuming a global perspective and focusing on the geometric and topological properties of this collection rather than the surviving sets of individual holes, we obtain a novel topological invariant for one-dimensional dynamics. We provide a detailed description of this invariant in the realm of transitive maps and observe that it carries fundamental dynamical information. In particular, for transitive non-minimal piecewise monotone maps, the bifurcation set encodes the topological entropy and strongly depends on the behavior of the critical points. [ABSTRACT FROM AUTHOR]

Based on a quadratic transformation and a switching function, a novel multi-wing chaotic switched system is proposed. First, a 4-wing chaotic system is constructed from a 2-wing chaotic system on the basis of a quadratic transformation. Then, a switching function is designed and by adjusting the switching function, the number and the distribution of the saddle-focus equilibrium points of the switched system can be regulated. Thus, a set of chaotic switched systems, which can produce 6-to-8-12-16-wing attractors, are generated. The Lyapunov exponent spectra, bifurcation diagrams, and Poincaré maps are given to verify the existence of the chaotic attractors. Besides, the digital circuit of the multi-wing chaotic switched system is designed by using the Verilog HDL fixed-point algorithm and the state machine control. Finally, the multi-wing chaotic attractors are demonstrated via FPGA platform. The experimental results show that the number of the wings of the chaotic attractors can be expanded more effectively with the combination of the quadratic transformation and the switching function methods. [ABSTRACT FROM AUTHOR]

To improve the complexity of chaotic signals, in this paper we first put forward a new three-dimensional quadratic fractional-order multi-scroll hidden chaotic system, then we use the Adomian decomposition algorithm to solve the proposed fractional-order chaotic system and obtain the chaotic phase diagrams of different orders, as well as the Lyaponov exponent spectrum, bifurcation diagram, and SE complexity of the 0.99-order system. In the process of analyzing the system, we find that the system possesses the dynamic behaviors of hidden attractors and hidden bifurcations. Next, we also propose a method of using the Lyapunov exponents to describe the basins of attraction of the chaotic system in the matlab environment for the first time, and obtain the basins of attraction under different order conditions. Finally, we construct an analog circuit system of the fractional-order chaotic system by using an equivalent circuit module of the fractional-order integral operators, thus realizing the 0.9-order multi-scroll hidden chaotic attractors. [ABSTRACT FROM AUTHOR]

A new method is presented to generate two-directional (2D) grid multi-scroll chaotic attractors via a specific form of the sine function and sign function series, which are applied to increase saddle points of index 2. The scroll number in the x-direction is modified easily through changing the thresholds of the specific form of the sine function, while the scroll number in the y-direction is controlled by the sign function series. Some basic dynamical properties, such as equilibrium points, bifurcation diagram, phase portraits, and Lyapunov exponents spectrum are studied. Furthermore, the electronic circuit of the system is designed and its simulation results are given by Multisim 10. [ABSTRACT FROM AUTHOR]

Localized planar patterns in spatially extended bistable systems are known to exist along intricate bifurcation diagrams, which are commonly referred to as snaking curves. Their analysis is challenging as techniques such as spatial dynamics that have been used to explain snaking in one space dimension no longer work in the planar case. Here, we consider bistable systems posed on square lattices and provide an analytical explanation of snaking near the anti-continuum limit using Lyapunov–Schmidt reduction. We also establish stability results for localized patterns, discuss bifurcations to asymmetric states, and provide further numerical evidence that the shape of snaking curves changes drastically as the coefficient that reflects the strength of the spatial coupling crosses a finite threshold. [ABSTRACT FROM AUTHOR]

The tumor suppressor p53 plays a key role in protecting genetic integrity. Its dynamics have important physiological significance, which may be related to the cell fate. Previous experiments have shown that the wild-type p53-induced phosphatase 1 (Wip1) protein could maintain p53 oscillation. Therefore, we add Wip1 to remodel the p53 network. Firstly, we use the binomial τ-leap algorithm to prove our model stable under internal noise. Then, we make a series of bifurcation diagrams, that is, p53 levels as a function of p53 degradation rate at different Wip1 generation rates. The results illustrate that Wip1 is essential for p53 oscillation. Finally, a two-dimensional bifurcation diagram is made and the stability of some p53 dynamics under external noise is analyzed by potential landscape. Our results may have some implications for artificially interfering with p53 dynamics to achieve tumor suppression. [ABSTRACT FROM AUTHOR]

The work studies the transition from periodic to chaotic oscillations in a dc-driven planar gas discharge with a high-ohmic electrode. The applied voltage and resistance of the semiconductor layer act as control parameters. The oscillations occur in the subnormal discharge regime. The bifurcation diagram and Lorenz map characterizing transition of this system to chaos through period-doubling bifurcations are obtained. Numerical models employed are based on the drift-diffusion theory of gas discharges. The effect of two modelling approaches, namely the 'simple' fluid model and the more detailed 'extended' fluid model, is considered. The calculations showed that the results obtained by these two approaches are different in quantitative terms, however, qualitatively similar in terms of the dynamic behavior of the system as a function of the control parameters. [ABSTRACT FROM AUTHOR]

The self-excited attractors and hidden attractors in a memcapacitive system which has three elements are studied in this paper. The critical parameter of stable and unstable states is calculated by identifying the eigenvalues of Jacobian matrix. Besides, complex dynamical behaviors are investigated in the system, such as coexisting attractors, hidden attractors, coexisting bifurcation modes, intermittent chaos, and multistability. From the theoretical analyses and numerical simulations, it is found that there are four different kinds of transient transition behaviors in the memcapacitive system. Finally, field programmable gate array (FPGA) is used to implement the proposed chaotic system. [ABSTRACT FROM AUTHOR]

Codimension-two bifurcations are fundamental and interesting phenomena in dynamical systems. Fold-Hopf and double-Hopf bifurcations are the most important among them. We study the unfoldings of these two codimension-two bifurcations, and obtain sufficient conditions for their nonintegrability in the meaning of Bogoyavlenskij. We reduce the problems of the unfoldings to those of planar polynomial vector fields and analyze the nonintegrability of the planar vector fields, based on Ayoul and Zung's version of the Morales–Ramis theory. New useful criteria for nonintegrability of planar polynomial vector fields are also obtained. The approaches used here are applicable to many problems including circular symmetric systems. [ABSTRACT FROM AUTHOR]

Continuing Chen and Chen (2015 Nonlinearity 28 3535) and (2016 Nonlinearity 29 1798) which deal with the cases of two equilibria and three equilibria respectively, in this paper we investigate the global dynamics of a cubic Liénard system with global parameters in the case of exact one equilibrium. After analyzing qualitative properties of all equilibria and judging the number of limit cycles, we give the bifurcation diagram and all global phase portraits. Our method in judging the number of limit cycles is to construct a parameter transformation such that in new parameter space the vector field is rotated about multiple parameters and, hence, is essential different from the methods used in previous publications. Associated with the results of last two publications, we get a positive answer to conjecture 3.2 of Khibnik et al (1998 Nonlinearity 11 1505) for general parameters about the existence of some function whose graph is exactly the surface of the double limit cycle bifurcation and therefore solve this conjecture completely. [ABSTRACT FROM AUTHOR]

We report on the experimental observation and characterization of the bifurcation diagram, dynamical properties and fluctuations of spatially modulated kinks in a shallow one-dimensional fluidized granular layer subjected to a periodic air flow. We show the appearance of these solutions as the layer undergoes a parametric instability. Due to the inherent fluctuations of the granular layer, the kink profile exhibits an effective wavelength, termed the precursor, which modulates spatially the homogeneous states and drastically modifies the kink dynamics. We characterize the average and fluctuating properties of this solution. The long-term evolution of these kinks is dominated by a hopping dynamics, related directly to the underlying spatial structure and inherent fluctuation. The properties of this motion can be described by a Brownian particle in a symmetric periodic potential. Both the noise intensity of the Brownian fluctuations and the amplitude and periodicity of the potential arise from the underlying precursor structure. [ABSTRACT FROM AUTHOR]

Neurons in the brain utilize various firing trains to encode the input signals they have received. Firing behavior of one single neuron is thoroughly explained by using a bifurcation diagram from polarized resting to firing, and then to depolarized resting. This explanation provides an important theoretical principle for understanding neuronal biophysical behaviors. This paper reports the novel experimental and modeling results of the modification of such a bifurcation diagram by adjusting small conductance potassium (SK) channel. In experiments, changes in excitability and depolarization block in nucleus accumbens shell and medium-spiny projection neurons are explored by increasing the intensity of injected current and blocking the SK channels by apamin. A shift of bifurcation points is observed. Then, a Hodgkin—Huxley type model including the main electrophysiological processes of such neurons is developed to reproduce the experimental results. The reduction of SK channel conductance also shifts the bifurcations, which is in consistence with experiment. A global bifurcation paradigm of this shift is obtained by adjusting two parameters, intensity of injected current and SK channel conductance. This work reveals the dynamics underpinning modulation of neuronal firing behaviors by biologically important ionic conductance. The results indicate that small ionic conductance other than that responsible for spike generation can modify bifurcation points and shift the bifurcation diagram and, thus, change neuronal excitability and adaptation. [ABSTRACT FROM AUTHOR]

The interesting task here is to study the frequency-current (f—I) curve and phase response curve (PRC), subject to neural temperature variations, because the f—I curve and PRC are important measurements to understand dynamical mechanisms of generation of neural oscillations, and the neural temperature is widely known to significantly affect the oscillations. Nevertheless, little is discussed about how the temperature affects the f—I curve and PRC. In this study, frequencies of the neural oscillations, modulated with the temperature variations, are quantified with an average of the PRC. The frequency gradient with respect to temperature derived here gives clear classifications of the PRC, regardless of the form. It is also indicated that frequency decreases with an increase in temperature, resulted from bifurcation switching of the saddle-homoclinic to the saddle-node on an invariant circle. [ABSTRACT FROM AUTHOR]

We investigate analytically and numerically the dynamics of the Rikitake system. The Routh—Hurwitz criterion is used to study the stability of the equilibrium points of the differential equation system model, as functions of two parameters. The dynamics of the model are numerically studied using diagrams that associate colors to the largest Lyapunov exponent value, in two-dimensional parameter spaces. Additionally, phase-space plots and bifurcation diagrams are used to distinguish periodic and chaotic attractors. [ABSTRACT FROM AUTHOR]

The saddle-node bifurcation on an invariant circle (SNIC) is one of the codimension-one routes to creation or destruction of a periodic orbit in a continuous-time dynamical system. It governs the transition from resting behaviour to periodic spiking in many class I neurons, for example. Here, as a first step towards theory of networks of such units the effect of weak coupling between two systems with a SNIC is analysed. Two crucial parameters of the coupling are identified, which we call δ1 and δ2. Global bifurcation diagrams are obtained here for the ‘mutualistic’ case δ1δ2 > 0. According to the parameter regime, there may coexist resting and periodic attractors, and there can be quasiperiodic attractors of torus or cantorus type, making the behaviour of even such a simple system quite non-trivial. In a second paper we will analyse the mixed case δ1δ2 < 0 and summarize the conclusions of this study. [ABSTRACT FROM AUTHOR]

Complex dynamical phenomenon was studied in the single phase H-bridge inverter which was controlled by either a peak current or a valley current. The state functions and the discrete iterative map equations were established to analyze the dynamical phenomenon in the single phase H-bridge inverter. The dynamical characteristics of the single phase H-bridge inverter, such as time domain waveform diagram, bifurcation diagram, and folding map, were obtained by using the numerical calculation when the circuit parameters varied in specific range. Moreover, the simulation results were obtained by using the OrCAD-PSpice software to validate the numerical calculation. Both the numerical calculation and the circuit simulation show that the symmetrical dynamical phenomenon occurs in the single phase H-bridge inverter controlled by the peak current or the valley current. [ABSTRACT FROM AUTHOR]

By introducing weakly periodic perturbation given by light, the three-variable model of photosensitive Belousov-Zhabotinsky reaction with two time scales may possess two slow processes. The codimension-2 bursting called cusp bursting is novel, especially the phenomenon that is of the transiently quiescent state mixed among repetitively large amplitude oscillations. The slow-fast dynamics analysis with two slow parameters, based on overlapping the codimension-2 bifurcation diagram of the fast subsystem with the 3-dimensional transition phase portrait of the whole system, is proposed to reveal the bifurcation mechanism of bursting oscillation with two slow variables. The adjustment behavior of slow parameters is used to explain the transformation between the spiking state and quiescent state. [ABSTRACT FROM AUTHOR]

We model recruitment in adaptive social networks in the presence of birth and death processes. Recruitment is characterized by nodes changing their status to that of the recruiting class as a result of contact with recruiting nodes. Only a susceptible subset of nodes can be recruited. The recruiting individuals may adapt their connections in order to improve recruitment capabilities, thus changing the network structure adaptively. We derive a mean-field theory to predict the dependence of the growth threshold of the recruiting class on the adaptation parameter. Furthermore, we investigate the effect of adaptation on the recruitment level, as well as on network topology. The theoretical predictions are compared with direct simulations of the full system. We identify two parameter regimeswith qualitatively different bifurcation diagrams depending on whether nodes become susceptible frequently (multiple times in their lifetime) or rarely (much less than once per lifetime). [ABSTRACT FROM AUTHOR]

Mechanical buckling principles of conical film/substrate systems are explored in this study using extensive finite element simulations. The effects of geometrical and material parameters, including cone angle, normalized cone size, film/substrate thickness ratio, and modulus ratio between film and substrate, are explored systematically in terms of the resulting stress anisotropy (in particular the ratio between hoop and longitudinal stresses) and buckling morphology (longitudinal ridges, latitudinal stripes and reticular/herringbone patterns, as well as their transition and branching behaviours). The trends of buckling shape and wave numbers are also explored through analytical solutions of similar hollow cylindrical film/substrate systems. Finally, the mechanical buckling principles are employed to reproduce the morphological features of several types of mollusks, where each distinctive appearance corresponds to certain ranges of material and geometrical parameters. In addition to shedding some light on the morphogenesis of certain natural and biological systems, the study may also provide some useful insights on biomedical engineering and three-dimensional micro-fabrications. [ABSTRACT FROM AUTHOR]

We consider 2D flows with a homoclinic figure-eight to a dissipative saddle. We study the rich dynamics that such a system exhibits under a periodic forcing. First, we derive the bifurcation diagram using topological techniques. In particular, there is a homoclinic zone in the parameter space with a non-smooth boundary. We provide a complete explanation of this phenomenon relating it to primary quadratic homoclinic tangency curves which end up at some cubic tangency (cusp) points. We also describe the possible attractors that exist (and may coexist) in the system. A main goal of this work is to show how the previous qualitative description can be complemented with quantitative global information. To this end, we introduce a return map model which can be seen as the simplest one which is 'universal' in some sense. We carry out several numerical experiments on the model, to check that all the objects predicted to exist by the theory are found in the model, and also to investigate new properties of the system. [ABSTRACT FROM AUTHOR]

In this paper, a new hyperchaotic system is proposed, and the basic properties of this system are analyzed by means of the equilibrium point, a Poincaré map, the bifurcation diagram, and the Lyapunov exponents. Based on the passivity theory, the controllers are designed to achieve the new hyperchaotic system globally, asymptotically stabilized at the equilibrium point, and also realize the synchronization between the two hyperchaotic systems under different initial values respectively. Finally, the numerical simulation results show that the proposed control and synchronization schemes are effective. [ABSTRACT FROM AUTHOR]

Recently, effects of nonlinearity on topologically nontrivial systems have attracted attention and the stability of topologically protected edge states has been studied for a quantum walk with nonlinear effects, which is akin to time-periodically driven systems (Floquet systems). In the previous work, it has been found that the edge states can be stable attractors or unstable repellers depending on their intrinsic topological property, while the stability is not affected by the strength of nonlinearity. In the present work, we find additional bifurcations at which edge states change from stable attractors to unstable repellers with increasing the strength of nonlinearity in nonlinear quantum walks, for the first time. The new bifurcations are unique to Floquet systems, since we take dynamical properties of Floquet systems into consideration by directly applying the time-evolution operator of the quantum walks to the linear stability analysis. Our results shed new light on nonlinear effects on topological edge states in Floquet systems. [ABSTRACT FROM AUTHOR]

To date, there are very few studies on the second Hopf bifurcation in a driven square cavity, although there are intensive investigations focused on the first Hopf bifurcation in literature, due to the difficulties of theoretical analyses and numerical simulations. In this paper, we study the characteristics of the second Hopf bifurcation in a driven square cavity by applying a consistent fourth-order compact finite difference scheme recently developed by us. We numerically identify the critical Reynolds number of the second Hopf bifurcation located in the interval of (11093.75,11094.3604) by bisection. In addition, we find that there are two dominant frequencies in its spectral diagram when the flow is in the status of the second Hopf bifurcation, while only one dominant frequency is identified if the flow is in the first Hopf bifurcation via the Fourier analysis. More interestingly, the flow phase portrait of velocity components is found to make transition from a regular elliptical closed form for the first Hopf bifurcation to a non-elliptical closed form with self-intersection for the second Hopf bifurcation. Such characteristics disclose flow in a quasi-periodic state when the second Hopf bifurcation occurs. [ABSTRACT FROM AUTHOR]

The coupled angular momenta are a family of completely integrable systems that depend on three parameters and have a compact phase space. They correspond to the classical version of the coupling of two quantum angular momenta and they constitute one of the fundamental examples of so-called semitoric systems. Pelayo and Vũ Ngọc have given a classification of semitoric systems in terms of five symplectic invariants. Three of these invariants have already been partially calculated in the literature for a certain parameter range, together with the linear terms of the so-called Taylor series invariant for a fixed choice of parameter values. In the present paper we complete the classification by calculating the polygon invariant, the height invariant, the twisting-index invariant, and the higher-order terms of the Taylor series invariant for the whole family of systems. We also analyse the explicit dependence of the coefficients of the Taylor series with respect to the three parameters of the system, in particular near the Hopf bifurcation where the focus–focus point becomes degenerate. [ABSTRACT FROM AUTHOR]

We present a study of time-independent solutions of the two-dimensional discrete Allen–Cahn equation with cubic and quintic nonlinearity. Three different types of lattices are considered, i.e. square, honeycomb, and triangular lattices. The equation admits uniform and localised states. We can obtain localised solutions by combining two different states of uniform solutions, which can develop a snaking structure in the bifurcation diagrams. We find that the complexity and width of the snaking diagrams depend on the number of ‘patch interfaces’ admitted by the lattice systems. We introduce an active-cell approximation to analyse the saddle-node bifurcation and stabilities of the corresponding solutions along the snaking curves. Numerical simulations show that the active-cell approximation gives good agreement for all of the lattice types when the coupling is weak. We also consider planar fronts that support our hypothesis on the relation between the complexity of a bifurcation diagram and the number of interface of its corresponding solutions. [ABSTRACT FROM AUTHOR]

In this paper we report on the experimentally measured dynamics exhibited by a system comprised of two coupled circuits whose input–output relation follow the logistic-map function. The circuit takes in two external voltages that control the initial conditions, and we employ this capability to examine the phenomenon of symmetry breaking and to submit theoretical/numerical results on this dynamical system to experimental test. We demonstrate that symmetry-broken solutions manifest in this circuit for appropriately chosen initial conditions, and proceed to investigate experimentally the basins of attraction of these solutions, as well as their dependence on the coupling strength, ϵ. We illustrate the full power of this circuit by investigating the chaotic regime and by constructing experimental bifurcation diagrams. One intriguing phenomenon captured here involves the transition from synchronized chaos to decoherent chaos as the coupling is increased. Finally, we experimentally implement uni-directional coupling and explore the dynamics of a driven logistic map. [ABSTRACT FROM AUTHOR]

We analyze the characteristics of the global attractor of a type of dissipative nonautonomous dynamical systems in terms of the Sacker and Sell spectrum of its linear part. The model gives rise to a pattern of nonautonomous Hopf bifurcation which can be understood as a generalization of the classical autonomous one. We pay special attention to the dynamics at the bifurcation point, showing the possibility of occurrence of Li–Yorke chaos in the corresponding attractor and hence of a high degree of unpredictability. [ABSTRACT FROM AUTHOR]

A new Willis aneurysm system is proposed, which contains the Atangana–Baleanu(AB) fractional derivative. we obtain the numerical solution of the Atangana–Baleanu fractional Willis aneurysm system (ABWAS) with the AB fractional integral and the predictor–corrector scheme. Moreover, we research the chaotic properties of ABWAS with phase diagrams and Poincare sections. The different values of pulse pressure and system order are used to evaluate and compare their effects on ABWAS. The simulations verify that the changes of pulse pressure and system order are the significant reason for ABWAS’ states varying from chaotic to steady. In addition, compared with Caputo fractional WAS (FWAS), ABWAS shows less state that is chaotic. Furthermore, the results of bifurcation diagrams of blood flow damping coefficient and reciprocal heart rate show that the blood flow velocity tends to stabilize with the increase of blood flow damping coefficient or reciprocal heart rate, which is consistent with embolization therapy and drug therapy for clinical treatment of cerebral aneurysms. Finally, in view of the fact that ABWAS in chaotic state increases the possibility of rupture of cerebral aneurysms, a reasonable controller is designed to control ABWAS based on the stability theory. Compared with the control results of FWAS by the same method, the results show that the blood flow velocity in the ABWAS system varies in a smaller range. Therefore, the control effect of ABWAS is better and more stable. The new Willis aneurysm system with Atangana–Baleanu fractional derivative provides new information for the further study on treatment and control of brain aneurysms. [ABSTRACT FROM AUTHOR]

Using multi-scale beyond all order methods, we investigate stationary spatially localized solutions of a Schnakenberg system, a prototype reaction–diffusion system, near the onset of a subcritical Turing bifurcation. These solutions are homoclinic orbits to a homogeneous solution and passing near a periodic solution. In bifurcation diagrams, branches of these solutions commonly show two interwining snaking curves. Here we calculate the maximal range of existence for these solutions and compare our findings with numerical computations. We derive and optimally truncate a (divergent) asymptotic series of a front solution. The remainder of the truncated series is exponentially small if and only if a specific parameter range is met. This complements work on Swift–Hohenberg equations where similar results have been obtained. [ABSTRACT FROM AUTHOR]

A saddle to saddle-focus homoclinic transition when the stable leading eigenspace is three-dimensional (called the 3DL bifurcation) is analyzed. Here a pair of complex eigenvalues and a real eigenvalue exchange their position relative to the imaginary axis, giving rise to a 3D stable leading eigenspace at the critical parameter values. This transition is different from the standard Belyakov bifurcation, where a double real eigenvalue splits either into a pair of complex-conjugate eigenvalues or two distinct real eigenvalues. In the wild case, we obtain sets of codimension 1 and 2 bifurcation curves and points that asymptotically approach the 3DL bifurcation point and have a structure that differs from that of the standard Belyakov case. We give an example of this bifurcation in a perturbed Lorenz–Stenflo 4D ordinary differential equation model. [ABSTRACT FROM AUTHOR]

The role of cooperative effects (i.e. synergy) in transmission of infection is investigated analytically and numerically for epidemics following the rules of susceptible-infected-susceptible (SIS) model defined on random regular graphs. Non-linear dynamics are shown to lead to bifurcation diagrams for such spreading phenomena exhibiting three distinct regimes: non-active, active and bi-stable. The dependence of bifurcation loci on node degree is studied and interesting effects are found that contrast with the behaviour expected for non-synergistic epidemics. [ABSTRACT FROM AUTHOR]

This paper outlines our studies of bifurcation, quasi-periodic road to chaos and other dynamic characteristics in an external-cavity multi-quantum-well laser with delay optical feedback. The bistable state of the laser is predicted by finding theoretically that the gain shifts abruptly between two values due to the feedback. We make a linear stability analysis of the dynamic behavior of the laser. We predict the stability scenario by using the characteristic equation while we make an approximate analysis of the stability of the equilibrium point and discuss the quantitative criteria of bifurcation. We deduce a formula for the relaxation oscillation frequency and prove theoretically that this formula function relates to the loss of carriers transferring between well regime and barrier regime, the feedback level, the delayed time and the other intrinsic parameters. We demonstrate the dynamic distribution and double relaxation oscillation frequency abruptly changing in periodic states and find the multi-frequency characteristic in a chaotic state. We illustrate a road to chaos from a stable state to quasi-periodic states by increasing the feedback level. The effects of the transfers of carriers and the escaping of carriers on dynamic behavior are analyzed, showing that they are contrary to each other via the bifurcation diagram. Also, we show another road to chaos after bifurcation through changing the linewidth enhancement factor, the photon loss rate and the transfer rate of carriers. [ABSTRACT FROM AUTHOR]

In two-parameter bifurcation diagrams of piecewise-linear continuous maps on , mode-locking regions typically have points of zero width known as shrinking points. Near any shrinking point, but outside the associated mode-locking region, a significant proportion of parameter space can be usefully partitioned into a two-dimensional array of annular sectors. The purpose of this paper is to show that in these sectors the dynamics is well-approximated by a three-parameter family of skew sawtooth circle maps, where the relationship between the skew sawtooth maps and the N-dimensional map is fixed within each sector. The skew sawtooth maps are continuous, degree-one, and piecewise-linear, with two different slopes. They approximate the stable dynamics of the N-dimensional map with an error that goes to zero with the distance from the shrinking point. The results explain the complicated radial pattern of periodic, quasi-periodic, and chaotic dynamics that occurs near shrinking points. [ABSTRACT FROM AUTHOR]

A fast-slow coupled model of the hydro-turbine governing system (HTGS) is established by introducing frequency disturbance in this paper. Based on the proposed model, the performances of two time scales for bursting oscillations in the HTGS are investigated and the effect of periodic excitation of frequency disturbance is analyzed by using the bifurcation diagrams, time waveforms and phase portraits. We find that stability and operational characteristics of the HTGS change with the value of system parameter kd. Furthermore, the comparative analyses for the effect of the bursting oscillations on the system with different amplitudes of the periodic excitation a are carried out. Meanwhile, we obtain that the relative deviation of the mechanical torque mt rises with the increase of a. These methods and results of the study, combined with the performance of two time scales and the fast-slow coupled engineering model, provide some theoretical bases for investigating interesting physical phenomena of the engineering system. [ABSTRACT FROM AUTHOR]