Bifurcation Diagrams Research Articles

Bifurcation and Chaos in a Fractional-Order Cournot Duopoly Game Model on Scale-Free Networks

In this study, a Cournot duopoly model describing Caputo fractional-order differential equations with piecewise constant arguments is discussed. We have obtained two-dimensional discrete dynamical system as a result of applying the discretization process to the model. By using the center manifold theory and the bifurcation theory, it is shown that the discrete dynamical system undergoes flip bifurcation about the Nash equilibrium point. Phase portraits, bifurcation diagrams, and Lyapunov exponents show the existence of many complex dynamical behaviors in the model such as the stable equilibrium point, period-2 orbit, period-4 orbit, period-8 orbit, period-16 orbit, and chaos according to changing the speed of the adjustment parameter [Formula: see text]. The discrete Cournot duopoly game model is also considered on two scale-free networks with different numbers of nodes. It is observed that the complex dynamical networks exhibit similar dynamical behaviors such as the stable equilibrium point, flip bifurcation, and chaos depending on changing the coupling strength parameter [Formula: see text]. Moreover, flip bifurcation and transition chaos take place earlier in more heterogeneous networks. Calculating the largest Lyapunov exponents guarantees the transition from nonchaotic to chaotic states in complex dynamical networks.

Nonreciprocal feedback induces migrating oblique and horizontal banded vegetation patterns in hyperarid landscapes

In hyperarid environments, vegetation is highly fragmented, with plant populations exhibiting non-random biphasic structures where regions of high biomass density are separated by bare soil. In the Atacama Desert of northern Chile, rainfall is virtually nonexistent, but fog pushed in from the interior sustains patches of vegetation in a barren environment. Tillandsia landbeckii, a plant with no functional roots, survives entirely on fog corridors as a water source. Their origin is attributed to interaction feedback among the ecosystem agents, which have different spatial scales, ultimately generating banded patterns as a self-organising response to resource scarcity. The interaction feedback between the plants can be nonreciprocal due to the fact that the fog flows in a well-defined direction. Using remote sensing analysis and mathematical modelling, we characterise the orientation angle of banded vegetation patterns with respect to fog direction and topographic slope gradient. We show that banded vegetation patterns can be either oblique or horizontal to the fog flow rather than topography. The initial and boundary conditions determine the type of the pattern. The bifurcation diagram for both patterns is established. The theoretical predictions are in agreement with observations from remote sensing image analysis.

The Two-Parameter Bifurcation and Evolution of Hunting Motion for a Bogie System

The complex service environment of railway vehicles leads to changes in the wheel–rail adhesion coefficient, and the decrease in critical speed may lead to hunting instability. This paper aims to reveal the diversity of periodic hunting motion patterns and the internal correlation relationship with wheel–rail impact velocities after the hunting instability of a bogie system. A nonlinear, non-smooth lateral dynamic model of a bogie system with 7 degrees of freedom is constructed. The wheel–rail contact relations and the piecewise smooth flange forces are the main nonlinear, non-smooth factors in the system. Based on Poincaré mapping and the two-parameter co-simulation theory, hunting motion modes and existence regions are obtained in the parameter plane consisting of running speed v and the wheel–rail adhesion coefficient μ. Three-dimensional cloud maps of the maximum lateral wheel–rail impact velocity are obtained, and the correlation with the hunting motion pattern is analyzed. The coexistence of periodic hunting motions is further revealed based on combined bifurcation diagrams and multi-initial value phase diagrams. The results show that grazing bifurcation causes the number of wheel–rail impacts to increase at a low-speed range. Periodic hunting motion with period number n = 1 has smaller lateral wheel–rail impact velocities, whereas chaotic motion induces more severe wheel–rail impacts. Subharmonic periodic hunting motion windows within the speed range of chaotic motion, pitchfork bifurcation, and jump bifurcation are the primary forms that induce the coexistence of periodic motion.

Bifurcation analysis of a two-dimensional magnetic Rayleigh–Bénard problem

We perform a bifurcation analysis of a two-dimensional magnetic Rayleigh–Bénard problem using a numerical technique called deflated continuation. Our aim is to study the influence of the magnetic field on the bifurcation diagram as the Chandrasekhar number Q increases and compare it to the standard (non-magnetic) Rayleigh–Bénard problem. We compute steady states at a high Chandrasekhar number of Q=103 over a range of the Rayleigh number 0≤Ra≤105. These solutions are obtained by combining deflation with a continuation of steady states at low Chandrasekhar number, which allows us to explore the influence of the strength of the magnetic field as Q increases from low coupling, where the magnetic effect is almost negligible, to strong coupling at Q=103. We discover a large profusion of states with rich dynamics and observe a complex bifurcation structure with several pitchfork, Hopf, and saddle–node bifurcations. Our numerical simulations show that the onset of bifurcations in the problem is delayed when Q increases, while solutions with fluid velocity patterns aligning with the background vertical magnetic field are privileged. Additionally, we report a branch of states that stabilizes at high magnetic coupling, suggesting that one may take advantage of the magnetic field to discriminate solutions.

Dynamical properties of a small heterogeneous chain network of neurons in discrete time

We propose a novel nonlinear bidirectionally coupled heterogeneous chain network whose dynamics evolve in discrete time. The backbone of the model is a pair of popular map-based neuron models, the Chialvo and the Rulkov maps. This model is assumed to proximate the intricate dynamical properties of neurons in the widely complex nervous system. The model is first realized via various nonlinear analysis techniques: fixed point analysis, phase portraits, Jacobian matrix, and bifurcation diagrams. We observe the coexistence of chaotic and period-4 attractors. Various codimension-1 and -2 patterns for example saddle-node, period-doubling, Neimark–Sacker, double Neimark–Sacker, flip- and fold-Neimark–Sacker, and 1 : 1 and 1 : 2 resonance are also explored. Furthermore, the study employs two synchronization measures to quantify how the oscillators in the network behave in tandem with each other over a long number of iterations. Finally, a time series analysis of the model is performed to investigate its complexity in terms of sample entropy.

An impact-damping model of collision body with multi-point contact and external load, and its application in gear transmission

The damping term is an important component of the calculation model of the contact force of collision bodies because it simulates the energy dissipation of collision bodies during the contact-impact process. Therefore, an impact-damping model of a collision body with multi-point contact and external load is established by considering the restitution coefficient. The algorithm for solving this model is proposed based on iterative methods. Then, the accuracy of the established model is verified by the consistency between the calculated and tested results. The effect of multi-point contact and external load on the contact-impact process of the collision body is studied. The results show that the hysteresis damping factor increases from small to large when the collision body enters a stable state through repeated impacts. Further, the established damping model is applied to the dynamics model of a spur gear pair with backlash, extended teeth contact, and structure coupling effect of the gear body. The nonlinear dynamic characteristics, multi-state mesh, and teeth impact of the system are studied through the root-mean-square of transmission error, bifurcation diagrams with multi-mapping sections, and contact forces. The calculation results of the gear dynamics model with the impact-damping model are more consistent with the tested results than the calculation results of the gear dynamics model with the traditional damping model. Meanwhile, the connection between system resonance, response coexistence, teeth disengagement, and teeth impact is revealed.

Transition to oscillatory instability of lean methane–air flames in microchannels

Micro-flow reactors and porous media burners are prone to instability in the form of flame with repetitive extinction and ignition (FREI) near the region of a temperature gradient. Although significant progress has been made in this field, there is still a lack of understanding regarding the nature of such stable-to-unstable transition. Some studies reported a smooth and continuous instability transition in microchannels, interpreting this phenomenon as a supercritical bifurcation. Meanwhile, others have observed that the transition is subcritical, with rapid, stepwise evolution of the stable flame to FREI. This study aims to rigorously determine the transition character theoretically and numerically using a two-dimensional model with a precise resolution of the bifurcation parameter (flow velocity) near the critical point and gradual wall temperature ramp. The results indicate that the transition is a subcritical bifurcation with strong hysteresis. However, the amplitude of the limit cycle born in the bifurcation point could be small compared to FREI-like oscillations. Moreover, further development of the instability depends significantly on the channel width. In relatively wide channels, the stable regime transforms immediately into the extinction/ignition one with rapid growth in amplitude. In moderate channels, complex transitional dynamics were observed with an evident pulsating regime, which evolved into FREI through mixed-mode oscillations. In narrow channels, the amplitude of transitional pulsating flame becomes comparable to the fully-developed extinction/ignition cycles, and the bifurcation diagram has a continuous character with no significant jumps or discontinuities. Such qualitative variation of transition character provides a possible explanation for contradictory interpretations presented in the literature.Novelty and Significance StatementCurrently, there is no consensus about the character of transition between the stable and extinction/ignition regimes in microchannels. Some studies have reported a continuous instability transition with an evident pulsating regime, interpreting it as a supercritical bifurcation, while others have observed subcritical bifurcation with a rapid transition. The novelty of this research lies in the rigorous determination of the bifurcation type and further instability development in microchannels of different widths based on an accurate simulation of flame dynamics near the critical point with very small steps of the flow velocity variation, along with the bifurcation theory. The study contributes to understanding the instability transition nature and may explain the different interpretations of the transitional phenomenon found in the literature. The results offer valuable insights for designing micro-scale and porous media combustion devices.

Constructing a New Multi-Scroll Chaotic System and Its Circuit Design

Multi-scroll chaotic systems have complex dynamic behaviors, and the multi-scroll chaotic system design and analysis of their dynamic characteristics is an open research issue. This study explores a new multi-scroll chaotic system derived from an asymptotically stable linear system and designed with a uniformly bounded controller. The main contributions of this paper are given as follows: (1) The controlled system can cause chaotic behavior with an appropriate control position and parameters values, and a new multi-scroll chaotic system is proposed using a bounded sine function controller. Meanwhile, the dynamical characteristics of the controlled system are analyzed through the stability of the equilibrium point, a bifurcation diagram, and Lyapunov exponent spectrum. (2) According to the Poincaré section, the existence of a topological horseshoe is proven using the rigorous computer-aided proof in the controlled system. (3) Numerical results of the multi-scroll chaotic system are shown using Matlab R2020b, and the circuit design is also given to verify the multi-scroll chaotic attractors.

Dynamical analysis and circuit realization of a high complexity fourth-order double-wing chaotic system with transient chaos and its application in image encryption

This paper proposes a fourth-order double-wing chaotic system with high complexity. After conducting a dynamic analysis, it is found that the system exhibits transient chaos and a rare inverse period-doubling bifurcation phenomenon in the bifurcation diagram. The system also exhibits attractor coexistence, with periodic, quasi-periodic, indicating high sensitivity to initial values. These phenomena sufficiently demonstrate the rich dynamical characteristics of chaotic systems. By introducing an impulse function with a cosine function in the foundation of the proposed system, it is found that controllable wing number and staircase burst oscillations occur. Furthermore, the number of wings and oscillation periods vary with changes in parameters, which has significant implications in engineering applications. The circuit design and construction are carried out using the Multisim simulation software, and the digital circuit is realized by using a Field-Programmable Gate Array (FPGA). It is found that the simulation results and the actual implementation results are highly consistent with the phase portrait of the system, thus demonstrating the feasibility of the circuit. Finally, by combining the proposed system with a DNA encryption algorithm, a novel image encryption algorithm with multiple layers of encryption is designed, greatly enhancing the security of encrypted images. The security of this encryption algorithm is analyzed in terms of information entropy, key space, correlation, and resistance to attacks. It is found that the proposed encryption algorithm exhibits high confidentiality and resistance to attacks. The proposed system has significant reference value in secure communication when applied to image encryption.

Cross-channel image encryption algorithm on the basis of a conservative hyperchaotic system

In image encryption, the traditional encryption means of scrambling and diffusion are widely used, but they fail to completely eliminate the correlation between the channels of the ciphertext image and still exist security risks. The selecting channel and position scrambling and cross-channel S-shaped diffusion methods proposed in this study push the encryption to a new depth. Compared with ordinary methods, it is more thorough in disrupting pixel positions, increases the complexity of pixel relationships between different channels, and substantially improves the unpredictability of encryption. To support this approach, we design a four-dimensional conservative hyperchaotic system with a very large hyperchaotic interval, which combines the high randomness of the conservative system and the broad key-space property of the hyperchaotic system, effectively defending against the risk of phase-space reconstruction aroused by missing attractors and reducing the possibility of being cracked. We deeply analyze the dynamic properties of the system by means of phase diagrams, bifurcation diagrams and Lyapunov exponents. The cross-channel scrambling and diffusion encryption algorithm designed based on this system not only ensures the security of image information during transmission, but also greatly enhances the ability to resist various attacks. The proposal of this method undoubtedly brings a secure solution to the field of image encryption.

Medical image encryption using novel sine-tangent chaotic map

Medical images play a crucial role in the diagnostic, planning, and monitoring of various types of diseases and physical problems. These images contain confidential details about the patient's condition. This confidential data can be safeguarded by using techniques like image encryption that transforms a given image into an unidentifiable format. This research paper proposes a medical image cryptosystem that uses a one-dimensional (1-D) novel Sine-Tangent Chaotic (STC) map and a shared key. Firstly, a private shared key is generated using the XOR operations, on which permutation and substitution operations are carried out. Secondly, a chaotic sequence is generated using the novel STC map. Finally, the permuted-substituted private shared key, the chaotic sequence, and one of the plain image components (i.e. Red, blue or Green) are XORed to construct the cipher image. To test the randomness of the proposed novel 1-D STC, we have used bifurcation diagram, Lyapunov Exponent (LE) and Shannon entropy (SE) parameters. The results show that the proposed mathematical chaotic equation has a wider chaotic range, and is more complex than some of the existing chaotic maps. Also, the observations prove that the proposed medical image encryption method performs well in comparison to the existing state of the art approaches, by obtaining values 7.99977, 99.66023 and 33.80065 for parameters Entropy, Number of Changing Pixel Rates (NPCR) and Unified Averaged Changed Intensity (UACI), respectively.

Dynamics modeling of a memristor-based Rucklidge chaotic system: Multistability, offset boosting control and FPGA implementation

The complex dynamics greatly contributes to the application of chaos, especially the memristor chaotic systems that have emerged very recently. In this paper, a novel four-dimensional continuous chaotic system is developed by introducing a generalized memristor model into the classical Rucklidge system, and the rich dynamical traits are confirmed by theories and numerical simulations. System Hamilton energy is deduced and analyzed to expose the underlying energy evolution and chaos mechanism. And then, realizing the offset boosting control of the system promotes its engineering applications. Numerical simulations verify that the offset boosting controller can directly change the polarity of the chaotic signal. Specifically, bifurcation diagram, Lyapunov spectra, system complexity evolution diagram and dynamic evolution map are adopted to expound the rich dynamics. Meanwhile, the extreme multi-stability with various initial values resulting in an infinite number of coexisting attractors is also revealed. Finally, the proposed attractor is implemented using the Field Programmable Gate Array (FPGA) platform and building analog circuit based on Multisim. Among which, the former is potentially used to design the Pseudo-random signal generator in information encryption, while the latter is for verifying the existence of the proposed chaotic system.

Exploiting 2D-SDMCHM and matching embedding driven by flag-shaped hexagon prediction for visually meaningful medical image cryptosystem

With the proliferation of advanced medical technologies and smart wearable devices, a considerable volume of patient data and medical images is being generated and stored, leading to an escalating concern over patient privacy violations and manipulation. Therefore, safeguarding the secrecy and integrity of medical images is paramount for preserving patient privacy and ensuring accurate medical diagnostics. To tackle this challenge, this paper introduces a novel approach, the Visually Meaningful Medical Image Cryptosystem (VMMIC), leveraging a 2-Dimensional Sinusoidal Discrete Memristor Coupled Hyperchaotic Map (2D-SDMCHM) to protect the privacy of medical images. The innovation of this study is primarily reflected in three aspects: Firstly, a novel Discrete Memristor (DM) model is proposed, upon which a new 2D-SDMCHM is constructed. Comparative analysis showcases its superior chaos robustness, as evidenced by evaluations through bifurcation diagrams, phase trajectories, Lyapunov exponents, and Shannon entropy. Secondly, we introduce a measurement matrix optimization algorithm based on Optimal Matrix Arrangement (OMA) for Compressive Sensing (CS), aimed at minimizing the coherence of the measurement matrix, thereby enhancing reconstruction accuracy. Lastly, a Matching Embedding Method Driven by Flag-Shaped Hexagon Prediction (MEMFHP) is introduced to ensure lossless concealment of confidential information. Experimental findings and comparative evaluations affirm the VMMIC's capability to meet high standards of reconstruction quality, steganography, and efficiency in handling medical images, while also demonstrating robustness against various attacks. Consequently, the proposed VMMIC holds significant promise for practical implementation in engineering applications.

A stochastic hormesis Ricker model and its application to multiple fields

Random noise pervades ecosystems and has the potential for having major impacts on the growth processes of pest populations. In this paper, we aim to investigate the impact of random perturbations on the hormesis Ricker model by considering control measures applied within each generation which can generate hormetic effects, where randomness is characterized by a uniform discrete distribution and white noise, respectively. The main results indicate that the addition of discrete randomness will make the model appear with blurred orbits when the intrinsic growth rate is large enough. The position of the random variable, at the end of each generation or within each generation, cause the blurred orbits to exhibit various forms. Moreover, the effects of variances and expectations of the discrete uniform random variable on the dynamics are evaluated. Further, the introduction of randomness increases the hormetic zone and the maximum response, but can increase or decrease the monotonically increasing interval under different parameter values. In contrast, the stochastic model characterized by white noise, exhibits only a small effect on the bifurcation diagrams with respect to the intrinsic growth rate and the hormesis, which may be attributed to the mean of its noise being zero. Finally, we fit the stochastic model to experimental hormetic data sets observed in multiple fields, and our results demonstrate that the stochastic hormesis Ricker model can capture the characteristics of these data accurately. These findings could provide valuable insights into the understanding of the complexities of pest population dynamics, with implications for better pest control, resource management and for other complex biological systems such as toxicology and drug development.

Effects of inclination angle and fluid parameters on binary fluid convection in a tilted rectangular cavity

Thermal convection of binary fluid mixtures with negative separation ratios in a tilted rectangular cavity heated from below is numerically investigated using a high-accuracy finite-difference method. The fluids under consideration are Newtonian fluids with the Prandtl number (Pr) and Lewis number (Le) in the ranges of 0.01≤Pr≤10 and 0.001≤Le≤1, respectively. Extensive direct numerical simulations are conducted to explore the dynamics of the system by varying the control parameters, namely, the relative Rayleigh number (r) and the inclination angle of the cavity (β) together with the separation ratio (ψ) in the ranges of 1.0≤r≤2.0, 0°≤β≤90°, and −0.6≤ψ≤0. It is found that the inclination angle leads to a distinct trend for heat and mass transfer: the Nusselt number first decreases for small β, then gradually increases for moderate β, and finally decreases again for large β when r varies from 1.4 to 2.0. The optimal inclination angle lies in the range of 54°≤β∗≤64°, where the heat transport peaks. The Sherwood number initially decreases for small β, then increases with increasing β, and finally remains constant for large β. We present the bifurcation diagram of the solutions for different separation ratios, focusing on the distinct variation trends of the inclination angle. The corresponding heat- and mass-transfer properties of the flow on different branches of the bifurcation diagram are investigated. Furthermore, we explore the effect of the relative Rayleigh number, Prandtl number and Lewis number as well as the separation ratio on heat and mass transfer for various inclination angles, and obtain the Nusselt number as a function of the separation ratio and inclination angle.

Delayed self-feedback echo state network for long-term dynamics of hyperchaotic systems.

Analyzing the long-term behavior of hyperchaotic systems poses formidable challenges in the field of nonlinear science. This paper proposes a data-driven model called the delayed self-feedback echo state network (self-ESN) specifically designed for the evolution behavior of hyperchaotic systems. Self-ESN incorporates a delayed self-feedback term into the dynamic equationof a reservoir to reflect the finite transmission speed of neuron signals. Delayed self-feedback establishes a connection between the current and previous m time steps of the reservoir state and provides an effective means to capture the dynamic characteristics of the system, thereby significantly improving memory performance. In addition, the concept of local echo state property (ESP) is introduced to relax the conventional ESP condition, and theoretical analysis is conducted on guiding the selection of feedback delay and gain to ensure the local ESP. The judicious selection of feedback gain and delay in self-ESN improves prediction accuracy and overcomes the challenges associated with obtaining optimal parameters of the reservoir in conventional ESN models. Numerical experiments are conducted to assess the long-term prediction capabilities of the self-ESN across various scenarios, including a 4D hyperchaotic system, a hyperchaotic network, and an infinite-dimensional delayed chaotic system. The experiments involve reconstructing bifurcation diagrams, predicting the chaotic synchronization, examining spatiotemporal evolution patterns, and uncovering the hidden attractors. The results underscore the capability of the proposed self-ESN as a strategy for long-term prediction and analysis of the complex systems.

Bifurcation Analysis and Chaos Control of Carrier Phase-Shifted Cascaded H-Bridge Inverter

Cascaded H-bridge inverters, which belong to strongly nonlinear time-varying system with complex dynamics, have been widely used in high-voltage and high-power engineering fields. In this paper, a carrier phase-shifted sinusoidal pulse-width modulation cascaded H-bridge inverter is taken as an example with piecewise analysis to establish the discrete iterative mathematical model. Furthermore, the nonlinear dynamic behavior of this inverter system is investigated using folding diagrams, spectrum diagrams, bifurcation diagrams, Lyapunov exponents, etc. We find that this inverter can exhibit multiperiod bifurcation, tangential bifurcation and paroxysmal chaos within certain ranges of control coefficients and other circuit parameters, which will influence the system’s stability seriously. On the basis of this discovery, the sinusoidal time-delay feedback control method is proposed. The difference between the load current of the controlled object and its own delay is used to form a feedback term, which is then fed into the sine function and proportional link to obtain the control term. Moreover, the Jacobian matrix of this feedback inverter system is derived, and the constraints on the feedback control coefficient are obtained based on the Jury criterion. Simulation results have verified the effectiveness of such a sinusoidal time-delay feedback control method, which can improve the stability and anti-interference of the system. The research can be used to provide some reference and guidance for the circuit design, debugging and control of cascaded H-bridge inverters.

Cap like trajectories in 5D chaotic tangent hyperbolic memristive model: fractional calculus and encryption

This research aims to investigate the influence of model parameters and fractional order on a novel mathematical model with tangent hyperbolic memristor. This investigation conducted by applying Lyapunov exponents and bifurcation analysis. We utilize the Lyapunov exponent theory to understand and characterize these chaotic behaviors under fractional indices. The Lyapunov exponent, bifurcation, and phase diagrams have been depicted to explore the intricate dynamics of the chaotic system governed by the chaotic equation. A novel approach termed Atangana-Baleanu-Caputo (ABC) fractional derivative (FD) to generate phase portraits and gain insights into the system’s behavior. The random numbers generated by the chaotic system are employed to distort the image through an amalgamated image encryption (AIE) algorithm. Subsequently, the integrity of the scrambled image has been assessed using various image security evaluation methods to reinforce the notion that combining the chaotic system and image can constitute a valuable encryption key. Finally, the chaotic model circuit realization uses active and passive components, and the outcomes are compared with the numerical simulations.

An asymmetric rotor model under external, parametric, and mixed excitations: Nonlinear bifurcation, active control, and rub-impact effect

This article explores the nonlinear dynamical analysis and control of a [Formula: see text] DOF system that emulates the lateral vibration of an asymmetric rotor model under external, multi-parametric, and mixed excitation. The linear integral resonant controller ([Formula: see text]) has been coupled to the rotor as an active damper through a magnetic actuator. The complete mathematical model, governing the nonlinear interaction among the rotor, controller, and actuator, is derived based on electromagnetic theory and the principle of solid mechanics. This results in a discontinuous [Formula: see text] DOF system coupled with two [Formula: see text] DOF systems, incorporating the rub-impact effect between the rotor and stator. The complicated mathematical model is investigated using analytical techniques, employing the perturbation method, and validated numerically through time response, basins of attraction, bifurcation diagrams, [Formula: see text] chaotic test, and Poincaré return map. The main findings indicate that the asymmetric system model may exhibit nonzero bistable forward whirling motion under external excitation. Additionally, it can whirl either forward or backward under multi-parametric excitation, besides the trivial stable solution. Furthermore, in the case of mixed excitation, the rotor displays nontrivial tristable solutions, with two corresponding to forward whirling orbits and the other one corresponding to backward whirling oscillation. These findings are validated through the establishment of different basins of attraction. Finally, the performance of the [Formula: see text] in mitigating rotor vibrations and averting nonlinear catastrophic bifurcations under various excitation conditions. Furthermore, the rotor’s dynamical behavior and stability are explored in the event of an abrupt failure of one of the connected controllers. The outcomes demonstrate that the proposed [Formula: see text] effectively eliminates dangerous nonlinearities, steering the system to respond akin to a linear system with controllable oscillation amplitudes. However, the sudden controller failure induces a local rub-impact effect, leading to a nonlocal quasiperiodic oscillation and restoring the dominance of the nonlinearities on the system’s response.

Snap-back repellers and chaos in a class of discrete-time memristor circuits

In the last decade the flux-charge analysis method (FCAM) has been successfully used to show that continuous-time (CT) memristor circuits possess for structural reasons first integrals (invariants of motion) and their state space can be foliated in invariant manifolds. Consequently, they display an initial condition dependent dynamics, extreme multistability (coexistence of infinitely many attractors) and bifurcations without parameters. Recently, a new discretization scheme has been introduced for CT memristor circuits, guaranteeing that the first integrals are preserved exactly in the discretization. On this basis, FCAM has been extended to discrete-time (DT) memristor circuits showing that they also are characterized by invariant manifolds and they display extreme multistability and bifurcations without parameters. This manuscript considers the maps obtained via DT-FCAM for a circuit with a flux-controlled memristor and a capacitor and it provides a thorough and rigorous investigation of the presence of chaotic dynamics. In particular, parameter ranges are obtained where the maps have snap-back repellers at some fixed points, thus implying that they display chaos in the Marotto and also in the Li–Yorke sense. Bifurcation diagrams are provided where it is possible to analytically identify relevant points in correspondence with the appearance of snap-back repellers and the onset of chaos. The dependence of the bifurcation diagrams and snap-back repellers upon the circuit initial conditions and the related manifold is also studied.