International Journal of Bifurcation and Chaos

Symmetric Aerostatic Cavities Bearing (SACB) systems have attracted increasing attention in the field of high-precision machinery, particularly rotational mechanisms applied at ultra-high speeds. In an air bearing system, the air bearing serves as the main support, and the load-carrying capacity is not as high as that of oil film bearings. However, the aero-spindle can operate at considerably high rotational speeds with relatively lower heat generated from rotation compared with that of oil film bearings. In addition, the operating environment of air bearings does not easily cause the rotor to deform. Hence, through adequate design, air pressure systems exhibit a certain level of stability. In general, the pressure distribution function of air bearings exhibits strong nonlinearity when there are changes in the rotor mass or rotational speed, or when the bearing system is inadequately designed. These issues may lead to instabilities in the rotor, such as unpredictable nonperiodic movements, rotor collisions, or even chaotic movements under certain parameters. In this study, rotor oscillation was analyzed using the maximum Lyapunov exponent to identify whether chaotic behavior occurred. Machine learning methods were then used to establish models and predict the rotor behavior. Especially, random forest and extreme gradient boosting were combined to develop a new model and confirm whether this model offered higher prediction performance and more accurate results in predicting tendencies with considerable changes compared with other models. The results can be effectively used to predict the SACB system and prevent nonlinear behavior from occurring.

Deep Neural Networks (DNNs) have been successfully applied to investigations of numerical dynamics of finite-dimensional nonlinear systems such as ODEs instead of finding numerical solutions to ODEs via the traditional Runge–Kutta method and its variants. To show the advantages of DNNs, in this paper, we demonstrate that the DNNs are more efficient in finding topological horseshoes in chaotic dynamical systems.

One of the promising applications of locally-active memristors (LAMs) is to construct oscillators for oscillatory neural networks. By using two current-controlled (CC) LAMs, a fully CC LAM-based oscillator is designed in this paper. The oscillator principle originates from the small-signal inductive and capacitive impedance characteristics of two different CC LAMs, and thus extra reactance element is not required in the circuit. Based on bifurcation theory and small-signal analysis method, conditions of the equilibrium point instability are quantitatively derived. Theoretical analysis indicates that the circuit oscillation is dependent on three critical parameters. Then, according to the conditions of the equilibrium point instability, parameters design methods of the two LAMs are proposed, including the static and dynamic parameters. A simple NbO_x CC LAM model is taken as an example to conduct detailed simulation analysis. The simulation results verify the feasibility of the proposed circuit and analysis methods. Finally, the effects of the LAM model parameters on the oscillator performance are investigated, which is helpful for optimal design of the oscillator.

The effect of time-periodic two-frequency rotation modulation on Rayleigh–Bénard convection in water with either AA7072 or AA7075 nanoparticles is investigated. The single-phase description of the Khanafer–Vafai–Lightstone model is used for modeling the nanoliquids. An asymptotic expansion procedure is adopted in the case of the linear stability to obtain the correction (due to modulation) to the Rayleigh number at marginal stability of unmodulated convection. A nonlinear regime of convection is considered with a nonautonomous generalized Lorenz model as the governing equation. The method of multiscales is then employed to obtain the coupled nonautonomous Ginzburg–Landau equations with cubic nonlinearity from the Lorenz model. These equations are presented in the phase-amplitude form and the amplitude is used to quantify the heat transport. The modulation amplitude is considered to be small (of order less than unity) and moderate frequencies of modulation are considered. We found that there is a threshold frequency beyond which the system behavior reverses. At frequencies below the threshold, the mean Nusselt number increases with an increase in the amplitude of modulation while an opposite influence is seen for values above the threshold. Such a behavior is a consequence of what is analogously seen in the case of the critical Rayleigh number. The influence of two-frequency modulation is more pronounced on the results of the linear and nonlinear regimes compared to that of the single-frequency one. The heat transport is enhanced due to the presence of dilute concentration of suspended nanoparticles (either AA7072 or AA7075 nanoalloys) in water. The influence of nanoparticles is to modify the threshold values generating chaos but it does not qualitatively alter the dynamical behavior of the system. The plots of Lyapunov exponents reveal that there is no possibility of hyper-chaos in the generalized Lorenz model when there is a rotational modulation.

In previous works [Inaba & Kousaka, 2020; Inaba & Tsubone, 2020; Inaba et al., 2023], significant bifurcation structures referred to as nested Mixed-Mode Oscillations (MMOs) were found to be present in forced Bonhoeffer–van der Pol (BVP) oscillators. It is well known that unnested Mixed-Mode Oscillation-Incrementing Bifurcations (MMOIBs) can generate $[A_0,B_0\times m]$ oscillations (i.e. $A_0$ followed by $B_0$ repeated $m$ times) for successive values of $m$, where $A_0$ and $B_0$ are adjacent fundamental simple MMOs, e.g. $A_0=1^s$ and $B_0=1^{s+1}$, where $s$ is an integer. Furthermore, it has been confirmed that MMOIBs can generate nested MMOs. Let two adjacent unnested MMOIB-generated MMOs be denoted $A_1(=[A_0,B_0\times m])$ and $B_1(=[A_0,B_0\times (m+1)])$. Then, singly nested MMOIBs can generate $[A_1,B_1\times p]$ for successive values of $p$, i.e. $A_1$ followed by $B_1$ repeated $p$ times, between the $A_1$- and $B_1$-generating regions. The sequential generation of singly nested MMOs has been investigated in detail in previous work [Ito et al., 2021]. Nested MMOs can, however, be at least doubly nested. In this study, we investigate doubly nested MMOs considering a constrained nonautonomous BVP oscillator containing an idealized diode. Based on the observed dynamics of this system, Poincaré return maps are rigorously constructed in one dimension. Therefore, we can solve the successive saddle-node bifurcations using a nested (i.e. double-loop) bisection method. We track 60 successive doubly nested MMOIBs and we do not rule out the possibility that the 58 scaling constants corresponding to the MMOIB intervals converge to unity. We note that because we solve the bifurcation equation avoiding the use of a method that requires the careful selection of the initial conditions (e.g. the Newton–Raphson), we can accurately track the saddle-node bifurcations without missing any doubly nested MMO sequences.

In this paper, we investigate the impact of functional shifts in a time-discrete cross-catalytic system. We use the hypercycle model considering that one of the species shifts from a cooperator to a degrader. At the bifurcation caused by this functional shift, an invariant curve collapses to a point $P$ while, simultaneously, two fixed points collide with $P$ in a transcritical bifurcation. Moreover, all points of a line containing $P$ become fixed points at the bifurcation and only at the bifurcation in a degenerate scenario. We provide a complete analytical description of this degenerate bifurcation. As a result of our study, we prove the existence of the invariant curve arising from the transition to cooperation.

In this research paper, we consider a Leslie–Gower Reaction–Diffusion (RD) model with a predator-driven Allee term in the prey population. We derive conditions for the existence of nontrivial solutions, uniform boundedness, local stability at co-existing equilibrium points, and Hopf bifurcation criteria from the temporal system. We identify sufficient conditions for Turing instability with no-flux boundary condition for the spatial system. Our investigation delves into the analysis of diffusion-induced Turing instability, incorporating stability conditions for the constant steady-state in the spatial model. We also investigate the conditions for the existence and nonexistence of nonconstant steady states in the diffusion-induced model. During numerical simulations, we observe that the predator-driven Allee term is essential for the model to generate Turing structures. Our findings reveal intriguing properties within the RD system, demonstrating its ability to produce patterns within the Turing domain. The simulation confirms that cold–hot spots and stripes-like patterns (a mixture of spots and strips) arises for different strengths of the predation parameter and Allee parameter. In contrast, we observe that for the above threshold value of the Allee parameter, the above-mentioned patterns may disappear from the system. Interestingly, we also observe that the stationary system produces patterns for both large and small amplitudes of perturbation in the vicinity of the Turing boundary. Our research may contribute valuable insights into the Allee effect and enhance our understanding of predator–prey interactions in naturalistic environments.

This paper investigates the nonlinear bifurcation and chaos characteristics of piezoelectric composite lattice sandwich plates at 1:3 internal resonance. The nonlinear vibration partial differential equation is first discretized to become an ordinary differential equation by applying the Galerkin method. The main resonance modulation equations and the parametric resonance for the external excitation frequency that is close to the system’s second-order modal frequency are then obtained by using the multiscale method. The Newton–Raphson method is subsequently applied to obtain the bifurcation diagram of the steady-state equilibrium of modulation equation with varying system parameters. The equilibrium stability is finally analyzed. We confirm the existence of static bifurcation such as saddle-node bifurcation, pitchfork bifurcation, and Hopf dynamic bifurcation. A detailed analysis is also conducted on the complex nonlinear jump phenomenon caused by the presence of multiple nonlinear steady-state solution regions. In the dynamic Hopf bifurcation interval, the fourth-order Runge–Kutta method is used to continue to track the dynamic periodic solution of the modulation equation in Cartesian coordinates. It is found that there are multiple boundary crises and attractor merging crises in the vibration system for piezoelectric composite lattice sandwich plates.

Multistep prediction of high-dimensional time series is an essential and challenging task. In this study, we propose an integrated reservoir predictor for making accurate and robust multistep-ahead forecasts based on short-term high-dimensional time series. Initially, a conjugated pair of Spatiotemporal Information (STI) equations is derived using Takens’ embedding theory to transform the spatial information of high-dimensional variables into one-dimensional temporal information of the target variable and vice versa. Next, by exploiting reservoir networks, reservoir-based STI equations are established to efficiently capture nonlinear dynamics of the target system with only linear optimization. Then, through an integration phase, the integrated reservoir predictor can output precise and robust predictions of the multistep-ahead states of any target variable. The integrated reservoir predictor outperforms some other prediction methods (including reservoir computing, long-short-term-memory network, convolutional neural network and support vector regression), when applied to classical dynamic systems (e.g. 60D double scroll model, 40D Lorenz 96 model, and 60D Rössler model) and real-world datasets (solar generation data and PM2.5 concentration records), as indicated by evaluation metrics such as Pearson correlation coefficients exceeding 0.9 and root-mean-square errors below 0.3, even in the presence of noise in training data.

In this paper, we investigate the existence and number of crossing limit cycles in a class of planar piecewise linear systems with node–node type critical points defined in two zones separated by a nonregular line formed by two rays emanated from the origin $(0,0)$, which are the positive $x$- and $y$-axes. We focus our attention on the existence of two-point crossing limit cycles, which intersect the switching line at two points. We obtain sufficient conditions under which the system has two two-point crossing limit cycles which intersect only one of the two rays. Moreover, we construct examples to show that this class of systems can have two, three or four two-point crossing limit cycles.

Memristor-type feedback provides a unique passage for chaos produced with easy control. In this work, a novel memristive map with amplitude control and coexisting hyperchaotic attractors is designed, in which two nonbifurcation parameters are extracted for partial amplitude control crossing the origin and total amplitude control. Attractor splitting and attractor merging are captured, which lead to five regimes of attractor self-reproducing. In this case, double-cavity attractors extend in the direction of 0${}^{\circ }$, 45${}^{\circ }$, 90${}^{\circ }$ in negative $x$-axis, 90${}^{\circ }$ in positive $x$-axis, 135${}^{\circ }$ in $x$-axis forming a specific wall of self-reproduction. Furthermore, the FPGA-based hardware implementation is carried out showing consistent results with numerical simulation. Finally, a high-security chaotic encryption scheme is proposed for the orthogonal frequency division multiplexing transmission system, where the power division multiplexing technique and two-dimensional region joint encryption are effectively utilized. The security of the transmitted information is improved by the various coexisting attractors and nonbifurcation amplitude controllers.

Tri-rhythmical nature has attracted extensive attention from scholars in describing the dynamical behaviors of self-sustained systems. In this paper, we consider a tri-rhythmic van der Pol system and give a bifurcation analysis of a stochastic tri-rhythmic self-sustained system under joint noise perturbation. Based on an approximate approach, we give the stationary probability density function of amplitudes, and we find that the noise and time-delay feedback, regulating the velocity time-delay feedback strength parameter, may not cause transitions among unimodal, bimodal and trimodal in the tri-rhythmic system. More stochastic bifurcations appear by regulating the time delay in this system. The system, surprisingly, undergoes five times of stochastic bifurcations when the time delay is monotonically increased. It is shown that the time delay is more sensitive to the tri-rhythmic system and a much stronger dependence on the system. From a biological point of view, the reaction rate of biological molecules can be enhanced or diminished by the change of the noise intensity or correlation time of Gaussian colored noise. More surprisingly, an increase in displacement feedback will delay the reaction rate; however, the effect of an increase in velocity feedback on the reaction rate depends on the time delay. A detailed research on the parameter space indicates that time delay and colored noise parameters can effectively control the multirhythmicity of the system. Finally, this paper verifies the effectiveness of the theoretical solution through the numerical results of Monte Carlo simulation of the original system. These results may help to further explore forking bifurcations in real-world applications. This research provides new insight into the understanding of nontrivial effects of joint noise on the tri-rhythmic system.

We take into consideration two different kinds of two-parameter bifurcations in a class of 3D linear Filippov systems, namely pseudo-Bautin bifurcation and boundary equilibrium bifurcations for two scenarios. The bifurcation conditions for generating rich dynamic behaviors are established. The main objective is to investigate the effects of two parameters interacting simultaneously on a variety of dynamic phenomena. In order to analyze the pseudo-Bautin bifurcation, we build the Poincaré map and analyze the number of fixed points whose types are related to the crossing limit cycles. In order to analyze boundary equilibrium bifurcations for two scenarios, we perform an analysis on the existence and admissibility of equilibria. Besides, a comprehensive investigation on hidden attractors induced by boundary equilibrium bifurcations is conducted. The novelty resides in overcoming the constraints of previous studies that solely take into account the dynamics of individual parameter variations. We innovatively characterize the two-parameter bifurcation mechanism of a new class of Filippov systems, and qualitatively demonstrate the coexistence of hidden attractor and stable pseudo-equilibrium.

Benefiting from trigonometric and hyperbolic functions, a nonlinear megastable chaotic system is reported in this paper. Its nonlinear equations without linear terms make the system dynamics much more complex. Its coexisting attractors’ shape is diamond-like; thus, this system is said to have diamond-shaped oscillators. State space and time series plots show the existence of coexisting chaotic attractors. The autonomous version of this system was studied previously. Inspired by the former work and applying a forcing term to this system, its dynamics are studied. All forcing term parameters’ impacts are investigated alongside the initial condition-dependent behaviors to confirm the system’s megastability. The dynamical analysis utilizes one-dimensional and two-dimensional bifurcation diagrams, Lyapunov exponents, Kaplan–Yorke dimension, and attraction basin. Because of this system’s megastability, the one-dimensional bifurcation diagrams and Kaplan–Yorke dimension are plotted with three distinct initial conditions. Its analog circuit is simulated in the OrCAD environment to confirm the numerical simulations’ correctness.