International Journal of Bifurcation and Chaos

This research delves into simple Jerk-type dynamical networks, constructed using a specific bidirectional coupling scheme that influences the sub-oscillators and compound nonlinearity gradient functions that perturb each sub-unit of the network. The novelty of this work is to demonstrate a new pedagogical method able to stimulate the formation of higher-order multiscroll dynamics in the Jerk sub-oscillator. This is done by modeling a simple dynamic network built by applying a special bidirectional coupling between the Jerk sub-systems. Another novelty of this approach is that it allows us to study the collective dynamics of Jerk system networks, which has not yet been done in the literature. Multiscroll systems are exceedingly complex dynamically, making them valuable in chaos-based applications. The pedagogical approach used in this study is exceptional because it produces many additional equilibrium points (from 5 to 25 in dynamical network 1, and from 3 to 15 in dynamical network 2) in each Jerk sub-unit of the network. These equilibria elevate the complexity of Jerk systems by emulating higher-order multiscroll dynamics. The methodology used in this study is efficient and differs from those used in the literature, which mainly uses nonlinear multizero functions in dissipative systems. This research further explores the dynamic system characterization tools and conducts an experimental investigation on a microcontroller (ATMEGA2560) to confirm the predictions.

This paper expands the concept of periodic orbit dividing surfaces within rotating Hamiltonian systems possessing three degrees of freedom. Initially, we detail the implementation of our second method for constructing these surfaces, as outlined in [, , ], for such systems. Subsequently, we analyze the configuration of these surfaces within a rotating quadratic normal-form Hamiltonian system characterized by three degrees of freedom.

Aiming to explore the influence of ultrasonic burnishing technology on the wear behaviors of Al7075 alloy friction pairs, the recurrence analysis method is employed to study the evolutions of recurrence plot (RP) and recurrence quantification analysis (RQA) parameters of friction coefficient signals. The results show that the RPs of burnishing Al7075 alloy specimens evolve from an abrupt structure to a homogeneous structure in the wear process. The average value of determinism $(\overline{\mathrm{\text{DET}}})$ and the average value of entropy $(\overline{\mathrm{\text{ENTR}}})$ of burnishing specimens as a whole are larger than those of nonburnishing specimens. For the burnishing specimens, the $\overline{\mathrm{\text{DET}}}$ and $\overline{\mathrm{\text{ENTR}}}$ are the highest under the load of 25N, with the values of 34.2008 and 4.3129, respectively. It demonstrates that the friction system of burnishing Al7075 alloy under the load of 25N has superior determinism and stability.

This paper proposes a novel Switching Chaotic Coupled Map Lattices (SCCML) system based on Elementary Cellular Automata (ECA) to address issues found in Coupled Map Lattices (CML) systems such as period windows, poor lattice correlation, and limited chaotic behavior parameters. The system uses a bidirectional coupling method and introduces two ECA with different rules to continuously alter the indexes of the coupled lattices and the underlying chaotic mapping. This approach effectively mitigates correlation issues in CML systems. At the same time, the iterative results of ECA are cascaded with the chaotic mapping as a perturbation of the SCCML system to extend the parameter range of chaotic behavior. The Kolmogorov–Sinai entropy density and universality, bifurcation diagrams, uniformity and correlation are analyzed to investigate the dynamical properties of the proposed system. Theoretical analysis and numerical simulations demonstrate that, compared with other novel spatiotemporal chaotic systems, the proposed system eliminates periodic windows, expands the chaotic behavior parameter range, and decreases the correlation between different lattices. The above experimental results demonstrate the proposed system’s excellent potential for cryptosystems and secure communication. Furthermore, an Encrypted Code Index Modulation (E-CIM) scheme based on reversible ECA encryption is proposed to address the problems of limited pseudo-random sequence resources and low spectrum utilization in direct sequence spread spectrum systems. The proposed scheme uses a chaotic sequence with good correlation and randomness generated by the SCCML spatiotemporal chaos system as the spreading code sequence. In the scheme, the information bits are divided into two parts, namely modulation bits and mapping bits. The modulation bits are mapped to points on the modulation constellation diagram. The mapping bits are encrypted using reversible ECA, and the mapped bits before and after encryption are mapped as spreading sequences with different indexes. Simulation and analysis show that under the same spectrum efficiency conditions, the BER performance of E-CIM is superior to that of the Code Index Modulation (CIM) and Generalized Code Index Modulation (GCIM) schemes by about 2–4dB and superior to that of the Nonorthogonal Code Shift Keying and Code Index Modulation (N-CSK-CIM) scheme by about 0.5dB in an additive Gaussian white noise channel when the BER is 10${}^{-5}$.

This paper studies the bifurcation and stability of a parabolic–ordinary-parabolic system to a model with three reactive species: molecular Anchors A>, Nanoparticulates P and Matrix constituents M (MAP). Global asymptotic stability is studied for both the ODE system and the reaction–diffusion system. The results show that the positive equilibrium is globally asymptotically stable in both the ODE and the corresponding reaction–diffusion systems. Furthermore, by selecting the haptotaxis coefficient as a bifurcation factor, we analyze the existence, direction, and stability of the Hopf bifurcation. The results also reveal that haptotaxis is crucial in determining the stability and bifurcation behavior of the model, suggesting that it has a destabilizing effect and can undergo Hopf bifurcation. Moreover, numerical simulations are presented to verify and illustrate the theoretical results.

This paper investigates the bifurcation problem associated with a ring-shaped one-way coupled Toda oscillator incorporating distributed time delays. We define precise criteria to assess the stability and bifurcation conditions of the positive equilibrium within the model incorporating distributed time delays, applying the Routh–Hurwitz criterion. Our study enriches the characterization of Toda oscillators to better capture their dynamic properties. This improvement takes into account the infinite memory feature associated with distributed time delays, thereby refining our understanding of their behavior. To substantiate our theoretical analysis, we conduct comprehensive numerical simulations.

Chameleon chaotic systems have a special property to display various types of chaotic attractors by tuning system parameters, thereby allowing for the generation of diverse chaotic signals that are suitable for various applications. In this paper, we propose and study a class of three-dimensional quadratic chameleon systems capable of transitioning between self-excited and hidden chaotic regimes. Through systematic analysis of the systems, we can identify hidden chaotic attractors in parameter regions where no equilibria exist, or where there is a line equilibrium, or where a single stable equilibrium exists. In order to study the basic properties of the system, we carried out both local stability analysis and Hopf bifurcation analysis. Further bifurcation analysis and Lyapunov exponent calculation uncovered intricate transitions among periodic, chaotic, and hidden chaotic regimes as the system parameters varied. Through the research, we find that antimonotonicity holds significant implications for creating various types of chaotic dynamics in the chameleon systems. Furthermore, we find that by adjusting the values of parameters, the system can display a self-excited chaotic attractor, a hidden chaotic attractor with no equilibrium, a hidden chaotic attractor with a line equilibrium, or a hidden chaotic attractor with a single stable hyperbolic/nonhyperbolic equilibrium point. We are interested in a hidden chaotic system with a stable nonhyperbolic equilibrium point, for which the practical feasibility is verified through circuit simulations. The chameleon chaotic systems studied in this paper expand our understanding of the chaotic mechanisms with various equilibrium configurations.

This paper aims to investigate the stability and Hopf bifurcation for a three-species prey–predator model with Holling type-II and type-III functional responses under fractional Proportional–Derivative (PD) control. The prey’s fear of the middle predator and the super predator as well as gestation delay is also incorporated in our proposed model for its wider applicability. Based on the stability theorem of linear system, Hopf bifurcation theorem and implicit function array solution method, some sufficient order-induced bifurcation conditions are given. The application and effectiveness of the presented theoretical results are illustrated through a simulation example. Especially, the impact of the feedback control gains as well as the fear effect on the stability and Hopf bifurcation is explored.

In this study, we aim to explore the birth mechanism of coexisting Strange Nonchaotic Attractors (SNAs) in a multistability system. The underlying mechanism behind the emergence of coexisting SNAs is understood by examining the interruption of coexisting torus-doubling bifurcations. We identify three different routes that lead to the creation of coexisting SNAs, namely, the intermittent route, Heagy–Hammel route, and fractalization route. To effectively characterize SNAs, we employ several metrics, including the nontrivial largest Lyapunov exponents, phase sensitivity exponents, and power spectrum analysis. Notably, SNAs exhibit a fractal (strange) nature, which has inspired us to develop a novel optimization algorithm termed the Strange Nonchaotic Optimization (SNO) algorithm. This algorithm is used to optimize the Proportional-Integral-Derivative (PID) controller parameters in time-delay systems based on a defined objective function. Simulation results validate that the proposed SNO algorithm presents better performance than the available algorithms in the literature for time-delay systems.

This paper examines the variable-order fractional Lorenz system with distinct types of delays. The application of the Arzela–Ascoli theorem is made to prove the existence of solutions for the given problem, while the Banach fixed point theorem is employed to derive the uniqueness results. The utilization of the Adams–Bashforth–Moulton technique facilitates the exploration and resolution of the approximation solution, complemented by a thorough error analysis of particular approaches. The computational simulations showing chaotic behaviors in various delayed systems with different variable orders demonstrate the effectiveness of the approach.

The stability and Hopf bifurcation of a class of flywheel governor systems subject to stochastic excitation under time-delay feedback are studied. The model is simplified using center manifold reduction to obtain a paradigm for a two-dimensional central subsystem. The system converges asymptotically to an Itô stochastic differential equation based on polar transformations and stochastic averaging method. The stochastic stability of the system is analyzed by applying the maximum Lyapunov exponent and the singular boundary theory, and its probability density function is calculated. In addition, it is found through Monte Carlo numerical simulations that the effects of noise intensity and time-delay on the stochastic P-bifurcation of the system are opposite, and proper noise intensity makes the flywheel governor system converge to the equilibrium position, while a large time-delay leads to stochastic bifurcation and makes the flywheel governor system unstable. Finally, the two-parameter bifurcation diagram is used to further analyze the impact of time-delay and noise intensity on the bifurcation of the system under variable load torque. It is found that the influence of noise intensity and time-delay on the periodic oscillation state is the same, and the stability of the flywheel governor system will decrease with both of their increases. This also further demonstrates the duality of the impact of noise intensity on the same system under different conditions. The above research provides better theoretical references for the design and safe operation of speed control systems.

The intricate relationship between temporal and spatiotemporal dynamics in a Crowley–Martin predator–prey model, enriched with fear effect and anti-predator behavior, is investigated in this study. A careful mathematical analysis is conducted to explore the feasible equilibria of the model system, followed by an examination of their stability, instability, and all possible bifurcation scenarios. Asymptotic stability, bistability, and various codimension-1 and codimension-2 bifurcations, including transcritical, saddle–node, Hopf, cusp, and Bogdanov–Takens bifurcations, are demonstrated by the model. The analytical findings and the model’s applications are validated through numerical simulations, employing a biparameter bifurcation diagram for quantitative analysis. The study also observes bubbling phenomena and a scenario leading to the density-dependent hydra effect. The diffusion effect is investigated with special attention to the system’s nonlinearity and chosen parameter values. Numerical simulations reveal the emergence of spatiotemporal patterns both within and beyond the Turing space. The evolution of diffusion-driven pattern generation, including cold spots, stripes, labyrinthines, mixtures of stripes and cold spots, and complex non-Turing patterns, is demonstrated on the plane. These spatial patterns are shown to be influenced biologically by both the fear effect and anti-predator behavior.

In the reaction–diffusion systems, the Laplacian is usually a self-adjoint operator. Incorporating advection terms generally destroys this and yields a quite complicated decomposition of phase space in the Center Manifold Reduction (CMR) method. Considering that the Multiple Time Scales (MTS) method can be directly applied to bifurcation analysis and normal form derivation based on algebraic operations, and the computational process is relatively universal, we apply the MTS method to analyze the Hopf bifurcation problem of reaction–diffusion–advection systems with time delay. First, we introduce the MTS method for the system with a specific boundary condition, which is used to derive the normal form of Hopf bifurcation. Calculating the normal form using the MTS method can be summarized as determining the bifurcation parameters, conducting Taylor expansion based on the MTS idea, and eliminating secular terms. Then, the key coefficients in the normal form are explicitly given, which are also compared with the results obtained by the CMR method, and both methods lead to the same bifurcation results. Finally, we use the MTS method to calculate the normal form of Hopf bifurcation for a predator–prey model with mixed boundary conditions. We find that the spatially nonhomogeneous periodic solutions appear near the equilibrium in the system when the time delay exceeds the critical value, and the theoretical results are illustrated by numerical simulations.

This paper presents two representative classes of Impulsive Fractional Differential Equations defined with generalized Caputo’s derivative, with fixed lower limit and changing lower limit, respectively. Memory principle is studied and numerical examples are considered. The problem of the memory principle of the MATLAB code for Lyapunov exponents of fractional-order systems [, ] is analyzed.

For the cantilever beam vibration model with small damping and forced terms, under some special parameter conditions, the corresponding unperturbed system can be described by a planar dynamical system. In this paper, by using the dynamical system method to analyze the unperturbed differential system and find bifurcations from the corresponding phase portraits, the dynamical behavior can be derived. Exact explicit homoclinic and heteroclinic solutions, and periodic solution families can be found. For the periodic perturbation system, the chaotic behavior of system is revealed. Numerical simulations are carried out to illustrate the theoretical results.