International Journal of Bifurcation and Chaos

This work proposes a generalization of the family of chaotic maps without fixed points, proposed by [, ] and termed the Vertigo maps. The original map family is parameterized by four control parameters, which can be used to scale the function used as a seed and control its domain. Several theoretical results are provided regarding the existence of the fixed points, the periodic cycles, and the Lyapunov exponents of the maps. Furthermore, two map examples are provided based on the logistic and tent seed functions, which are then studied using a series of numerical tools, like phase portraits, bifurcation diagrams, and Lyapunov exponent diagrams. Finally, an application to a Pseudo-Random Bit Generator is considered. The generator utilizes an exponential-based hash function in combination with the remainder operator.

Chaos is thought to be random, but a few constants, such as Lyapunov exponent and fractal dimension, can be obtained by statistic measures based on a large amount of data. This paper presents the local regularity in chaotic bursting oscillations caused by the coupling of two scales in frequency domain. When slow-varying external excitation is introduced in a well-known Nosé–Hoover oscillator, called also Sprott-A system, different types of bursting attractors, such as periodic and chaotic bursting oscillations, can be observed. By regarding the external excitation as a slow-varying parameter, the dynamical evolution of the generalized autonomous fast subsystem is obtained. Two typical cases for the physical parameters are considered, in which codimension-1 fold and codimension-2 fold–Hopf bifurcations take place, respectively. Several types of coexistent attractors in the fast subsystem, such as the coexistence of equilibrium points, the coexistence of equilibrium point and sequence of period-doubling bifurcation routine to chaos, may appear, which may lead to different bursting oscillations, the mechanism of which is presented by overlapping the transformed phase portrait and the behaviors of the fast subsystem. The trajectory of periodic bursting oscillations visits the equilibrium points in turn via fold bifurcations, while fold–Hopf bifurcation may result in quasi-periodic bursting oscillations. Two coexisted asymmetric bursting attractors may expand in the phase space with the increase of exciting amplitude. The interaction forms an enlarged symmetric bursting attractor. The influence of the periodic windows on the full vector field may appear, which causes the spiking stage to be divided into segments with different behaviors. Specially spiking state can oscillate according to different regular cycles as well as chaos in turn, which suggests that regular attractors are embedded in chaos. More periodic windows exist in the fast subsystem, more fine structures can be observed in the spiking stage. Accordingly, the coexisted attractors, together with the related bifurcations, in the fast subsystem, may be one of the main reasons for the complexity of the world with the coupling of multiple scales.

The use of fractals and chaos theory has opened many new research possibilities in science, engineering, and technology. This paper adds to these studies by using numerical methods, analytical techniques, and experimental circuit components to study and simulate a modified stretch–twist–fold (STF) flow. This flow is important for understanding how particles move in fluids and is related to the dynamo theorem. We first check the basic properties of the system to ensure it is well defined. Then, we create a numerical method, analyze its errors, and run simulations that include the fractal–fractional operators. These simulations show how the system combines self-replication with chaotic behavior for different parameter values. We also build the system on a Field-Programmable Gate Array (FPGA) board, which produces results similar to the numerical simulations. This research helps improve our understanding of fractals and chaos theory and their role in explaining the complex fluid dynamics, especially the STF flow and its link to particle movement in fluids connected to the dynamo theorem.

In this work, the influences of geometry and domain size on spatiotemporal pattern formation are investigated to establish the parameter spaces for a cross-diffusive reaction–diffusion model on an annulus. By applying the linear stability theory, we derive the conditions which can give rise to Turing, Hopf and transcritical types of diffusion-driven instabilities. We explore whether the selection of a sufficiently large domain size, together with the appropriate selection of parameters, can give rise to the development of patterns on nonconvex geometries, e.g. annulus. Hence, the key research methodology and outcomes of our studies include a complete analytical exploration of the spatiotemporal dynamics in an activator-depleted reaction–diffusion system; a linear stability analysis to characterize the dual roles of cross-diffusion and domain size of pattern formation on an annulus region; the derivation of the instability conditions through the lower and upper bounds of the domain size; the full classification of the model parameters; and a demonstration of how cross-diffusion relaxes the general conditions for the reaction–diffusion system to exhibit pattern formation. To validate the theoretical findings and predictions, we employ the finite element method to reveal the spatial and spatiotemporal patterns in the dynamics of the cross-diffusive reaction–diffusion system within a two-dimensional annular domain. These observed patterns resemble those found in ring-shaped cross-sectional scans of hypoxic tumors. Specifically, the cross-section of an actively invasive region in a hypoxic tumor can be effectively approximated by an annulus.

Chaos is considered to be essential for life, since it can bring diversity to the world. However, it is difficult to detect the pattern of chaos due to its apparent randomness. In this paper, a nonlinear system based on a Hénon-like map is studied to show the semi-ordered structure of its chaotic sequences. The periodic sequences of such a map are solved analytically, and the stability is analyzed. Multiple coexisting unstable sequences with different periods and the chaotic attractor are obtained. An evidence is found that chaos is not purely apparently random, and it is weakly attracted by different unstable periodic sequences. Due to the weak attraction, it transits from one unstable periodic sequence to another at random instances, and it appears unpredictable.

This paper presents an investigation into the dynamics of a Memristor-Coupled Heterogeneous Tabu Learning Neuronal Network (MCTLNN) comprising distinct types of Tabu learning neurons. The network adopts both monotone and nonmonotone composite hyperbolic tangent functions as activation mechanisms. A sinusoidal function-based memristor interconnects the neurons, leading to intriguing dynamical interactions. A theoretical analysis reveals the existence of infinitely many equilibria in the MCTLNN. Utilizing multiple numerical simulation methods, including phase portraits, Lyapunov exponents, bifurcation diagrams, and local attraction basin analyses, we systematically investigate the system’s complex dynamics. This study discovers the coexistence of diverse attractors, such as heterogeneous, symmetric, and initial offset-boosted attractors. Notably, we explore the dependence of the system’s dynamics on the initial conditions, specifically highlighting the initially offset boosting phenomenon through extensive simulations. Analog circuit-based hardware implementation not only confirms the model’s theoretical predictions, but also provides a platform for brain-inspired computing applications.

This paper presents a bipedal model with asymmetric leg motion under a two-parameter pulse thrust, represented by an impulsive hybrid system. By exploring the successive and immediate transitions in motion states, a Poincaré map with explicit form is designed to reveal the dynamic characteristics of walking. The stability of periodic orbits during bipedal walking under two-parameter regulation, as well as the bifurcation mechanism of locomotion, is discussed. The simulation results about periodic gait, flip bifurcation, chaotic gait, and bifurcation control are in agreement with the theoretical analysis. The reliability of bipedal walking for collision angle is demonstrated through an error analysis comparing the nonlinear impulsive hybrid system and its linearized system. The research findings may help to predict and control the motion behavior of bipedal robots. Furthermore, it constitutes part of a theoretical analysis for the pulse propulsion method.

Chaos-based learning is a popular method that combines with optimized techniques to achieve the best fitness values by preventing early convergence and improving initial conditions. Chaos enhances the search process, while opposition learning helps improve the population development matrix, ultimately boosting result quality. This paper introduces two new concepts in machine learning. It proposes an improved Opposition and Chaos-based Water Cycle Algorithm (OCWCA) to effectively achieve the best fitness value through cost calculation and to validate the results of objective functions on 15 engineering benchmark problems with constraints. The Water Cycle Algorithm (WCA), a hydrology-based method, provides global search locations based on the flow of streams and rivers toward the sea using predefined control parameters to generate a population matrix. A convergence plot for OCWCA compared to WCA demonstrates significant improvement in results, incorporating learnings from the WCA method and yielding the best fitness data.

Higher-order networks represent complex multinode interactions beyond what traditional pairwise models can capture, offering deeper insights into many real-world system dynamics. Synchronization of complex networks is essential for achieving coordinated behavior, promoting efficient communication, stability, and overall functionality across various domains, including neuroscience. Extreme events, marked by abrupt and significant deviations from typical behavior that surpass a statistical threshold, can severely affect system stability and performance. Understanding and managing these events are vital for improving resilience and preventing catastrophic failures within complex networks and dynamical systems. This paper explores synchronization and extreme events in the synchronous solutions of a higher-order Chialvo neuron map network model, utilizing inner-linking functions and chemical synapses to represent pairwise and nonpairwise interactions, respectively. By applying Master Stability Functions and assessing synchronization error, we derive synchronization criteria and investigate the potential for extreme events within stable synchronous regions. Our findings indicate that the emergence of extreme events is contingent on the values of the higher-order coupling parameter, while the pairwise coupling parameter is responsible for sustaining synchronization. In response to these events, we develop a control strategy to suppress extreme events and chaotic behavior. Our results demonstrate that connecting all nodes to an external reference source is necessary to stabilize the network and maintain synchronization, as partial control proved insufficient.

The alteration in concentration of atmospheric pollutants is influencing the functionality and growth of forests. Also, the growing human demand for forestry resources is detrimentally affecting the sustainability of these valuable resources. In this study, we present a mathematical model that incorporates the influence of atmospheric pollutants on the intrinsic growth rate of forests, while concurrently addressing the utilization of forestry land by the human population for diverse purposes which diminishes the carrying capacity of forestry resources. We establish sufficient conditions under which all relevant dynamic variables stabilize at their equilibria. Upon scrutinizing the model system, we observe multiple bifurcations concerning certain key parameters. Additionally, numerical simulations have been conducted to corroborate the analytically derived findings. Moreover, we fortify the proposed model through the integration of a time delay in the impact of pollutants on the intrinsic growth rate of forestry resources. Despite the conventional belief that introducing a time delay tends to destabilize systems, our resolute delayed model system showcases that a time delay in the effect of pollutants on intrinsic growth rate of forestry resources can, in fact, stabilize the unstable interior equilibrium.

Due to the unique structure and transmission ratio of the Ravigneaux planetary gearset, this paper investigates the nonlinear vibration characteristics of the Ravigneaux planetary gearset transmission systems. First, a dynamic model of the Ravigneaux planetary gearset transmission systems is established, considering nonlinear factors such as time-varying meshing stiffness, comprehensive equivalent errors, tooth clearance, and time-varying friction. Next, the dynamic equations are formulated and solved. The nonlinear behavior of the system is characterized through phase diagrams, Poincaré diagrams, time-domain plots, and frequency-domain plots. The bifurcation diagram and three-dimensional spectrum diagram illustrate how different external load excitation frequencies affect the system’s nonlinear behavior. The results indicate that as the external load excitation changes, the system’s motion state transits from chaotic to bifurcation and ultimately to periodic motion. Finally, the multiscale method is employed to examine the effects of various parameters on the main resonance characteristics of the system, determining the stability conditions for the primary resonance. The principal resonance analysis reveals that increasing damping, stiffness, and fluctuations in the external load can enhance the system’s stability.

An expansion of the periodic solution of the van der Pol oscillator for small values of the damping parameter $𝜖$ is shown via a frequency domain method and harmonic balancing technique. An approximation of the amplitude of the first harmonic and the frequency $\omega$ in terms of $𝜖$ is obtained by an analytical form instead of the conventional algorithmic approach used before with this methodology. Its decomposition under different harmonics $sink𝜗$ and $cosk𝜗$ where $k=3,5,7$ is written in terms of their leading coefficients in power series of $𝜖$ to make easy comparisons with other methods in the literature.

Although the electroencephalogram (EEG) activity of Autism Spectrum Disorder (ASD) has extremely complicated dynamics, the basic research on its dynamic modeling remains unexplored. This paper aims to investigate the mechanisms underlying EEG abnormalities in ASD from the perspective of dynamic modeling, and delve into the therapeutic principles of neuromodulation. First, by adjusting the average inhibitory synaptic gain and the average excitatory synaptic connection strength in the Wendling model, we reproduce clinical EEG characteristics of ASD, including Interictal Epileptiform Discharges (IEDs), increased relative power of the $\delta +𝜃$ band, decreased relative power of the $\alpha$ band, and reduced EEG complexity. This indicates that Excitation–Inhibition (EI) imbalance is potentially the trigger of abnormal EEG in ASD. Then, we demonstrate that both monophasic and biphasic repetitive Transcranial Magnetic Stimulation (rTMS) effectively control pathological discharges in ASD, with biphasic rTMS showing a reduced risk of over-control at the same frequency and intensity, consistent with clinical findings. Additionally, we show that Deep Brain Stimulation (DBS) plays a similar role to rTMS in the treatment of ASD, which proves the low-frequency DBS to be more effective in enhancing the complexity of brain activity, suggesting that DBS may hold promise as an innovative neuromodulation strategy for ASD. Our work helps to promote the development of dynamic modeling for ASD and to inspire therapeutic approaches for regulating the EI balance.

In this paper, we use the techniques from dynamical systems and singular traveling wave theory developed by [, ] to investigate the exact explicit solutions for the perturbed Gerdjikov–Ivanov (GI) equation. By considering the corresponding dynamical system and finding the bifurcations of phase portraits for the amplitude component of the traveling solutions, the dynamical behavior can be revealed. Under different parameter conditions, exact explicit solutions of the perturbed GI equation are found.

For a porous media equation, the existence of uncountably infinite many global monotonic kink wave solutions with bounded supports is proved. By using the method of planar dynamical systems, the dynamical behavior of the corresponding traveling wave system is discussed. A few exact explicit parametric representations of the kink wave solution families are given.

A high data rate multi-carrier differential chaos shift keying system with hybrid index modulation, referred to as HDR-HIM-MC-DCSK, is proposed in this paper. In the proposed system, multiple data-bearing wavelets are parallelly transmitted with one reference signal. Different Walsh codes are utilized for separating these data-bearing wavelets. Some subcarriers are used for hybrid index modulation, in which the chaotic signal and its Hilbert transform are generated for representing the active and inactive subcarriers. In addition, the analytical Bit-Error-Rate (BER) expressions of the proposed HDR-HIM-MC-DCSK system are derived over the AWGN and the multipath Rayleigh fading channels. The hardware complexity, computational complexity, energy efficiency, spectral efficiency and data rate of the proposed system are analyzed. The analytical accuracy and the advantages of the proposed system are verified by relevant simulations.