Narrow Results

This study explores the synchronization and control of the Five Elements Chaotic Systems, grounded in a mathematical model that accounts for both promotion and restriction dynamics. An active controller is employed to achieve synchronization between two identical Traditional Chinese Medicine Five Elements Chaotic Systems. Additionally, a linear feedback controller is integrated into the nonlinear Traditional Chinese Medicine Five Elements Chaos system to facilitate system regulation. These approaches rely on Lyapunov stability theory to establish conditions for synchronization and control. Numerical simulations are conducted to verify the effectiveness of the proposed methods. The simulations demonstrated that our proposed synchronization and control techniques are robust and effective in managing the dynamic equilibrium of the Five Elements. By achieving synchronization, we were able to effectively restore balance, which has direct applications in healthcare for the prevention and treatment of diseases related to elemental imbalances.

This work proposes a generalization of the family of chaotic maps without fixed points, proposed by Jafari et al.; 2016 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.

The location and basins of attraction of attractors involved in nonlinear smooth dynamical systems play a crucial role in understanding the dynamics of these systems across a wide range of parameter values. Nonlinear dynamical systems, described by autonomous nonlinear coupled ordinary differential equations with three or more variables, often exhibit chaotic dynamics within specific parameter ranges. In the literature, three distinct routes to chaos are well known: period doubling, crisis, and intermittency. Recent research in the field of nonlinear dynamical systems focuses on the characterization and identification of hidden attractors. The primary objective of this study is to investigate the dynamics and the successive local and global bifurcations that give rise to chaotic behavior in a Modified Lorenz System (MLS). Our findings reveal that chaos emerges via the type II intermittency route. Furthermore, we identify the chaotic attractor as a hidden attractor, with its existence corroborated by the dissipative nature of the model beyond the subcritical Hopf bifurcation. Key contributions of this research include the derivation of analytical criteria for a degenerate pitchfork bifurcation, the numerical calculation of the first Lyapunov exponent, and a semi-analytic proof of the homoclinic bifurcation. These results enhance our understanding of the complex dynamics within the MLS and contribute to the broader field of nonlinear dynamics and chaos theory.

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.

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.

Analyzing tumor growth dynamics improves treatment strategies for cancer. Many models have been proposed to analyze cancer development dynamics, which do not always exhibit chaotic behavior. This research aims to create and examine a unique dynamical cancer model that exhibits chaotic behavior for certain parameters. Introducing chaos into the model allows for the exploration of irregular tumor growth patterns and the identification of critical thresholds that can influence treatment outcomes. The model is examined, and each system parameter’s impact on the model’s dynamics is evaluated. The analysis of the bifurcation and Lyapunov diagrams demonstrates chaos in three populations of tumor, healthy, and immune system cells. By suppressing the immunological response, the cancer cell gains control of the chaotic attractor and establishes a stable state. This might lower the cancer condition by altering the appropriate parameter range assisting in tumor treatment.

The paper is devoted to analyzing the mechanisms of spread and prevention of epidemics, based on a discrete model that takes into account the infection spread because of contacts among infected and susceptible individuals, disease-induced mortality, and the factor of treatment. Pathways leading to the complete extinction, complete recovery, and nontrivial coexistence of susceptible and infected individuals are revealed by bifurcation analysis. Parametric conditions of nontrivial modes of coexistence in the form of equilibrium, discrete cycle, quasiperiodic closed invariant curve and chaos are found. An extended analysis on transformation scenarios of these regimes in dependence of the variation of the rate of the infection spread and the treatment intensity is performed. Phenomena of infection-induced chaos and its suppression by treatment are discussed.

In this paper, we investigate the circular motion and chaotic behavior of charged particles in the dyonic global monopole spacetime surrounded by a perfect fluid. We classified the black hole into three special regimes: dark matter, dust, and radiation. We precisely calculated the Lyapunov exponent for each regime as an eigenvalue of the Jacobian matrix. We examine, through numerical and graphical analysis, the circular motion and chaos bound violation across all the regimes. In the dark matter regime, stable orbits conform to the chaos bound. Even though the bound brings orbits with small charges and those far from the event horizon closer, they never violate it. In the dust regime, there can be more than one orbit for a fixed mass, charge, topological defect, and fluid parameter, especially when the angular momentum is small. At this point, the orbits are unstable, and those that are closer to the event horizon violate the bound. Similarly, in the radiation regime, orbits that are closer to the event horizon are unstable and chaotic, especially with greater angular momentum. In fact, regardless of the charge, topological defect, and fluid parameter, all orbits, whether they are far from or close to the horizon, become unstable and violate the bound when the angular momentum is significantly large.

In this work, we study the Nanoparticles (NPs) impact on a phytoplankton–zooplankton interaction model with Ivlev-like and Holling type-II functional responses. We found that the growth rate of phytoplankton reduces due to NPs. In the non-spatial model, we investigated boundedness, stability, bifurcation and chaos. The stability criteria is determined using the Routh–Hurwitz criterion. Hopf bifurcation is demonstrated with parameter $K$, which represents the NPs carrying capacity while interacting with phytoplankton. The normal theory is used to examine the Hopf bifurcation direction and the stability of bifurcating periodic solutions. Moreover, the stability of non-hyperbolic equilibrium points have been determined using the Center Manifold theorem. Also, the parameter $\beta$, which represents the interaction rate between NPs and phytoplankton, exhibits chaotic behavior. Furthermore, we also investigated Hopf bifurcation and Turing instability in spatial model systems. This study demonstrates that NPs can influence the dynamics of the system in a balanced environment.

The aim of this study is to analyze the dynamics of an initial value problem associated with a conformable spatial partial differential equation within the space $𝔛=C_0([0,+\infty );ℂ)$. First, we demonstrate that the solution generates a strongly continuous semigroup on $𝔛$. Next, we construct a quasiconjugacy between this solution and another semigroup defined on a conformable weighted function space. Under specific conditions, we demonstrate that the problem exhibits both hypercyclicity and chaos.

It is known that statistical model selection as well as identification of dynamical equations from available data are both very challenging tasks. Physical systems behave according to their underlying dynamical equations which, in turn, can be identified from experimental data. Explaining data requires selecting mathematical models that best capture the data regularities. The existence of fundamental links among physical systems, dynamical equations, experimental data and statistical modeling motivate us to present in this paper our theoretical modeling scheme which combines information geometry and inductive inference methods to provide a probabilistic description of complex systems in the presence of limited information. Special focus is devoted to describe the role of our entropic information geometric complexity measure. In particular, we provide several illustrative examples wherein our modeling scheme is used to infer macroscopic predictions when only partial knowledge of the microscopic nature of a given system is available. Finally, limitations, possible improvements, and future investigations are discussed.

We introduce a stimulus-response scheme that supports plastic variation of synapse weights in neural networks, and analyze how memory formation evolves under external stimulation. In so doing, chaotic networks and stochastic networks that have very different dynamics are compared. Experimental results suggest that chaotic activity remarkably outperforms stochastic activity in stimulus-response memorization. This seems to be indicative of effectiveness of the chaos in dynamic learning by stimulus-response scheme oriented to natural learning.

We consider different versions of exploration in reinforcement learning. For the test problem, we use navigation in a shortcut maze. It is shown that chaotic ∊-greedy policy may be as efficient as a random one. The best results were obtained with a model chaotic neuron. Therefore, exploration strategy can be implemented in a deterministic learning system such as a neural network.

To determine the attractor dimension of chaotic dynamics, the box-counting method has the difficulty in getting accurate estimates because the boxes are not weighted by their relative probabilities. We present a new method to minimize this difficulty. The local box-counting method can be quite effective in determining the attractor dimension of high-order chaos as well as low-order chaos.

Synchronization is introduced into a chaotic neural network model to discuss its associative memory. The relative time of synchronization of trajectories is used as a measure of pattern recognition by chaotic neural networks. The retrievability of memory is shown to be connected to synapses, initial conditions and storage capacity. The technique is simple and easy to apply to neural systems.

Multiobjective oligopoly models are constructed. The objective of the first two models are to maximize profits and to maximize sales. In the third model, the objectives are to maximize profits and to minimize risk. Giving more weight to risk minimization decreased the profits. In all the three models, we found that the weight of the profit maximization has to be higher than a given threshold. Sufficient conditions for persistence of some multiobjective oligopolies are derived. Again, they require that the weight of profit maximization to be higher than certain value.

We study the behavior of one-dimensional chaotic signals and filtering using the discrete Haar wavelet and Daubechies wavelets.

Dynamic mean field theory is applied to the problem of forest fires. The starting point is the Monte Carlo simulation in a lattice of a million cells. The statistics of the clusters is obtained by means of the Hoshen–Kopelman algorithm. We get the map p_n → p_{n + 1}, where p_n is the probability of finding a tree in a cell, and n is the discrete time. We demonstrate that the time evolution of p is chaotic. The arguments are provided by the calculation of the bifurcation diagram and the Lyapunov exponent. The bifurcation diagram reveals several windows of stability, including periodic orbits of length three, five and seven. For smaller lattices, the results of the iteration are in qualitative agreement with the statistics of the forest fires in Canada in the years 1970–2000.

We present a stability and stochastic analysis of the zigzag map which describes the dynamical behavior of an electrical switching circuit. Analytical expressions are derived for the invariant density and the corresponding Lyapunov exponent for specific parameter values.