In this paper, we investigate a class of KdV-like advection-dispersion equations and derive explicit exact solutions describing various nonlinear wave phenomena, including peakon, periodic peakon, and smooth wave solutions. By using the method of dynamical systems, under given parameter conditions, we obtain bifurcations of the phase portraits for the corresponding traveling wave dynamical system. Corresponding to different level curves, we derive exact explicit parametric representations of the periodic peakon, peakon, smooth periodic wave solution as well as solitary wave solutions.
We study local Hopf bifurcations in an activator–inhibitor system without diffusion which can be modeled as a delay differential equation with two discrete delays. The main result of this paper is the local existence of the Poincaré–Lindstedt series to all orders for the bifurcating periodic solutions. The model has a nonpolynomial type of nonlinearity, and yet this allows us to exploit the use of Fourier–Taylor series to develop order-by-order calculations that lead to linear recurrence equations for the coefficients of the Poincaré–Lindstedt series. As applications, we implement the computation of the coefficients of these series for any finite order, and use a pseudo-arclength continuation to compute local branches of small-amplitude periodic solutions.
In this work, we perform a bifurcation analysis of a planar piecewise-smooth continuous system with eight parameters, previously introduced to model neural decision-making in social behaviors. While previous studies of related models have relied mainly on numerical simulations, there is a lack of rigorous analytical results describing their global organization in parameter space. We provide a structured two-parameter bifurcation characterization of this system, showing how a suitable parameter selection allows to uncover the global dynamics. We prove that the model is topologically equivalent to a reduced formulation that preserves the qualitative dynamics while simplifying its bifurcation structure. Within this reduced framework, we characterize equilibria and derive analytical conditions for saddle-node and boundary equilibrium bifurcations in the two-parameter [Formula: see text]-plane. Using numerical continuation and targeted search strategies, we compute Hopf, grazing, and homoclinic bifurcation curves. These curves organize the [Formula: see text]-plane into regions with distinct dynamics, revealing in particular small parameter regions where multistability occurs. In particular, the diagram identifies regions where equilibria and limit cycles coexist. The resulting bifurcation diagram provides a unified and rigorous description of the global dynamics, complementing previous numerical studies and offering a consistent interpretation of the multistability observed in the original model.
Neuronal dynamics have been widely studied using both continuous-time and discrete-time models. Among these, the Rulkov map neuron model — a two-dimensional discrete-time dynamical system with slow–fast dynamics — effectively captures complex firing patterns. This study investigates the influence of nonlinear terms on the dynamics of such systems through a comparative analysis of three distinct two-dimensional map-based neuron models derived from the Rulkov framework: the Rulkov map (Lorentzian nonlinearity), the Cubic map (polynomial nonlinearity), and a newly proposed Cubic/Quartic rational map (rational nonlinearity). The proposed map integrates key features of the Rulkov and Cubic maps, reproducing typical burst firing of the Rulkov map and characteristical long-interval spike firing of the Cubic map. Bifurcation analysis of fixed points reveals distinct geometric features among the three models. In particular, the rational nonlinearity of the Cubic/Quartic rational map generates a torsional structure on the manifolds of the Neimark–Sacker and period-doubling bifurcations. This torsional structure prevents the theoretical intersection of Neimark–Sacker and period-doubling bifurcation curves that typically appears at the edge of the fixed-point existence region in the Cubic map. Furthermore, we demonstrate that the emergence of burst firing is underpinned by a symmetry-breaking transition and that the map’s rational structure ensures the system’s boundedness by effectively preventing divergence.
This paper discusses dynamical system synthesis with strange chaotic attractors. Such dynamical systems have abstract mathematical meaning but can be used to solve different applied problems. Two approaches for the formation of strange attractors are proposed. The first approach is a direct approach based on searching the coefficients of the nonlinear differential equation system that satisfy the required conditions for Lyapunov exponents and Kaplan–Yorke dimension. The second approach is an indirect approach based on methods and models of spacecraft attitude dynamics. The search for strange attractors is conducted using the differential evolution algorithm. Some new strange attractors are synthesized and analyzed. The analysis involves obtaining Lyapunov exponents, Kaplan–Yorke dimensions, and bifurcation diagrams.
The dynamics of questioning have been a subject of profound inquiry throughout history, rooted in the Socratic tradition and Platonic dialogues. Maieutics, the “midwifery” of ideas, remains a cornerstone of knowledge discovery in both scientific discourse and contemporary pedagogy. Today, the rapid evolution of Artificial Intelligence (AI) has reinvigorated this topic, necessitating models capable of evaluating whether a dialogue facilitates a genuine increase in knowledge or remains trapped in recursive loops. In this paper, we propose a nonlinear dynamical model of Maieutics based on the Chua’s circuit paradigm. Recognized as a universal model for chaotic dynamics, the Chua system provides an elegant formalization for transitions between certainty, doubt, and synthesis, making real the isomorphism between the Socratic Maieutic and the Chua circuit design. A case study to demonstrate the procedural application of this model, illustrating its suitability for mapping the evolution of cognitive states in both human and AI interactions, is presented.
This paper addresses the problem of simultaneous estimation of faults and attacks in networked Takagi–Sugeno fuzzy chaotic systems. To this end, an enhanced multiple intermediate estimator framework is developed to reconstruct both fault and attack signals in a unified manner without relying on restrictive observer matching conditions. To further improve the performance of the estimations, an optimization strategy is proposed based on the genetic algorithm to optimize the most important parameters of the estimators in a structured way. The tuning problem is outlined as an optimization problem that attempts to minimize the estimation errors of fault and attack signals. Using the global search feature of the genetic algorithm, the proposed approach does not require manual parameter selection and has a higher level of estimation accuracy. Based on the obtained estimates, a compensation mechanism is incorporated to mitigate the adverse effects of faults and attacks on system performance. Sufficient conditions ensuring the stability of the closed-loop system are derived using Lyapunov theory and expressed in terms of linear matrix inequalities. Finally, numerical simulations are provided to validate the effectiveness of the proposed approach. The results demonstrate that the genetic algorithm-optimized estimator significantly improves estimation accuracy and enhances the resilience of the system against simultaneous faults and cyber-attacks.
This study presents a novel approach to designing chaotic systems devoid of linear terms, leveraging a modified Thomas circulant system. By introducing a nonlinear dissipation in the original Thomas system and adopting a strictly nonlinear function, a new family of chaotic systems without linear terms is constructed. Accordingly, five new examples of such systems are presented by selecting five different nonlinear functions. We conduct a comprehensive theoretical analysis of a prototypal system, the model with piecewise cubic nonlinearities, focusing on fixed points and bifurcation behaviors to uncover the dynamical features of the system. Notably, we explore the impact of bidirectional connections between variables, which enhances the complexity and richness of the chaotic dynamics. Additionally, we investigate the effects of circulant symmetry breaking, revealing new insights into stability and bifurcation patterns. The findings indicate potential pathways for generating chaos in systems typically constrained by linear interactions. To validate the theoretical framework, we implement the modified systems on an Arduino module, demonstrating their practical applicability. This implementation not only confirms the theoretical predictions but also paves the way for future explorations in chaotic system design, particularly in fields requiring nonlinear dynamics. Our work offers significant contributions to both theoretical and applied aspects of chaotic systems, with implications for engineering and computational applications.
Sustained chaotic motion systems based on active materials have potential applications in energy harvesting, artificial hearts, medical instruments and other fields. This paper proposes a novel light-fueled sustained chaotic oscillation system consisting of a Liquid Crystal Elastomer (LCE) fiber, a mass ball and a rigid substrate. Under periodic illumination, the fiber in the system expands and contracts continuously, as the mass ball collides with the rigid substrate. Ultimately, the system achieves sustained oscillation motion through the fiber contraction work to compensate for the energy dissipation. With the help of the dynamic constitutive model of LCE, the nonlinear dynamic theoretical model of the system is established. The numerical results demonstrate that the fiber can oscillate in two modes, i.e. periodic oscillation and chaotic oscillation, under periodic illumination. The corresponding mechanisms of sustained periodic oscillation and chaotic oscillation of the fiber system are revealed. Moreover, the effect of system parameters on the sustained oscillation behavior is studied in depth, and the effect of parameter variations on the oscillation mode is illustrated by bifurcation diagrams. The research results of this work can deepen the understanding of chaotic motion and provide ideas for chaotic system design and chaotic machine development.
In this paper, we extend the bifurcation analysis of the classical DeLisi–Rescigno minimal ordinary differential equation model in cancer dynamics with the immune system by explicitly incorporating capillary vascularization into a spherical tumor. Unlike prior treatments where higher-codimension phenomena remained implicit, we provide an explicit and comprehensive bifurcation diagram that identifies Bautin and nondegenerate Bogdanov–Takens bifurcations. In contrast to the nonvascular model, there is a lymphocyte suppression regime, an immune-deactivation state driven by vascular–immune coupling, and introduces the concept of a Bautin bifurcation at infinity within this oncological context. A description of the general bifurcation analysis is provided, thus supplying a biological interpretation.
This paper employs a concise method as a streamlined analytical framework for deriving exact analytical solutions with free parameters for the higher-dimensional ZK-BBM equation and the (2 + 1)-dimensional Bogoyavlenskii system. While structurally related to traditional ansatz-based techniques, this approach reduces the reliance on predefined auxiliary equations or extensive symbolic computation by naturally deducing the governing linear ordinary differential equations during the algebraic balancing process. This offers a direct and manually verifiable derivation mechanism. A set of exact analytical solutions, including bell-shaped bright and kink solitons, are obtained and physically characterized through three-dimensional graphical representations. Dynamical analysis further reveals that these analytical solutions correspond geometrically to the homoclinic or heteroclinic orbits of the unperturbed systems, serving as baseline states for investigating nonlinear wave evolution. By introducing periodic external perturbations, the sensitivity, the coexistence of invariant orbits, and the chaotic behaviors of the corresponding two-dimensional dynamical systems are systematically investigated. The finite-time maximum Lyapunov exponent is used as a quantitative criterion to distinguish between regular quasi-periodic regimes and chaotic states. These results provide detailed insights into complex wave phenomena and the topological dynamics of higher-dimensional nonlinear evolution equations.
In this paper, we propose a spatiotemporal model incorporating memory-driven toxicant-taxis and nonlocal accumulation delay to investigate the interaction between the population and the toxicant in a polluted aquatic environment. Through linear stability theory and bifurcation analysis, we examine how the taxis coefficient, spatial memory delay, and average accumulation delay influence the stability of spatially homogeneous steady states and the formation of spatial patterns. Theoretical analysis reveals that the spatial memory delay does not affect the stability of the coexisting steady state, whereas the taxis coefficient and the average accumulation delay can destabilize it, triggering spatially heterogeneous patterns. We numerically validate these findings and illustrate how key toxicant-related parameters govern the distributions of the population and the toxicant. This study highlights the significant effects of memory-driven avoidance behavior and toxicant cumulative effects on shaping population distributions in polluted aquatic ecosystems.
There are only a few works focused on the design of nonlinear oscillators without linear terms in a lower-order polynomial system. In this paper, a new 4D-autonomous chaotic system with only quadratic nonlinearities is proposed. The preliminary analysis of the introduced model shows that it displays 16 equilibrium points, two of which can undergo Hopf bifurcation. Its complete analysis, based on quantitative and qualitative nonlinear dynamics tools, is presented. We demonstrate that when changing the system parameters, very rich and interesting phenomena are obtained including, for instance, chaos, period-doubling bifurcation, periodic window, periodic oscillations, offset boosting property, and anti-monotonicity. More interestingly, in some ranges of system parameters, the value of the Lyapunov exponent remains unchanged and thus our model exhibits the exciting characteristic of a constant Lyapunov exponent. Furthermore, a pair of symmetric chaotic attractor-repellor is found on the proposed system when the time is reversed. To our best knowledge, this work represents the first report on a 4D-polynomial chaotic system without linear terms investigated up to date. An appropriate analog circuit is constructed to verify and demonstrate the theoretical study.
Safety critical human-robot collaboration requires inverse kinematics schemes that simultaneously maintain tracking accuracy in task space, exploit redundancy, and guarantee collision avoidance in human proximal workspaces. This paper proposes a Chaos-Aware Inverse Kinematics (CAIK) framework in which a low-dimensional deterministic chaotic map generates a structured and repeatable exploration signal injected into the Jacobian null space, while a safety critical quadratic program with higher-order control barrier function constraints enforces admissible motion. Simulation studies on a seven degree-of-freedom collaborative manipulator using a synthetic VR taught trajectory dataset show that CAIK preserves millimeter-scale Cartesian tracking accuracy, achieving RMS(x )of 8.45 & times; 10(-3), 5.80 & times; 10(-3), and 1.00 & times; 10(-2) mm on circular, line, and figure eight tasks, respectively, while the corresponding minimum norm IK baseline yields 6.70 & times; 10(-3), 1.09 & times; 10(-2), and 6.92 & times; 10(-3) mm. When the HoCBF constraints are enabled, collision violations are eliminated, and joint space dispersion increases markedly, reaching MADq = 1.51 rad on the circular task compared to 0.028 rad and 0.071 rad on line and figure eight. These results indicate that deterministic chaos can provide posture diversity without degrading task tracking when coupled with barrier certified optimization, offering a computationally tractable pathway to safer and more versatile redundancy utilization in collaborative robotics.
To address the poor adaptability of traditional fixed control-gain methods to complex topological dynamics, we propose a Dynamic Learning-based Synchronization (DLS) strategy for memristive Hindmarsh-Rose (HR) neuronal networks. By integrating Multifractal Detrended Fluctuation Analysis, we investigate the synchronization mechanisms and multifractal characteristics of the neuronal networks under four topologies: regular, random, small-world, and scale-free. The results demonstrate that the proposed DLS strategy can significantly enhance the steady-state synchronization factor R. The regulation of topological structure on the synchronization performance shows obvious heterogeneity, among which the small-world network that balances local aggregation and global efficiency has the best steady-state synchronization performance, while the synchronization stability of the regular network is relatively poor. It is observed that the synchronization factor R tends to decrease as the singularity spectrum width increases across the four topologies examined. Moreover, the noise experiment shows that a topology with strong synchronization robustness does not necessarily have better fractal robustness. This study clarifies the regulatory mechanism of topological structure on the synchronization behavior and multifractal characteristics of HR neuronal networks, which provides theoretical support for the design of brain-like network control strategies and the evaluation of the network synchronization states.
This research focuses on the study of the bifurcation behavior of the solution for an Infinite System of Partial Differential Continuity Equations (ISPDCEs) appearing in the simulation of stochastic energy exchange between an Open Thermodynamic System (OTS) and incoming raw energy flow. Following the declared aim, ISPDCE reduces to a single randomized nonlinear differential equation. The closed-form solution obtained assumes the existence of the energy infrastructure in the form of two interconnected discrete energy families of admissible values. Analysis of the appropriate families reveals the key role of a phenomenon of symmetry breaking. The interconnected families are composed of the Points of Equilibrium (PoE) that are the placeholders for the energy bifurcations. The primary family is based on natural limitations imposed on the rate of energy exchange, while the secondary family is defined by the fundamental relation connecting the functionally dissimilar parts of the exchange energy. In this sense, the primary family creates the quantitative basis for the realization of the solution, whereas the secondary family presents an optimal way for the evolution of OTS marked by suitable bifurcations. Another important distinction of the secondary family is that it rails off by bifurcations in the energy area of constructive interaction of OTS with an energy flux and the remaining area with the nonevolutionary conditions.
In this work, we investigate the nonlinear dynamics of a Quasi-Zero Stiffness Energy Harvester (QZEH) under quasi-periodic forcing and report a successive localized torus-doubling route that leads to the formation of a Strange Nonchaotic Attractor (SNA), a phenomenon not previously documented in QZEH systems. The presence of the SNA is verified through multiple characterization techniques, including Lyapunov exponents, finite-time Lyapunov statistics, recurrence-based Shannon entropy, and singular continuous spectral analysis. A key methodological contribution of this study is the introduction of a finite-time energy-variance diagnostic that quantifies the consistency of harvested energy across different dynamical regimes. By analyzing the variance of the energy generated over sliding time windows, we show that tori yield low but highly stable energy output, chaotic attractors produce high yet strongly irregular energy, and SNAs provide an intermediate energy level with substantially lower variability than chaos. This reveals SNAs as a favorable compromise between energy yield and robustness.
This paper investigates the effect of fractional parameters on chaos and pattern formation in the fractional-in-time complex Swift-Hohenberg model. A novel higher-order numerical scheme is developed, combining a pth-order Gr & uuml;nwald-Letnikov discretization for the Caputo time-fractional derivative with a nine-point finite difference approximation for spatial operators, achieving fourth-order spatial accuracy and incorporating a short-memory principle for computational efficiency. Linear stability and Turing bifurcation analysis reveal that the fractional order alpha nonmonotonically modulates both the complexity measures and the Turing instability threshold, enabling multiple stability switches and transitions to globally unstable regimes. Numerical simulations demonstrate that alpha acts as a bifurcation parameter, inducing complex pattern selection and chaotic dynamics, with the proposed scheme effectively capturing rich spatiotemporal behaviors across various parameter regimes.
Infectious diseases constitute a multifaceted threat, affecting not just public health but also the social, economic, and psychological aspects of society. Here, we propose and analyze a diffusion-driven nonlinear epidemic model to investigate the impact of psychological fear on the direct transmission patterns of infectious diseases. First, we analyzed the equilibrium and stability of steady states derived from the temporal system. The temporal dynamics of the model system exhibit transcritical and Hopf-bifurcation under certain conditions. Moreover, the stability behavior of the endemic steady-state in a spatially extended setting has been examined. To fit our model, we used actual COVID-19 data from India and estimated essential epidemiological parameters. It is noteworthy to see that the increased psychological fear in a community can reduce the amplitude of bifurcating periodic solutions. Once the psychological fear factor reaches a certain threshold, the oscillations can be diminished, and endemic equilibrium stabilizes to a lower value. Our findings suggest that high levels of epidemic preparedness can mitigate infection in the community in the presence of awareness campaigns. The study additionally demonstrates that when the public migrates frequently, regulating infectious disease transmission patterns becomes challenging. We believe these findings could provide valuable insights into the spatial patterns of real epidemics and aid in the implementation of appropriate nonpharmaceutical interventions to suppress the disease burden.
In this study, we investigate the dynamical behavior of a discrete-time Leslie-Gower predator-prey model incorporating a predation-driven Allee effect in the prey population. The presence of the Allee term introduces nonlinear feedback that significantly alters system stability and long-term dynamics. Through rigorous analytical and numerical approaches, we explore the existence and stability of equilibria, along with bifurcations that govern transitions between different dynamical regimes. Our results demonstrate that the system undergoes rich dynamical phenomena, including flip and Neimark-Sacker bifurcations, leading to periodic oscillations and quasi-periodic solutions. Numerical simulations further reveal that a gradual increase in prey growth rate can trigger the onset of chaos, whereas predation-driven Allee effects may suppress chaotic oscillations. However, these Allee effects also enlarge the basin of attraction of the prey extinction steady state, implying an increased likelihood of prey extinction. Moreover, the interplay between the Allee threshold and intrinsic growth rate induces complex behaviors such as bistability, where the system may converge to distinct attractors depending on initial conditions. These findings provide deeper insights into predator-prey interactions and emphasize the ecological significance of incorporating Allee effects in discrete models.
