Search Results (164)

Understanding how information spreads in complex networks is essential for analyzing social influence, opinion formation, and the emergence of collective behavior. In many real-world systems, interactions are not instantaneous but involve delays due to communication, cognition, and response times. Motivated by this observation, the present paper investigates a delayed network model of information spreading, focusing on how time delay and interaction strength shape the system’s dynamical behavior. The novelty of the proposed approach lies in the formulation of a discrete-time network model that explicitly incorporates delayed interactions within a nonlinear dynamical framework. Using delay difference equations, the model captures both local coupling effects and memory-driven feedback, allowing for a systematic study of their combined impact on stability and complexity. Analytical results establish the existence of steady states and provide conditions for their local stability, revealing critical thresholds at which the system undergoes qualitative transitions. These findings are complemented by extensive numerical simulations. In particular, bifurcation analysis and the computation of the largest Lyapunov exponent demonstrate a progression from stable equilibria to oscillatory behavior, and further to chaotic dynamics as the delay and coupling strength increase. Our results highlight the fundamental role of delay as a mechanism that enhances nonlinear complexity and promotes unpredictable dynamics in networked systems. These insights contribute to a deeper understanding of information propagation processes, and may inform the design and control of spreading phenomena in social and technological networks. Full article

Cycle-to-cycle variability of switching parameters inherent to memristive devices introduces significant problems in the design of neuromorphic systems and non-volatile memory. This study investigates the dynamics of a second-order memristive system incorporating capacitive effects that model parasitic charge within individual memristors, addressing both the technical need for accurate analysis of complex regimes and the demand for exploratory environments. Simulations were performed using CUDAynamics, an interactive software suite developed by the authors, which utilizes parallel computing, primarily via NVIDIA Compute Unified Device Architecture (CUDA). It integrates multiple analysis tools for dynamical systems, including bifurcation diagrams, the largest Lyapunov exponent and periodicity mapping, and interactive navigation in multidimensional parameter spaces. The memristive system was discretized applying multiple integration methods with a fixed time step and various waveforms of the input signal. Analysis tools revealed well-defined regions of chaotic dynamics in the memristor resistance parameter space as functions of input signal properties. Sinusoidal and triangular waveforms produced topologically similar distributions of dynamical regimes, whereas the square waveform, mimicking digital inputs, generated distinct dynamical patterns while still preserving chaotic trajectories under specific conditions. Interactive visualization capabilities of CUDAynamics effectively demonstrate attractor evolution and hysteresis deformation, providing immediate visual feedback that significantly enhances conceptual comprehension of nonlinear feedback mechanisms. Beyond its practical implications for the design of analog and digital memristive devices, CUDAynamics offers a scalable, open-source toolkit to aid researchers and engineers in exploring complex dynamical phenomena. Full article

In this article, we use truncated M-fractional derivatives to analyze the bifurcation and chaotic behavior of and traveling-wave solutions to the Zhiber–Shabat equation. By introducing truncated M-fractional derivatives, the equation exhibits richer dynamic properties. Based on phase diagram analysis and dynamical system theory, the bifurcation behavior of the equilibrium point of a two-dimensional dynamical system is discussed. At the same time, the dynamical behavior of a two-dimensional dynamical system with periodic disturbances is considered, revealing the complex chaotic phenomena of the system under specific parameters. A planar phase diagram, a three-dimensional phase diagram, a sensitivity analysis, and a maximum Lyapunov exponent diagram of the perturbed two-dimensional dynamical system were employed. Furthermore, various forms of accurate analytical solutions were obtained through traveling-wave transformation and numerical simulation. The three-dimensional, two-dimensional, density, and polar coordinates of the solutions were plotted using mathematical software. The results indicate that the fractional order and system parameters have a significant impact on the morphology and chaotic characteristics of the solution. This study provides new theoretical insights into the nonlinear dynamics of fractional-order Zhiber–Shabat equations. Full article

The paper deals with the asymptotic behavior of a widely used correlation characteristic in large quantum systems. The correlation is quantum entanglement, the characteristic is entanglement entropy, and the system is an ideal gas of lattice fermions. If the one-body Hamiltonian of fermions is an ergodic finite difference operator with an exponentially decaying spectral projection, then the large-block form of the entanglement entropy is the so-called area law. However, the only class of one-body Hamiltonians for which this spectral condition was verified consists of discrete Schrödinger operators with random potential. In this paper, we prove the area law for several classes of Schrödinger operators whose potentials are ergodic but not random. We begin with quasiperiodic and limit-periodic operators and then move to a highly non-trivial case of potentials generated by subshifts of finite type. These arose in the theory of dynamical systems when studying chaotic phenomena. The corresponding asymptotic study requires involved spectral analysis, which therefore constitutes the bulk of the paper. Specifically, we prove uniform localisation of the eigenfunctions for the Maryland model and exponential decay of the eigenfunction correlator for various models. We believe these properties are of significant independent interest. Full article

In this paper, a supply–demand–price network model incorporating a marginal feedback mechanism is proposed to characterize the evolution of market prices. Unlike classical supply–demand models, the marginal effect of excess demand, defined as the rate of change in excess demand, is explicitly introduced into the price adjustment process. As the coefficient of the marginal feedback term varies, the system exhibits rich and complex nonlinear dynamics. In particular, the model gives rise to a centrally symmetric double-wing chaotic attractor, as well as a pair of coexisting single-wing chaotic attractors. The transition routes among different dynamical regimes are systematically analyzed using phase portraits, bifurcation diagrams, and Lyapunov exponents. Furthermore, multistability phenomena are observed, including the coexistence of equilibrium points, limit cycles, and chaotic attractors. The corresponding basins of attraction are illustrated to reveal their intricate and interwoven structures. In addition, the emergence of endogenous chaos is investigated through both theoretical analysis and numerical simulations. Finally, the consistency between the model dynamics and real market data provides empirical evidence supporting the validity and applicability of the proposed framework. Full article

This paper is devoted to the problem of identifying the mechanisms of hard excitation of oscillations in coupled systems of equilibrium neurons. In this study, a system of two coupled Chialvo neurons is used. For the deterministic model, we studied how increased coupling causes an abrupt transformation of the quiescent neurons into complex oscillations, both regular and chaotic. We show that even in the case when the deterministic system is in equilibrium, similar spike oscillations can be generated by noise. The important role of fractal basins of short and long deterministic transients is discussed. The potential of the principal directions and confidence domain methods for analyzing noise-induced excitation is demonstrated. The phenomena of coherence resonance and the global transition from order to chaos are explored. Full article

With the rapid advancement of exoskeletons and rehabilitation robotics, modern healthcare increasingly demands high dynamic accuracy and reliability from medical devices. However, the dynamic response and durability of mechanical systems are greatly influenced by the inevitable existence of clearances in kinematic joints. Existing studies predominantly focus on simplified planar or spatial mechanisms, offering limited guidance for complex mechanical structures in medical applications. To address this issue, a unified modeling framework is proposed in this study to explore the nonlinear dynamic behavior and wear properties of bionic humanoid rigid mechanisms incorporating revolute joint clearances. A dynamic model that accounts for revolute joint clearances is established, employing the Lankarani–Nikravesh contact model alongside a refined Coulomb friction approach to characterize contact behavior. To characterize the wear progression between the shaft and the bushing, the Archard wear model is employed, while the system’s dynamic equations are formulated using the Lagrange multiplier approach. Systematic simulations are conducted to examine the effects of clearance size, location, and multi-clearance coupling on dynamic response and wear behavior. The results reveal that clearances at the hip joint have the most pronounced impact on system performance, tibiofemoral joint clearances exacerbate precision disturbances, and foot-end clearances considerably undermine system robustness. Increased clearance sizes and the coexistence of multiple clearances aggravate wear and induce more severe nonlinear dynamic phenomena. Phase portraits and Poincaré maps further illustrate that the system may exhibit complex or chaotic behavior under certain conditions. This study offers theoretical insights into performance degradation mechanisms in humanoid robots with joint clearances and introduces a modular “driving–mid–terminal” structure that enhances model generality, enabling its application to exoskeletons and rehabilitation devices for design optimization, service life prediction, and health monitoring. Full article

This paper proposes a novel four-dimensional strongly dissipative nonlinearly coupled hyperchaotic system, investigates its dynamical characteristics, and demonstrates its applicability through Deoxyribonucleic Acid (DNA)-encoded RGB image encryption. First, a four-dimensional nonlinearly coupled hyperchaotic system with strong dissipativity is constructed. Nonlinear dynamics analysis methods, including phase trajectory diagrams, Lyapunov exponent spectra, and bifurcation diagrams, are employed to thoroughly reveal the system’s complex dynamical evolution mechanisms. The analysis indicates that the system not only possesses a wide range of chaotic parameters but also exhibits rich phenomena of multiple coexisting attractors, demonstrating a high degree of multistability. This characteristic offers potential advantages for image encryption, as it increases the diversity of dynamical behaviors and enhances sensitivity to initial conditions. The physical realizability of the chaotic behavior is further verified through an analog circuit implementation. Consequently, the system supports the design of encryption algorithms with larger key spaces, stronger resistance to phase space reconstruction, and improved pseudo-randomness, making it particularly suitable for applications with extremely high security requirements. Subsequently, leveraging the highly random chaotic sequences generated by this system, combined with various DNA coding rules and operations, the RGB image components are scrambled and diffused for encryption. Security analysis demonstrates that the algorithm effectively passes examinations across multiple dimensions, including histogram analysis, information entropy, adjacent pixel correlation, Number of Pixel Change Rate (NPCR), Unified Average Changing Intensity (UACI), and The Peak Signal-to-noise Ratio (PSNR). It achieves favorable encryption results, significantly enhances image resistance against attacks, and provides a reliable technical solution for the secure transmission of remote sensing and military images. Full article

In this study, we develop and investigate a novel fractional discrete-time computer virus dynamics model in two dimensions with a memristive nonlinear coupling mechanism. The memristor introduces nonlinearity by having memory regulation that depends on the state and enhances the propagation dynamics of virus spread. By investigating both matching and non-matching fractional orders, it is then possible to derive useful knowledge with respect to cooperating roles in terms of fractional memory and memristive effects. The complexity behind it is confirmed via 3D phase portraits, bifurcation analysis with LE_max calculation, 0–1 chaos test, and SE complexity. Numerical results reveal rich dynamical phenomena, including periodic oscillations, quasi-periodicity, and strong chaos. In fact, positive LE_max values, Brownian-like trajectories, and high-complexity SE corroborate the chaotic nature of the regimes. Thereby, the fractional-order separation in noncommensurate conditions is a marker of chaotic motion, magnified in the emergently high-dimensional space introduced by the memristive element. As these results indicate that the derivative model proposed here provides an excellent fit for complex viruses present in scaffolds, it may prove to be a useful modeling tool. Full article

The Rössler system is a paradigmatic chaotic oscillator widely used to investigate synchronization phenomena. Existing studies on monovariate coupling almost exclusively rely on the x or y variables, while coupling through z is commonly regarded as ineffective. In this work, we report that complete synchronization through the z variable is indeed possible, provided that specific parameter values are chosen. We further consider a parameter regime in which the Rössler system exhibits multistability and show that synchronization via z-coupling occurs only when the dynamics evolve on a particular attractor. Although synchronization can be achieved, the admissible range of coupling strengths is very narrow as determined by the master stability function. For small networks, full connectivity is required, whereas larger networks can tolerate the removal of a limited number of links without losing synchronization. An analytical expression predicting the fraction of connections that must be preserved as a function of network size is derived and validated, revealing that a very high average degree is necessary. This effectively excludes common topologies such as small-world and scale-free networks. Numerical examples with up to 100 oscillators are presented, and potential challenges that may yield new insights are discussed. Full article

This work explores the qualitative dynamics of the modified unstable time-fractional nonlinear Schrödinger equation (mUNLSE), a model applicable to nonlinear wave propagation in plasma and optical fiber media. By transforming the governing equation into a planar conservative Hamiltonian system, a detailed bifurcation study is carried out, and the associated equilibrium points are classified using Lagrange’s theorem and phase-plane analysis. A family of exact wave solutions is then constructed in terms of both trigonometric and Jacobi elliptic functions, with solitary, kink/anti-kink, periodic, and super-periodic profiles emerging under suitable parameter regimes and linked directly to the type of the phase plane orbits. The validity of the solutions is discussed through the degeneracy property which is equivalent to the transmission between the phase orbits. The influence of the fractional derivative order on amplitude, localization, and dispersion is illustrated through graphical simulations, exploring the memory impacts in the wave evolution. In addition, an externally periodic force is allowed to act on the mUNLSE model, which is reduced to a perturbed non-autonomous dynamical system. The response to periodic driving is examined, showing transitions from periodic motion to quasi-periodic and chaotic regimes, which are further confirmed by Lyapunov exponent calculations. These findings deepen the theoretical understanding of fractional Schrödinger-type models and offer new insight into complex nonlinear wave phenomena in plasma physics and optical fiber systems. Full article

This paper considers the solution behavior and dynamical properties of the variable-order fractional Newton–Leipnik system defined via Liouville–Caputo derivatives of variable order. In contrast to integer-order models, the presence of variable-order fractional operators in the Newton–Leipnik structure enriches the model by providing memory-dependent effects that vary with time; hence, it is capable of a broader and more flexible range of nonlinear responses. Numerical simulations have been conducted to study how different order functions influence the trajectory and qualitative dynamics: clear transitions in oscillatory patterns have been identified by phase portraits, time-series profiles, and three-dimensional state evolution. The work goes further by considering the development of bifurcations and chaotic regimes and stability shifts and confirms the occurrence of several phenomena unattainable in fixed-order and/or integer-order formulations. Analysis of Lyapunov exponents confirms strong sensitivity to the initial conditions and further details how the memory effects either reinforce or prevent chaotic oscillations according to the type of order function. The results, in fact, show that the variable-order fractional Newton–Leipnik framework allows for more expressive and realistic modeling of complex nonlinear phenomena and points out the crucial role played by evolving memory in controlling how the system moves between periodic, quasi-periodic, and chaotic states. Full article

Chaotic dynamical systems are ubiquitous in nature and modern technology, with applications ranging from secure communications and cryptography to the design of chaos-based sensors and modeling biological phenomena such as arrhythmias and neuronal behavior. Given their complexity, precise analysis of these systems is crucial for both theoretical understanding and practical implementation. The characterization of chaotic dynamical systems typically relies on conventional measures such as Lyapunov exponents and fractal dimensions. While these metrics are fundamental for describing dynamical behavior, they are often computationally expensive and may fail to capture subtle changes in the overall geometry of the attractor, limiting comparisons between systems with topologically similar structures and similar values in common chaos metrics such as the Lyapunov exponent. To address this limitation, this work proposes a geometric framework that treats chaotic attractors as spatial objects, using topological tools—specifically the $\alpha$-sphere—to quantify their shape and spatial extent. The proposed method was validated using Chua’s system (including two reported variations), the Rössler system (standard and piecewise-linear), and a fractional-order multi-scroll system. A parametric characterization of the Rössler system was also performed by varying parameter b. Experimental results show that this geometric approach successfully distinguishes between attractors where classical metrics reveal no perceptible differences, in addition to being computationally simpler. Notably, we observed geometric variations of up to 80% among attractors with similar dynamics and introduced a specific index to quantify these global discrepancies. Although this geometric analysis serves as a complement rather than a substitute for chaos detection, it provides a reliable and interpretable metric for differentiating systems and selecting attractors based on their spatial properties. Full article

Memristor is widely used to construct various memristive neural networks with complex dynamical behaviors. However, hidden multiple attractors have never been realized in memristive neural networks. This paper proposes a novel chaotic system based on a memristive Hopfield neural network (HNN) capable of generating hidden multiple attractors. A multi-segment memristor model with multistability is designed and serves as the core component in constructing the memristive Hopfield neural network. Dynamical analysis reveals that the proposed network exhibits various complex behaviors, including hidden multiple attractors and a super multi-stable phenomenon characterized by the coexistence of infinitely many double-chaotic attractors—these dynamical features are reported for the first time in the literature. This encryption process consists of three key steps. Firstly, the original chaotic sequence undergoes transformation to generate a pseudo-random keystream immediately. Subsequently, based on this keystream, a global permutation operation is performed on the image pixels. Then, their positions are disrupted through a permutation process. Finally, bit-level diffusion is applied using an Exclusive OR(XOR) operation. Relevant research shows that these phenomena indicate a high sensitivity to key changes and a high entropy level in the information system. The strong resistance to various attacks further proves the effectiveness of this design. Full article

In this study, we investigate a nonlinear Schrödinger equation relevant to the evolution of optical beams in weakly nonlocal media. Utilizing the modified F-expansion method, we construct a variety of novel soliton solutions, including dark, bright, and wave solitons. These solutions are illustrated through comprehensive graphical simulations, including 2D contour plots and 3D surface profiles, to highlight their structural dynamics and propagation behavior. The effects of the temporal parameter on soliton formation and evolution are thoroughly analyzed, demonstrating its role in modulating soliton shape and stability. To further explore the system’s dynamics, chaos and sensitivity theories are employed, revealing the presence of complex chaotic behavior under perturbations. The outcomes underscore the versatility and richness of the present model in describing nonlinear wave phenomena. This work contributes to the theoretical understanding of soliton dynamics in weakly nonlocal nonlinear optical systems and supports advancements in photonic technologies. This study reports a novel soliton structure for the weak nonlocal cubic–quantic NLSE and also details the comprehensive chaotic and sensitivity analysis that represents the unexplored dynamical behavior of the model. This study further demonstrates how the underlying nonlinear structures, along with the novel solitons and chaotic dynamics, reflect key symmetry properties of the weakly nonlocal cubic–quintic Schrödinger model. These results enhanced the theoretical framework of the nonlocal nonlinear optics and offer potential implications in photonic waveguides, pulse shape, and optical communication systems. Full article