Chaos最新文献

Digital memcomputing machines (DMMs) have been designed to solve complex combinatorial optimization problems. Since DMMs are fundamentally classical dynamical systems, their ordinary differential equations (ODEs) can be efficiently simulated on modern computers. This provides a unique platform to study their performance under various conditions. An aspect that has received little attention so far is how their performance is affected by the numerical errors in the solution of their ODEs and the physical noise they would be naturally subject to if built in hardware. Here, we analyze these two aspects in detail by varying the integration time step (numerical noise) and adding stochastic perturbations (physical noise) into the equations of DMMs. We are particularly interested in understanding how noise induces a chaotic transition that marks the shift from successful problem-solving to failure in these systems. Our study includes an analysis of power spectra and Lyapunov exponents depending on the noise strength. The results reveal a correlation between the instance solvability and the sign of the ensemble averaged mean largest Lyapunov exponent. Interestingly, we find a regime in which DMMs with positive mean largest Lyapunov exponents still exhibit solvability. Furthermore, the power spectra provide additional information about our system by distinguishing between regular behavior (peaks) and chaotic behavior (broadband spectrum). Therefore, power spectra could be utilized to control whether a DMM operates in the optimal dynamical regime. Overall, we find that the qualitative effects of numerical and physical noise are mostly similar, despite their fundamentally different origin.

Various extensions of evolutionarily stable strategy (ESS)-the central concept in evolutionary game theory-defined for asymmetric games differ in how they correspond to fixed points of the replicator equation, which models evolutionary dynamics of frequencies of strategies in a population. Along with reporting interesting new results, this paper is partially intended as a contextual mini-review of some of the most important definitions of ESS in asymmetric games. We present the definitions coherently and scrutinize them closely while establishing equivalences-some of them hitherto unreported-between them. Since it is desirable that a definition of ESS should correspond to asymptotically stable fixed points of the replicator dynamics, we bring forward the connections between various definitions and their dynamical stabilities. Furthermore, in this context, we use the principle of relative entropy minimization to gain information-theoretic insights into the concept of ESS, thereby establishing a threefold connection between game theory, dynamical system theory, and information theory.

This paper investigates the periodic modulation on phase and backgrounds for breathers and rogue waves by spin-orbit coupling (SOC) and Raman coupling in two-component Bose-Einstein condensate systems. First, linear stability analysis examines modulation instability and identifies modulation-stable and unstable parameter regimes across different parameter planes. Subsequently, we establish Lax pairs and (n,N-n)-fold generalized Darboux transformation to construct exact analytical solutions for diverse breathers and rogue waves featuring periodic phase distributions and periodic backgrounds. Based on these solutions, rigorous analysis reveals that SOC induces spatiotemporal periodic modulation on the phase and spin-density distributions of breathers and rogue waves, while Raman coupling generates periodic modulation in their backgrounds. By adjusting Raman coupling strength and initial plane-wave amplitudes and wavenumbers, backgrounds exhibiting double-periodic, single-periodic, or non-periodic modulation can be excited. Additionally, we discover higher-order breathers with curved trajectories and dual-rogue-wave structures. Nonlinear interactions among breathers and rogue waves of distinct structures and orders are systematically investigated. Finally, the stability of the analytical solutions for breathers was verified through numerical simulations. This study may facilitate a deeper understanding of periodic modulation mechanisms for localized waves under SOC and Raman coupling.

Conventional multi-scroll chaotic systems (MSCSs) typically exhibit uniform scroll distributions, limiting the diversity of attractor structures. In contrast, non-uniformly distributed MSCSs can overcome this constraint, which enables more flexible attractor configurations and enhances their potential in practical engineering applications. In this study, five modified sawtooth wave functions are proposed and embedded into a three-dimensional chaotic system to generate five types of multi-scroll attractors with irregular spatial distributions, including (1) attractors with enlarged scroll structures on both sides, (2) attractors with an enlarged central scroll structure, (3) attractors with a central separation structure, (4) attractors with enlarged scroll structures at the center and both sides, and (5) attractors with separated scrolls and enlarged side scrolls. Among these, the third and fifth types exhibit attractor coexistence. Furthermore, by selecting and combining two different modified sawtooth functions, four types of grid multi-scroll attractors are constructed: (1) attractors with separated structures and varying scroll sizes, (2) attractors with cross-shaped separated structures, (3) attractors with a double-chain structure, and (4) attractors with a triple-chain structure. Among them, the cross-shaped type also exhibits attractor coexistence. This study systematically analyses the generation mechanisms of these non-uniform multi-scroll attractors and examines their offset-boosting phenomenon. The chaotic characteristics of different types of attractors are analyzed using the largest Lyapunov exponent, bifurcation diagrams, and spectral entropy. In addition, the National Institute of Standards and Technology test is employed to validate the randomness of the proposed systems. Finally, hardware implementation on a digital signal processing platform confirms its applicability for practical engineering applications.

In the early stages of rumor dissemination, accurately locating the source of transmission is crucial for the management and control of information flow. However, the inherent uncertainty in information dissemination complicates precise source localization. Although incorporating transmission direction can alleviate some of this uncertainty, thereby facilitating source localization, traditional methods still depend on the often inaccurate informed timestamps of observed nodes. To address this limitation, this paper develops a method to infer all rumor sources using a single snapshot of observed nodes at a specific time and the direction of transmission toward these nodes, without requiring prior knowledge of the informed timestamps. First, we conduct a theoretical analysis demonstrating how the network structure can be pruned based on the status and transmission direction of the observers. Subsequently, we propose the Reduce Candidate Source algorithm, which operates within the pruned subgraph to identify the node with the minimum sum of shortest paths to the informed observers as the potential source of propagation. Additionally, we propose the Reduce Candidate Source by Deleting All algorithm for propagation models characterized by high certainty, retaining only the informed observers and those nodes indicating the transmission direction to further narrow the candidate source range. Finally, extensive experiments confirm that our two proposed methods align with the theoretical analysis, effectively identifying the sources of rumor dissemination in the early stages, even under conditions where the propagation model and informed are unknown.

Understanding and predicting how mechanical systems respond to environmental variability is essential for advancing next-generation robotic systems with physical intelligence. In this study, we investigated the use of echo state networks (ESNs), a representative class of reservoir computing (RC) models, to predict the bifurcation structures of real-world mechanical systems from limited observations. We examined two representative cases: a simulated passive dynamic walking (PDW) robot with hybrid continuous-discrete dynamics and a real-world soft pneumatic artificial muscle (PAM) actuator whose electrical resistance undergoes complex changes under varying loads. To address the challenges posed by the PDW's hybrid dynamics, we proposed a hybrid ESN (HESN) model that integrates a knowledge-based touchdown detection mechanism with an ESN module. The HESN successfully reproduced the route-to-chaos bifurcation structure of the PDW, captured its multi-attractor dynamics, and demonstrated robustness against imperfect domain knowledge. For the PAM, where no reliable physical model is available, a purely data-driven ESN accurately predicted resistance bifurcations across changing environmental conditions. These results highlight the potential of RC models as flexible digital twins for mechanical systems, enabling parameter-aware modeling of bifurcations with limited training data and supporting the design of robust, adaptive robots capable of operating in complex environments.

Following significant advances in microscopic and macroscopic single-particle tracking and supercomputing, the theoretical investigation of fluctuations and anomalous dynamics in complex systems is currently of high interest. Stochastic processes and their generalizations represent an important tool for the statistical description of such systems. Modeling random walks and stochastic processes in complex systems, including complex networks and graphs, requires an interdisciplinary approach due to the different applications in various fields, such as physics, biology, chemistry, engineering, computer science, and economy. Various studies of active and passive tracer diffusion, for instance, in biological cells and in heterogeneous and porous media showed that the underlying structure of the environment has a strong effect on the particle movement, leading to anomalous dynamics due to the constrained particle motion or the variation of the local diffusion coefficient and the potential energy function. Moreover, determining optimal search strategies is central in diverse fields, from physics to computer science, from biology to robotics. In particular, random search strategies have been widely observed for animal foraging, in reaction pathways in DNA-binding proteins, in intracellular transport, etc. Furthermore, it has been shown that the resetting of the searcher to its initial position can improve the search strategy by appropriate optimal resetting rate, which results in minimizing the mean first-passage time. This Editorial is meant to serve as an Introduction to this Focus Issue in the form of a mini-review of the field.

This study extends the concept of survivability from network security to coupled oscillators, introducing dynamic survivability as a novel notion to describe a system's capacity to sustain specific collective dynamics under attacks, which is crucial for ensuring functional integrity. A general analytical framework along with quantitative metrics is established to evaluate this capability. Focusing on synchronization as the key task, we theoretically analyze heterogeneous coupled oscillators under both the all-to-all network and complex networks, and derive closed-form expressions for critical attack cost. Numerical simulations show strong agreement with theoretical predictions, validating the proposed framework. Furthermore, we reveal a counterintuitive principle: increased heterogeneity in dynamical parameters-measured by the standard deviation of the Hopf bifurcation parameter-significantly enhances system survivability against attacks. This finding holds across the all-to-all network, Erdős-Rényi random network, and Barabási-Albert scale-free network, demonstrating generality beyond specific structures. Our work establishes a new analytical framework for dynamic survivability in oscillator networks and suggests that engineering parameter diversity rather than pursuing homogeneity offers a promising pathway for designing robust systems.

This work introduces a generalization of the Muthuswamy-Chua system of equations. The generalization allows us to embed memristive/hysteretic systems in a time-varying Hamiltonian, which acts as an energy-like Lyapunov function, whose growth or decay is governed by the memristor feedback and by whether the oscillator is momentarily kinetic- or potential-dominated. We perform mathematical analysis and detailed numerical investigations of the fourth-order Muthuswamy-Chua system based on bifurcation diagram and Lyapunov characteristic exponents. We then highlight bifurcation routes from torus breakdown to homoclinic chaos following the Newhouse-Ruelle-Takens scenario. The corresponding electronic oscillator is then analyzed and validated by SPICE (Simulation Program with Integrated Circuit Emphasis) simulations. To our knowledge, our work represents a novel memristive approach to describing a harmonic oscillator interacting with a finite "bath."

Stochastic resonance (SR) effect observed in biological, physical, and engineering systems is commonly described quantitatively by power spectral measures that require complex mathematical operations and long, continuous observation. Here, we propose two measures based on the switch-phase distribution to qualitatively describe the SR effect, namely, the power norm and the probability that the switch phase lies within a specific range around the peak of the switch-phase distribution. They are easy to be practically determined from a single long run or from multiple short runs. Further, theses metrics were used to quantitatively describe the SR effect observed experimentally in Chua's circuit, operating in chaotic single-scroll regime, forced by 1 kHz sinusoidal subthreshold internal electric or external magnetic signal with switches between attractors induced by internal electric Gaussian noise. The dependence of the switch-phase distributions on the noise intensity for two types of oriented switches are presented. The proposed measures give the optimal noise level as obtained with the widely used signal-to-noise ratio (SNR) measure. The dependence of the first measure on the noise intensity is the same as the SNR dependence. The second measure decreases with increasing noise intensity and has an inflection point at the optimal noise intensity, being almost linear in the vicinity of this point. This dependence on the noise intensity hints for many potential applications, e.g., to aperiodic signal coding and decoding. Both measures are particularly useful for adaptive stochastic resonance and parallel processing.