Chaos最新文献

This paper delves into a semi-discrete coherently coupled nonlinear Schrödinger equation characterized by a 4×4 matrix spectral problem. Our primary objective is to explore the modulation instability theory of this equation, elucidating its formation mechanism from its plane wave solutions. Second, we aim to demonstrate that this equation can be transformed into a new continuous equation in the context of the continuous limit. Notably, utilizing the established 4×4 matrix spectral problem, we establish a discrete generalized (m,N-m)-fold Darboux transformation, from which we theoretically derive novel rogue wave and periodic wave solutions, as well as their hybrid counterparts. In particular, we obtain discrete rogue waves featuring double peaks and double troughs on a plane wave background, as well as those that exhibit only peaks and lack troughs on a zero background, both of which incorporate arbitrarily controllable position parameters. Subsequently, we graphically analyze all these innovative structures. These findings may hold potential implications for describing the optical pulse propagation in the optical fiber.

The spread of epidemics is often accompanied by the spread of epidemic-related information, and the two processes are interdependent and interactive. A social reinforcement effect frequently emerges during the transmission of both the epidemic and information. While prior studies have primarily examined the role of positive social reinforcement in this process, the influence of negative social reinforcement has largely been neglected. In this paper, we incorporate both positive and negative social reinforcement effects and establish a two-layer dynamical model to investigate the interactive coupling mechanism of information and epidemic transmission. The Heaviside step function is utilized to describe the influence mechanism of positive and negative social reinforcements in the actual transmission process. A microscopic Markov chain approach is used to describe the dynamic evolution process, and the epidemic outbreak threshold is derived. Extensive Monte Carlo numerical simulations demonstrate that while positive social reinforcement alters the outbreak threshold of both information and epidemic and promotes their spread, negative social reinforcement does not change the outbreak threshold but significantly impedes the transmission of both. In addition, publishing more accurate information through official channels, intensifying quarantine measures, promoting vaccines and treatments for outbreaks, and enhancing physical immunity can also help contain epidemics.

Previously, we showed that the orbital motions of a binary system (e.g., two stars clusters in mutual interaction) can be modeled as a Brownian particle immersed in two heat baths (describing the thermodynamic incidence of the internal degrees of freedom). The fluctuations arising from the interaction of this effective particle with the baths lead to dynamical instabilities-escape and collapse events. Now, we focus on determining the quasi-stationary distribution of an ensemble of systems evolving under this stochastic model, specifically in the regime influenced by escape events. To this end, we develop numerical methods to compute the energy distribution of such an ensemble of systems. Notably, the resulting distribution exhibits lowered isothermal profiles akin to those observed in the structure of stellar clusters, such as the King distribution, which correspond to quasi-stationary states with positive heat capacities.

Analytical solutions of space-time fractional partial differential equations (fPDEs) are crucial for understanding dynamics features in complex systems and their applications. In this paper, fractional sub-equation neural networks (fSENNs) are first proposed to construct exact solutions of space-time fPDEs. The fSENNs embed the solutions of the fractional Riccati equation into neural networks (NNs). The NNs are a multi-layer computational models that are composed of weights and activation functions between neurons in the input, hidden, and output layers. In fSENNs, every neuron of the first hidden layer is assigned to the solutions of the fractional Riccati equation. In this way, the new trial functions are obtained. The exact solutions of space-time fPDEs can be obtained by fSENNs. In order to verify the rationality of this method, space-time fractional telegraph equation, space-time fractional Fisher equation, and space-time fractional CKdV-mKdV equation are investigated, and generalized fractional hyperbolic function solutions, generalized fractional trigonometric function solutions, and generalized fractional rational solutions are obtained. Since the fractional sub-equation is applied to the NNs model for the first time, more and new solutions can be obtained in this paper. The dynamic characteristics of some solutions corresponding to waves have been demonstrated through some diagrams.

The convergence toward asymptotic states at bifurcation points (BPs) r=rb of 1D mappings of a free parameter r presents scaling laws whose characteristic exponents in principle should depend on the maps non-linear features. Aiming to better understand such comportment, we investigated the logistic-like and sine-like family of maps by studying transcritical, pitchfork, period-doubling, and tangent BPs. For this, we employed the supertracks framework, where continuous functions of r are generated, having the 1D map critical point as the initial condition. Analyzing these functions we obtained, from numerical and analytical procedures, four exponents to describe the asymptotic behavior when r=rb as well as another exponent typifying the case of r>rb. Moreover, we confirmed the universality classes of transcritical and pitchfork BPs proposed in the literature and unveiled novel universality results for period-doubling and tangent BPs. Our findings highlighted the usefulness of the supertracks method, for instance, helping to uncover universality in dynamical systems and allowing to establish parallels with critical phenomena.

In neuroscience, delayed synaptic activity plays a pivotal and pervasive role in influencing synchronization, oscillation, and information-processing properties of neural networks. In small rhythm-generating networks, such as central pattern generators (CPGs), time-delays may regulate and determine the stability and variability of rhythmic activity, enabling organisms to adapt to environmental changes, and coordinate diverse locomotion patterns in both function and dysfunction. Here, we examine the dynamics of a three-cell CPG model in which time-delays are introduced into reciprocally inhibitory synapses between constituent neurons. We employ computational analysis to investigate the multiplicity and robustness of various rhythms observed in such multi-modal neural networks. Our approach involves deriving exhaustive two-dimensional Poincaré return maps for phase-lags between constituent neurons, where stable fixed points and invariant curves correspond to various phase-locked and phase-slipping/jitter rhythms. These rhythms emerge and disappear through various local (saddle-node, torus) and non-local (homoclinic) bifurcations, highlighting the multi-functionality (modality) observed in such small neural networks with fast inhibitory synapses.

Multi-state quantum molecular dynamics is one of the most accurate methodologies for predicting rates and yields of different chemical reactions. However, the generation of potential energy surfaces (PES), transition dipoles, and non-adiabatic couplings from ab initio calculations become a challenge, especially because of the exponential growth of computational cost as the number of electrons and molecular modes increases. Thus, machine learning (ML) emerges as a novel technique to compute molecular properties using fewer resources. Yet, the validity of ML methodologies continues in constant development, particularly for high-energy regions where conventional ab initio sampling is reduced. We test the accuracy of the potential energy surfaces interpolated with machine learning (ML) techniques in the solution of the time-dependent Schrödinger equation for the conventional IR+UV bond-breaking process of semi-heavy water. We perform a statistical analysis of the differences in expectation values and dissociation probabilities, which depend on the number of ab initio points selected to generate the machine learning potential energy surface (ML-PES). The energy differences of the electronic excited state modify population transfer from the ground state by driving with a UV laser pulse. We consider as the exact solution the photodynamics implemented with analytical expressions of the electronic ground X~1A1 and excited A~1B1 states. The results of the mean bond distance and dissociation probabilities suggest that ML-PES is suitable for dynamics calculations around the Franck-Condon region, and that standard interpolation methods are more efficient for multistate dynamics that involve dissociative and repulsive energy regions of the electronic states. Our work contributes to the continued inclusion of ML tools in molecular dynamics to obtain accurate predictions of dissociation yields with fewer computational resources and non-written rules to follow in multi-state dynamics calculations.

This paper presents a comprehensive analytical study of a two-degree-of-freedom vibrating system with impacts, which can model a kinematically forced cantilever beam with a substantial mass and a concentrated mass at its end that impacts a rigid base during motion. An analytical method, based on Peterka's approach and tailored to the specific features of the system, is developed to analyze periodic motions, with particular emphasis on their occurrence and stability. The influence of system parameters, including clearance, mass distribution, and excitation frequency, on the system behavior is investigated, and parameter ranges are identified that lead to stable periodic solutions. The analytical results are then compared with numerical simulations in which Lyapunov exponents are calculated using an adapted Müller approach for numerical verification of stability. The two methods yield consistent results, confirming the effectiveness and precision of the approaches employed. It is demonstrated that the location and extent of regions of stable periodic solutions are significantly influenced by the relationships between the excitation frequency and the system eigenvalues. These results provide important insights for the design of kinematically forced vibro-impact systems with significant masses of elastic elements.

Complex systems, such as biological networks, often exhibit intricate rhythmic behaviors that emerge from simple, small-amplitude dynamics in individual components. This study explores how significant oscillatory signals can arise from a minimal system consisting of just two interacting units, each governed by a simple non-autonomous delay differential equation with a recently obtained exact analytical solution. Contrary to the common assumption that large-scale oscillations require numerous units, our model demonstrates that rewiring two units from self-feedback to cross-feedback can generate robust, finite-amplitude dynamical oscillations. This phenomenon arises in this context when an appropriate amount of delay is present in the feedback line. Our findings highlight the potential of this minimalistic mechanism to generate high-amplitude dynamical oscillations from much smaller amplitude units, drawing a physical analogy to rewiring feedback lines.

Over the past few decades, many works have studied the evolutionary dynamics of continuous games. However, previous works have primarily focused on two-player games with pairwise interactions. Indeed, group interactions rather than pairwise interactions are usually found in real situations. The public goods game serves as a paradigm of multi-player interactions. Notably, various types of benefit functions are typically considered in public goods games, including linear, saturating, and sigmoid functions. Thus far, the evolutionary dynamics of cooperation in continuous public goods games with these benefit functions remain unknown in structured populations. In this paper, we consider the continuous public goods game in structured populations. By employing the pair approximation approach, we derive the analytical expressions for invasion fitness. Furthermore, we explore the adaptive dynamics of cooperative investments in the game with various benefit functions. First, for the linear public goods game, we find that there is no singular strategy, and the cooperative investments evolve to either the maximum or minimum depending on the benefit-to-cost ratio. Subsequently, we examine the game with saturating benefit functions and demonstrate the potential existence of an evolutionarily stable strategy (ESS). Additionally, for the game with the sigmoid benefit function, we observe that the evolutionary outcomes are closely related to the threshold value. When the threshold is small, a unique ESS emerges. For intermediate threshold values, both the ESS and repellor singular strategies can coexist. When the threshold value is large, a unique repellor displays. Finally, we perform individual-based simulations to validate our theoretical results.