Chaos最新文献

In this paper, we give a class of one-dimensional discrete dynamical systems with state space N+. This class of systems is defined by two parameters: one of them sets the number of nearest neighbors that determine the rule of evolution, and the other parameter determines a segment of natural numbers Λ={1,2,…,b}. In particular, we investigate the behavior of a class of one-dimensional maps where an integer moves to an other integer given by the sum of the nearest neighbors minus a multiple of b∈N+. We find the coexistence of fixed points and periodic cycles. Two single parameter families of maps are introduced and their dynamics in the segment of natural sequence Λ. Furthermore, an order of the numbers of the set Λ-b is given by these families. Last, we present a connection of the N+ generated by the orbits of a particular case.

Bifurcations are one of the most remarkable features of dynamical systems. Corral et al. [Sci. Rep. 8(11783), 2018] showed the existence of scaling laws describing the transient (finite-time) dynamics in discrete dynamical systems close to a bifurcation point, following an approach that was valid for the transcritical as well as for the saddle-node bifurcations. We reformulate those previous results and extend them to other discrete and continuous bifurcations, remarkably the pitchfork bifurcation. In contrast to the previous work, we obtain a finite-time bifurcation diagram directly from the scaling law, without a necessary knowledge of the stable fixed point. The derived scaling laws provide a very good and universal description of the transient behavior of the systems for long times and close to the bifurcation points.

We introduce mutations in the process of discrete iterations of complex quadratic maps in the family fc(z)=z2+c. More specifically, we consider a "correct" function fc1 acting on the complex plane. A "mutation" fc0 is a different ("erroneous") map acting on a locus of given radius r around a mutation focal point ξ∗. The effect of the mutation is interpolated radially to eventually recover the original map fc1 when reaching an outer radius R. We call the resulting map a "mutated" map. In the theoretical framework of mutated iterations, we study how a mutation affects the temporal evolution of the system and the asymptotic behavior of its orbits. We use the prisoner set of the system to quantify simultaneously the long-term behavior of the entire space under mutated maps. We analyze how the position, timing, and size of the mutation can alter the system's long-term evolution (as encoded in the topology of its prisoner set). The framework is then discussed as a metaphoric model for studying the impact of copying errors in natural replication systems.

Modeling how a shock propagates in a temporal network and how the system relaxes back to equilibrium is challenging but important in many applications, such as financial systemic risk. Most studies, so far, have focused on shocks hitting a link of the network, while often it is the node and its propensity to be connected that are affected by a shock. Using the configuration model-a specific exponential random graph model-as a starting point, we propose a vector autoregressive (VAR) framework to analytically compute the Impulse Response Function (IRF) of a network metric conditional to a shock on a node. Unlike the standard VAR, the model is a nonlinear function of the shock size and the IRF depends on the state of the network at the shock time. We propose a novel econometric estimation method that combines the maximum likelihood estimation and Kalman filter to estimate the dynamics of the latent parameters and compute the IRF, and we apply the proposed methodology to the dynamical network describing the electronic market of interbank deposit.

Reputation and punishment are significant guidelines for regulating individual behavior in human society, and those with a good reputation are more likely to be imitated by others. In addition, society imposes varying degrees of punishment for behaviors that harm the interests of groups with different reputations. However, conventional pairwise interaction rules and the punishment mechanism overlook this aspect. Building on this observation, this paper enhances a spatial public goods game in two key ways: (1) We set a reputation threshold and use punishment to regulate the defection behavior of players in low-reputation groups while allowing defection behavior in high-reputation game groups. (2) Differently from pairwise interaction rules, we combine reputation and payoff as the fitness of individuals to ensure that players with both high payoff and reputation have a higher chance of being imitated. Through simulations, we find that a higher reputation threshold, combined with a stringent punishment environment, can substantially enhance the level of cooperation within the population. This mechanism provides deeper insight into the widespread phenomenon of cooperation that emerges among individuals.

We consider a dynamical system undergoing a saddle-node bifurcation with an explicitly time-dependent parameter p(t). The combined dynamics can be considered a dynamical system where p is a slowly evolving parameter. Here, we investigate settings where the parameter features an overshoot. It crosses the bifurcation threshold for some finite duration te and up to an amplitude R, before it returns to its initial value. We denote the overshoot as safe when the dynamical system returns to its initial state. Otherwise, one encounters runaway trajectories (tipping), and the overshoot is unsafe. For shallow overshoots (small R), safe and unsafe overshoots are discriminated by an inverse square-root border, te∝R-1/2, as reported in earlier literature. However, for larger overshoots, we here establish a crossover to another power law with an exponent that depends on the asymptotics of p(t). For overshoots with a finite support, we find that te∝R-1, and we provide examples for overshoots with exponents in the range [-1,-1/2]. All results are substantiated by numerical simulations, and it is discussed how the analytic and numeric results pave the way toward improved risk assessments separating safe from unsafe overshoots in climate, ecology, and nonlinear dynamics.

An Ott-Antonsen reduced M-population of Kuramoto-Sakaguchi oscillators is investigated, focusing on the influence of the phase-lag parameter α on the collective dynamics. For oscillator populations coupled on a ring, we obtained a wide variety of spatiotemporal patterns, including coherent states, traveling waves, partially synchronized states, modulated states, and incoherent states. Back-and-forth transitions between these states are found, which suggest metastability. Linear stability analysis reveals the stable regions of coherent states with different winding numbers q. Within certain α ranges, the system settles into stable traveling wave solutions despite the coherent states also being linearly stable. For around α≈0.46π, the system displays the most frequent metastable transitions between coherent states and partially synchronized states, while for α closer to π/2, metastable transitions arise between partially synchronized states and modulated states. This model captures metastable dynamics akin to brain activity, offering insights into the synchronization of brain networks.

Traveling waves of excitation arise from the spatial coupling of local nonlinear events by transport processes. In corrosion systems, these electro-dissolution waves relay local perturbations across large portions of the metal surface, significantly amplifying overall damage. For the example of the magnesium alloy AZ31B exposed to sodium chloride solution, we report experimental results suggesting the existence of a vulnerable zone in the wake of corrosion waves where local perturbations can induce a unidirectional wave pulse or segment. The evolution of these segments, combined with the absence of rotating spiral waves, imply subexcitable dynamics for which the segments' open ends tangentially retract. Using a simple excitable reaction-diffusion model, we identify parameters that replicate these experimental observations. Under these conditions, small protected disks act as wavebreakers, disrupting continuous fronts, which then shrink and disappear. We further explore different placement schemes of these wavebreakers to optimize potential corrosion mitigation. For constant surface coverage, many small wavebreakers prove more effective than a few large ones. A comparison of triangular, square, rectangular, hexagonal, aperiodic Penrose, and random lattice geometries indicates that triangular placements of wavebreakers are generally the optimal choice, while rectangular and random lattices perform poorly. Although wavebreakers were not demonstrated experimentally in this study, these findings provide concrete design guidance for the protection of alloy surfaces prone to wave-mediated corrosion.

We consider the effect of the emergence of chimera states in a system of coexisting stationary and flying-through in potential particles with an internal degree of freedom determined by the phase. All particles tend to an equilibrium state with a small number of potential wells, which leads to the emergence of a stationary chimera. An increase in the number of potential wells leads to the emergence of particles flying-through along the medium, the phases of which form a moving chimera. Further, these two structures coexist and interact with each other. In this case, an increase in the local synchronization degree of the chimera is observed in the areas of the synchronous cluster location.

Human immunodeficiency virus (HIV) manifests multiple infections in CD4+ T cells, by binding its envelope proteins to CD4 receptors. Understanding these biological processes is crucial for effective interventions against HIV/AIDS. Here, we propose a mathematical model that accounts for the multiple infections of CD4+ T cells and an intracellular delay in the dynamics of HIV infection. We study the model system and establish the conditions under which the disease-free equilibrium point and the endemic equilibrium point are locally and globally asymptotically stable. We further provide the conditions under which these equilibrium points undergo forward or backward transcritical bifurcations for the autonomous model and Hopf bifurcation for both the delay model and autonomous models. Our simulation results show that an increase in the rate of multiple infections of CD4+ T cells stabilizes the endemic equilibrium point through Hopf bifurcation. However, in the presence of an intracellular delay, the model system evinces three types of stability scenarios at the endemic equilibrium point-instability switch, stability switch, and stability invariance and is demonstrated using bi-parameter diagrams. One of the novel aspects of this study is exhibiting all these interesting nonlinear dynamical results within a single model incorporating a single time delay.