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As governments implement low-carbon economy widely, boosting low-carbon transformation in industrial clusters has become a challenge. This study establishes an evolutionary game model of low-carbon technology collaborative innovation based on spatial public goods game to solve the free-riding problem effectively in research and development. By introducing a social exclusion mechanism, we explore the requirements for the emergence of cooperation between enterprises, and we consider the heterogeneity and scale-free characteristics of industrial clusters comprehensively. Simulation results confirm that social exclusion can significantly promote cooperation as a form of cooperation with additional cost. When exclusion cost decreases and probability increases, an excluder can survive in a lower enhancement factor, which guarantees a stable exclusion mechanism. Furthermore, this mechanism is key to forming and maintaining cooperative behavior. When a cluster follows a scale-free distribution, the sparse network structure can avoid cooperation collapse. Moreover, heterogeneous investment is a robust alternative in the face of invading defectors. This study provides a new understanding to promote the collaborative innovation of enterprises in industrial clusters.Numerical simulations reveal statistical distributions given by power laws resulting from movements of large quantities of phase points captured by strange attractors immersed in one-dimensional or two-dimensional phase spaces, attractors linked to ten specific dynamic systems. Unlike the characterization given by classical approaches as generalized dimensions and spectrum of singularities, the aforementioned distributions do not have their origin in observations of successive orbits, as consequence properties that would otherwise remain hidden are revealed. Specifically, occupancy times and occupancy numbers associated with small hypercubes that cover attractors obey well-defined statistical distributions given by power laws. One application concerns the determination of the intervals in which the most likely values of those numbers and times are located (effective intervals). The use of the effective interval with occupancy numbers to quantify the multifractalities (multifractality measures) is another application. The statistical approaches underlying the results consist of new paradigms that join the well-known classic paradigms to expand knowledge about strange attractors. The possibility that other attractors immersed in spaces with the same dimensions as those considered here exhibit analogous distributions is not ruled out due to the arbitrariness of the set taken.Recently, a family of nonlinear mathematical discrete systems to describe biological interactions was considered. Such interactions are modeled by power-law functions where the exponents involve regulation processes. Considering exponent values giving rise to hyperbolic equilibria, we show that the systems exhibit irregular behavior characterized by strange attractors. The systems are numerically analyzed for different parameter values. Depending on the initial conditions, the orbits of each system either diverge to infinity or approach a periodic orbit or a strange attractor. Such dynamical behavior is identified by their Lyapunov exponents and local dimension. Finally, an application to the biochemical process of bone remodeling is presented. The existence of deterministic chaos in this process reveals a possible explanation of reproducibility failure and variation of effects in clinical experiments.The structure of a social network plays a crucial role for dynamic analysis, which is invisible in most scenes. In this paper, we present a model for reconstructing the social network by taking into account the public opinion diffusion dynamic model for specific agenda. First, the initial polarity attitude of users i for the agenda, oi, is set in the range [-1,1], where negative and positive attitudes are set as -1 and 1, respectively, while 0 means that user i's attitude is uncertain. Second, we present an optimization model for detecting the relationship among each pair of users based on the generated public observable information. The experimental results for four synthetic networks and three real-world social networks show that the reconstruction accuracy depends on the uncertainty of the initial attitudes greatly. this website This work is helpful for revealing the structure of social networks in terms of public information.In this paper, emotions are classified into four types, namely, respect for the strong, envying the strong, sympathy for the weak, and bullying the weak. The corresponding relationship between the four emotion types and the two behaviors of competition and cooperation is then defined. The payoff matrices of the game based on emotions are obtained and the evolutionary dynamics of the four emotion types in a finite population based on the Moran process are studied. Next, we derive the absorption probabilities of a 4×4 symmetric evolutionary game of the population. The influence of the payoff parameters and the natural selection intensity on the result of the group evolution are then analyzed. The calculations indicate that there are differences in the absorption probabilities of the four absorption states of the system. At a steady state, individuals of the types envying the strong and bullying the weak have the highest probability of occupying the entire population, and individuals of the type respect for the strong and sympathy for the weak have the lowest one. By comparing the level of cooperation and average payoffs at a steady state, we observe that the level of cooperation and average payoffs based on the proposed model are better than those of the prisoner's dilemma game with two behaviors. Therefore, emotional evolution can promote cooperation and achieve better group fitness.History dependence of the evolution of complex systems plays an important role in forecasting. The precision of the predictions declines as the memory of the systems is lost. We propose a simple method for assessing the rate of memory loss that can be applied to experimental data observed in any metric space X. This rate indicates how fast the future states become independent of the initial condition. Under certain regularity conditions on the invariant measure of the dynamical system, we prove that our method provides an upper bound on the mixing rate of the system. This rate can be used to infer the longest time scale on which the predictions are still meaningful. We employ our method to analyze the memory loss of a slowly sheared granular system with a small inertial number I. We show that, even if I is kept fixed, the rate of memory loss depends erratically on the shear rate. Our study suggests the presence of bifurcations at which the rate of memory loss increases with the shear rate, while it decreases away from these points.
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