We scrutinize the curves of fold bifurcations, noting the appearance and subsequent combination of equilibrium pairs. Our research, therefore, reveals parameter regions that support novel stable fixed points, diverging from the single-population and dual-population equilibrium states characteristic of the original model. The heteroclinic connections, a hallmark of the original system, are absent in the linearly perturbed system. Linear perturbation, surprisingly, significantly affects the dynamics, even with small mutation rates.Seasonal fluctuations profoundly influence population dynamics, as species naturally adapt to periodic climate shifts. These impacts sometimes cause population dynamics to synchronize or generate unpredictable behavior. However, the synchronization of ecological processes is a critical factor in the survival of species in unstable natural systems; thus, the key is to identify and manipulate the elements that shift ecosystem dynamics towards a state of harmonious synchronization. Within this study, the interplay of ecological parameters and species' adaptive strategies for seasonal changes, leading to phase synchrony within ecosystems, is examined. Employing a periodic sinusoidal function, we model seasonal impacts within a tri-trophic food chain system, while acknowledging the presence of Allee and refugia effects. The presence of seasonal influences disrupts the inherent cyclical nature of the system's limit cycle, leading to a chaotic outcome. We additionally undertake a rigorous mathematical analysis to explore the dynamic and analytical attributes of the nonautonomous system. Included in these properties are sensitive dependence on initial conditions (SDIC), sensitivity analysis, bifurcation outcomes, the solution's positivity and boundedness, the concept of permanence, the ultimate boundedness of the system, and potential scenarios for species extinction. The SDIC indicates the existence of erratic fluctuations within the system. The outcome of numerical simulations is dependent on the parameters highlighted by sensitivity analysis. A study on seasonal bifurcations reveals that species are more reliant on the frequency of seasonal fluctuations than on the intensity of the seasons themselves. In the bifurcation analysis of bio-controlling parameters, dynamic states such as fold, transcritical branch points, and Hopf bifurcations are evident within the system. In addition, the mathematical study of our system, subject to seasonal fluctuations, demonstrates the periodic coexistence of all species and a globally attractive solution, dependent on specific parametric values. In the final analysis, we scrutinize the impact of essential parameters on phase synchrony. We numerically investigate the defining influence of the coupling dimension coefficient, biological control parameters, and seasonal factors. This study hypothesizes that species can harmonize their dynamic processes with seasonal fluctuations of low frequency, demonstrating a high degree of resilience against the intensity of the seasonal effects. The study demonstrates a link between the degree of phase synchrony, prey biomass levels, and the harshness of seasonal influences. This research uncovers crucial insights into the way seasonal changes affect ecosystem interactions, with direct bearing on conservation and management methodologies.Finite-size limitations can meaningfully impact the collective dynamical activities of large neuronal groups. We have recently shown, regarding globally coupled networks, that these effects manifest as an additional common noise term within the macroscopic dynamics of the thermodynamic limit, specifically termed 'shot noise'. Continuing our study, we investigate the role of shot noise in the dynamics of globally coupled neural networks. We examine how noise causes changes between different large-scale operational modes. The influence of shot noise on the infinitely large network's attractors is examined, revealing metastable states whose lifetimes are seamlessly connected to system parameters. The presence of shot noise has a surprising impact on the area where a certain macroscopic regime appears, unlike the prediction of the thermodynamic limit. The shot noise could potentially play a constructive role, making a particular macroscopic state present in a parameter space where it's not present in a hypothetical infinite network.A coupled oscillator network can spontaneously develop a symmetry-broken dynamical state, distinguished by the coexistence of coherent and incoherent segments. It is a chimera state, by all accounts. dpp2 signal Our analysis focuses on chimera states observed in a network built from six identical populations of Kuramoto-Sakaguchi phase oscillators. The populations are arranged in a ring, where oscillators within the same population are coupled uniformly to all oscillators in that population, and to those in each of the two adjacent populations. While this topology permits diverse configurations of coherent and incoherent populations distributed throughout the ring, these configurations exhibit linear instability across a broad range of parameter settings. In a substantial parameter range, starting with randomly chosen initial conditions, the chimera dynamic pattern manifests, involving one desynchronized population and five synchronized populations. Symmetrical saddle chimera variants form a heteroclinic cycle, which is responsible for the observed switching dynamics in these states. By applying the Ott-Antonsen ansatz, we analyze the dynamical and spectral properties of chimeras in the thermodynamic limit. Finite-sized systems are studied using the Watanabe-Strogatz reduction. Heterogeneity in a frequency distribution, when relatively small, leads to the asymptotic attraction of heteroclinic switching behavior. In spite of significant heterogeneity, the heteroclinic orbit is not sustainable; instead, it is replaced by various alluring chimera attractors.Coupled two-dimensional rotators in the Kuramoto system exhibit chimera states characterized by the coexistence of synchronous and asynchronous oscillator groups. Consequently, the average lifespan of these states grows exponentially as the size of the system increases. Recently, it was found that three-dimensional rotators in the Kuramoto model lead to short-lived chimera states, their lifespan scaling only with the logarithm of the dimension-increasing perturbation. To grasp the transient chimera states, we employ transverse-stability analysis. Long-lasting chimera states, occurring within the classic Kuramoto model, reside on the equator of the unit sphere depicting three-dimensional (3D) rotations; perturbations in latitude, which generate three-dimensional rotations, are orthogonal to it. We show that the largest transverse Lyapunov exponent, computed relative to these persistent chimera states, is generally positive, which implies their transient nature. Employing transverse-stability analysis, the previous numerical scaling law for transient lifetime is recast into an exact formula, determining the free proportional constant precisely using the largest transverse Lyapunov exponent. The results of our investigation suggest that, in physical systems, chimera states are frequently short-lived, being susceptible to any perturbations whose component is perpendicular to the invariant subspace.Complex network analysis explores the degree dynamics of neighbors around each node on average. Values of this quantity, within the framework of a Markov stochastic process, are chosen at each moment, governed by a probability distribution. We are keen to explore certain characteristics of this distribution, including its expected value, its variance, and its coefficient of variation. To grasp how these values shift over time within social networks, we initially examine numerous real-world communities. When examining the behavior of these quantities in real networks empirically, a high coefficient of variation is observed, remaining elevated as the network grows. The standard deviation and mean degree of neighboring nodes demonstrate a similar magnitude. Subsequently, we scrutinize the temporal progression of these three metrics across networks generated through simulations of one prominent variant of the Barabási-Albert model, the growth model incorporating nonlinear preferential attachment (NPA), characterized by a fixed number of connecting links per iteration. We analytically prove that the coefficient of variation of the average degree of a node's neighbors in these networks approaches zero, albeit at an exceptionally gradual rate. Hence, the behavior of the average degree of neighboring nodes in Barabási–Albert networks deviates from that seen in real networks. Our model, fundamentally grounded in the NPA mechanism with a randomized number of edges added per iteration, effectively reflects the dynamic evolution of the average degree of neighbor nodes in real-world networks.Using data, this paper constructs an algorithm to determine the piecewise non-linear dynamical system without pre-existing knowledge. Identification of the system does not require its representation in terms of a recognized model term or complete familiarity with its mechanisms. An unknown piecewise non-linear system's Riemann integrable motion equations facilitate its decomposition into a Fourier series representation. Leveraging this attribute simplifies the task of model selection to the process of finding the Fourier series approximation. Yet, the Fourier series' representation of the piecewise function is not precise. The new method capitalizes on this weakness to determine if the model exhibits piecewise features and to ascertain the location of the discontinuity set. On each segment, the dynamical system was subsequently recognized as a pure Fourier series representation. The identification of elaborate models can be undertaken using basic procedures. The equation of motion is accurately determined by the method, and its non-smooth characteristics are precisely captured, as shown by the results.