Our analysis reveals the curves of fold bifurcations, where equilibrium pairs form and subsequently combine. 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 original system's heteroclinic connections are not replicated in the linearly perturbed system's dynamics. Linear perturbation noticeably alters the dynamics, even when mutation rates are slight.Periodic climate changes powerfully influence population dynamics, a result of species' natural responses to seasonal variations. These consequences can induce synchronized population dynamics or lead to chaotic outcomes in population structures. Despite the inherent instability of natural environments, synchronized ecological rhythms contribute significantly to species survival; for this reason, determining the environmental variables capable of altering ecosystem dynamics towards synchronization is of paramount importance. Through the lens of this study, the influence of ecological parameters on species' dynamics as they adapt to seasonal variations and achieve phase synchronization within ecosystems is explored. To account for seasonality, a periodic sinusoidal function is integrated into a tri-trophic food chain system, which already accounts for Allee and refugia effects. It is observed that seasonal effects disrupt the system's limit cycle and cause chaotic conditions to arise. Rigorous mathematical analysis is further used to investigate the dynamical and analytical properties of the nonautonomous system instantiation. Key features of these properties are sensitive dependence on initial conditions (SDIC), sensitivity analysis, outcomes of bifurcations, the inherent positivity and boundedness of the solution, the permanence of the system, its ultimate boundedness, and the various possibilities of species extinction. The presence of chaotic oscillations in the system is elucidated by the SDIC. Through sensitivity analysis, the parameters that substantially impact the results of numerical simulations are identified. The bifurcation analysis concerning seasonal parameters suggests that species are more affected by the rate of seasonal fluctuations than by the magnitude of the seasonal variations. The bifurcation study, exploring bio-controlling parameters, discloses various dynamic states within the system, such as fold bifurcations, transcritical branch points, and Hopf points. Furthermore, our mathematical model of the seasonally changing system exposes the repeated existence of all species together, with a globally attractive outcome, only under specific parametric restrictions. Lastly, we consider the contribution of essential parameters in the process of phase synchrony. Using numerical methods, we investigate how the coupling dimension coefficient, bio-control parameters, and seasonal-related parameters influence the system. This study indicates that species' dynamics are adaptable to seasonal effects with low frequency, yet exhibit a high tolerance threshold for the intensity of seasonal variations. The study unveils a connection between prey biomass levels, phase synchrony, and the severity of seasonal impacts. Ecosystems, subject to seasonal perturbations, are analyzed in this study, with significant implications for strategic conservation and management plans.The collective dynamics of substantial neuronal populations are subject to the considerable influence of finite-size effects. Our recent study has demonstrated how effects observed in globally coupled networks within the thermodynamic limit can be interpreted as a supplementary common noise term, identified as 'shot noise,' influencing the macroscopic dynamics. This analysis further examines the contributions of shot noise to the collective dynamics within globally linked neural networks. We analyze how noise influences the switching among different macroscopic phases. We demonstrate how shot noise transforms the attractors of the infinitely large network into metastable states, with lifetimes that smoothly vary with system parameters. A noteworthy effect of shot noise is its influence on the spatial extent of the macroscopic regime, distinct from the thermodynamic limit's behavior. A constructive interpretation of shot noise is that it allows a certain macroscopic state to appear in a parameter region where this state would be absent in an unbounded 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. abcris Our analysis focuses on chimera states observed in a network built from six identical populations of Kuramoto-Sakaguchi phase oscillators. Uniform couplings exist between oscillators in the same population, and to those in the two adjacent populations, which are distributed in a ring formation. This topology enables varying coherent and incoherent population distributions along the ring, yet each distribution demonstrates linear instability over a significant segment of the parameter range. In a substantial parameter range, starting with randomly chosen initial conditions, the chimera dynamic pattern manifests, involving one desynchronized population and five synchronized populations. A heteroclinic cycle between symmetrical saddle chimera variants underlies the switching dynamics observed in these states. We examine the dynamic and spectral characteristics of the chimeras in the thermodynamic limit, leveraging the Ott-Antonsen approximation, and in systems of finite size, using the Watanabe-Strogatz simplification. Frequency distributions with heterogeneity, manifesting as a small degree of non-uniformity, exhibit asymptotically attractive heteroclinic switching dynamics. Yet, with a substantial level of diversity, the heteroclinic orbit does not endure; rather, a collection of attractive chimera states emerge.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. For an understanding of the short-lived chimera states, we apply transverse-stability analysis. The equator of the unit sphere representing three-dimensional (3D) rotations is the location of the persistent chimera states in the classic Kuramoto model, and latitudinal deviations that induce true three-dimensional rotations are perpendicular to it. Our findings indicate that the largest transverse Lyapunov exponent, when determined with respect to these long-lasting chimera states, is typically positive, thus confirming their short-lived behavior. By means of a transverse-stability analysis, the previous numerical scaling law governing the transient lifetime's duration is exactly formulated, allowing for the precise determination of the original scaling law's free proportional constant by using the largest transverse Lyapunov exponent. Our research highlights the possibility that chimera states within physical systems have limited duration, as they are vulnerable to any disturbances featuring a component transverse to their underlying invariant subspace.In intricate networks, we investigate the average degree of a node's surrounding neighbors. According to the rules of a Markov stochastic process, this quantity's value at any moment is probabilistically determined, following a specific distribution. Our interest lies in specific features of this distribution, including its expected value, its variance, and its coefficient of variation. An examination of several real communities serves as our initial step in comprehending the evolution of these values within social networks over time. 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 degrees of neighboring nodes, on average, exhibit a comparable standard deviation. Next, we investigate the dynamic behaviour of these three values over time for networks produced by simulations of a specific Barabási-Albert model, the growth model with non-linear preferential attachment (NPA) featuring a constant number of links attached at each iteration. Through analytical methods, we show that the coefficient of variation for the average degree of a node's neighboring nodes in these networks approaches zero, although this convergence occurs quite slowly. We conclude, then, that the average degree of neighboring nodes displays a different characteristic in Barabási–Albert networks relative to real-world networks. This model, built upon the NPA mechanism, proposes a stochastic approach to edge addition at each iteration, aiming to mirror the evolution of the average degree of neighbor nodes in real-world networks.An algorithm, designed in this paper, extracts the piecewise non-linear dynamical system from data, devoid of prior knowledge. The system under scrutiny does not require its representation to be framed by established model terms or detailed understanding. The Riemann integrability of a given unknown piecewise non-linear system's equations of motion enables its representation through the Fourier series. This inherent feature allows for a streamlined transition from model identification to the discovery of the Fourier series approximation. Nevertheless, the Fourier series' approximation of the piecewise function displays inaccuracies. The new approach exploits this limitation to discern whether the model displays piecewise characteristics and to identify the discontinuity set. A pure Fourier series was subsequently established as the dynamical system's representation for each segment. In a straightforward manner, intricate models can be identified. The method's results demonstrate its ability to precisely pinpoint the equation of motion while accurately highlighting the non-smooth aspects.