Pingla Thana Mahavidyalaya

This study introduces a novel three-dimensional nonlinear system featuring quadratic, cubic, quartic, and quintic nonlinearities, with inherent symmetry about the -axis. We analyze its fundamental dynamical properties, including equilibrium points and their stability, dissipative behavior, and high-periodicity limit cycles. The system's chaotic nature is rigorously examined through Lyapunov exponent spectra, fractal dimension analysis, and Poincar & eacute; maps, revealing sensitivity to initial conditions and parameter variations. A detailed bifurcation analysis identifies critical transitions, including Hopf bifurcations, using both numerical simulations and theoretical criteria such as the Routh-Hurwitz stability conditions. Notably, the system exhibits rich dynamics, from stable equilibria to chaotic attractors, with fractional Lyapunov dimensions confirming its complexity. Practical implications are underscored by parameter ranges yielding chaotic trajectories, validated through circuit-compatible amplitude constraints. The system's versatility suggests promising applications in cryptography, secure communications, and random number generation, motivating further exploration of its nonlinear phenomena.

Predator–prey interactions are governed not only by direct consumption but also by behavioral responses of prey to the perceived risk of predation. In this work, we formulate a discrete-time predator–prey model in which predator-induced fear modifies prey reproduction through an exponential suppression mechanism, while predator consumption follows a Holling type-II functional response. The resulting map combines Ricker-type prey growth, density-dependent predation, and a non-consumptive effect associated with predator presence. We establish fundamental dynamical properties of the system by proving positivity, boundedness, and persistence under appropriate parameter restrictions. The existence of biologically meaningful equilibria is determined, and their local behavior is characterized using the Jacobian matrix and Jury stability criteria. To examine the influence of ecological parameters on the dynamics, we employ a global sensitivity analysis based on the Partial Rank Correlation Coefficient (PRCC) approach. The analysis identifies the parameters that most strongly affect prey and predator abundance. Numerical investigations reveal a wide range of dynamical regimes, including stable coexistence, periodic oscillations, higher-period attractors, quasi-periodic motion, and chaos. In particular, flip and Neimark–Sacker bifurcations are observed as ecological parameters vary. Basin-of-attraction computations and two-parameter iso-spike diagrams further demonstrate the presence of multistability and strong dependence on parameter combinations and initial conditions. Finally, period-doubling control and pole-placement techniques are employed to suppress undesirable chaotic oscillations and recover stable coexistence. The results emphasize that predator-induced fear can substantially modify population fluctuations and may act as an important mechanism regulating coexistence and complex dynamics in discrete ecological systems.

This article presents a three-dimensional discrete-time ecological model to elucidate the intricate dynamics among three distinct species within an ecosystem. This approach extends traditional two-dimensional models, offering a more comprehensive perspective on ecological interactions. We identify all biologically feasible equilibria and perform a local stability analysis for each equilibrium point. Through bifurcation analysis (Neimark-Sacker and period-doubling bifurcations), we successfully demonstrate chaotic attractors via period doubling in the discrete-time model and implement chaos control through numerical simulations. By integrating this mathematical model, we derive ecological insights that contribute to informed conservation and management strategies, promoting sustainable biodiversity preservation.

Soft set theory generalizes fuzzy set theory and introduces a flexible approach to handling uncertainties. This article explores the definitions and properties of soft relations, soft nets, and soft filters, which serve as counterparts to ordinary relations, nets, and filters. It also examines the relationships between soft and ordinary structures using soft elements. Furthermore, the study presents key findings along with illustrative examples related to soft relations, soft nets, and soft filters.

Predator-prey interactions are fundamental to ecological systems, often exhibiting nonlinear and complex behaviors. The classical Rosenzweig-MacArthur (RMA) model has been instrumental in understanding these dynamics, but real-world ecosystems often involve higher trophic interactions. In this study, we modify the classical RMA model by incorporating a super-predator, extending it into a discrete-time three-species framework. This modification introduces additional complexity, leading to richer dynamical behaviors, including high-period oscillations and chaotic dynamics. We perform a rigorous local stability analysis of equilibrium points and examine the emergence of bifurcations, particularly Neimark-Sacker (NS) bifurcations, which indicate transitions to quasi-periodic and chaotic dynamics. Numerical simulations demonstrate that variations in interaction rates, particularly prey-predator ( β ) and prey-super-predator ( θ ), significantly influence system stability and dynamical behavior. Our findings reveal that controlled parameter adjustments can regulate bifurcations and prevent undesirable population fluctuations, offering practical implications for ecological management and conservation strategies. These results contribute to a deeper understanding of how multi-trophic interactions shape ecosystem stability and complexity in discrete-time predator-prey models.