This study includes the cosmic evolution and the potential periodicity of the universe. It employs a periodic varying deceleration parameter (PVDP) within the framework of $f(R,T)$ theory of gravity, with a specific focus on the Bianchi-II model. We explore the dynamic nature of the universe, with physical and geometrical properties within this theoretical framework. We have also analyzed the cosmographic parameters, including jerk, snap, and lerk, for deeper insights into the universe’s evolution and behavior. Utilizing state finder diagrams and the Om diagnostic (which illustrates the variation of $\mathrm{\text{Om}}(z)$ with redshift) signifying a shift from matter dominance to a stronger influence of dark energy (DE), we construct a comprehensive map of the universe’s trajectory and behavior. By employing the Bianchi-II model within the $f(R,T)$ theory, our proposed model helps us in understanding the universe’s oscillatory patterns and underlying mechanisms. This research significantly contributes to our understanding of cosmic evolution and periodicity within the $f(R,T)$ theory.

这项研究包括宇宙演化和宇宙潜在的周期性。它在以下框架内采用周期性变化减速参数（PVDP）：$F（右,\mathrm{时间}）$引力理论，特别关注 Bianchi-II 模型。我们在这个理论框架内探索宇宙的动态本质以及物理和几何特性。我们还分析了宇宙学参数，包括急动度、急动度和急动度，以更深入地了解宇宙的演化和行为。利用状态查找器图和 Om 诊断（说明了$\mathrm{\text{唵}}（z）$红移）意味着从物质主导转向暗能量（DE）的更强影响，我们构建了宇宙轨迹和行为的综合图。通过采用 Bianchi-II 模型$F（右,\mathrm{时间}）$理论上，我们提出的模型有助于我们理解宇宙的振荡模式和潜在机制。这项研究极大地促进了我们对宇宙演化和周期性的理解$F（右,\mathrm{时间}）$理论。