Stochastic Evolution of Galactic Star Formation with Halo Coupling, AGN Quenching and Hopf Bifurcation Dynamics

We present a computational framework for galactic evolution based on a coupled stochastic nonlinear oscillator, implemented with the \textbf{Stochastic Hopf Engine}. Gas density ($G$) and star formation rate ($S$) co-evolve through a supercritical Hopf bifurcation, capturing the transition from quiescent stability to merger-driven starbursts. Scatter in dark matter halo properties, modeled as multiplicative noise via the \textbf{Euler--Maruyama method}, broadens the bifurcation into a regime where noise-induced bursts occur below the deterministic threshold. Simulations reveal a periodic signature, the \textbf{Galactic Heartbeat}, emerging as a deterministic limit cycle validated by the \textbf{data3} resonance peak in the star-formation spectrum. A radial reduction yields an effective \textbf{Fokker--Planck equation} for burst amplitude; its stationary solution matches numerical PDFs, providing statistical closure. Including differential shear $\Omega(r)$ and spatially varying bifurcation fields reproduces spiral morphologies and AGN-driven quenching. Driving the growth parameter sub-critical ($r_{agn} < 0$) yields ``Red and Dead'' cores via attractor collapse. Dark matter halo scatter suppresses mean star formation while enhancing intermittency, offering a minimal yet interpretable framework linking local feedback and global potentials to macroscopic galactic evolution.