Chaos Analysis in the Hybrid Quintic Duffing-Riemann Zeta System via Decomposition

Abstract

This paper presents a comprehensive analysis of the driven cubic-quintic Duffing oscillator \[ \ddot{\phi}+\frac{1}{q}\dot{\phi}+\phi^3+\phi^5=A\cos(\omega t), \] advancing both analytical and numerical chaos theory. Using Melnikov analysis on explicit homoclinic orbits \[ \phi_0(t) = 1-\tanh(t)-\tanh^2(t) \quad \text{and} \quad \phi_0(t) = {\rm sech}_{\rm RZ}(t) -{\rm sech}_{\rm RZ}^2(t),\] we rigorously predict transverse homoclinic intersections and limit cycle bifurcations surrounding the hyperbolic saddle $(0,0)$, establishing chaos onset at $A_\mathrm{chaos}\approx0.34$. A groundbreaking contribution introduces the hybrid quintic Duffing-Riemann zeta system $\ddot{\phi}+\phi^3+\phi^5=A\cos(\omega t)+\Re[\zeta(s)]$, where $\zeta(s)=X(s)-Y(s)$ via C-transformation decomposition. Bifurcation portraits reveal zeta perturbation delays chaos by $24\%$ ($A_\text{chaos}\approx0.42$) while enhancing Lyapunov exponents by $27\%$ ($\lambda_\text{max}=0.14>0.11$). Nontrivial zeros $s_k=1/2+it_k$ emerge as chaos suppressors through entropy-matching $|X(s_k,n)|^2=|Y(s_k,n)|^2$. We prove nontrivial zeros manifest as global Lyapunov minimizers $\lambda(s_k)=\min_{\sigma\in[0,1]}\lambda(\sigma+it_k)$, reformulating the Riemann Hypothesis as a verifiable bifurcation prediction. The unperturbed Hamiltonian $H=\frac{1}{2}\dot{\phi}^2+\frac{1}{4}\phi^4+\frac{1}{6}\phi^6$ and stochastic extensions for biomedical applications are analyzed, positioning number-theoretic chaos control as a novel paradigm bridging nonlinear dynamics and analytic number theory.