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

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 $${\phi }_0(t)=1-\tanh (t)-{\tanh }^2(t)\text{and}{\phi }_0(t)={\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}}_{\mathrm{R}\mathrm{Z}}(t)-{\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}}_{\mathrm{R}\mathrm{Z}}^2(t),$$ we rigorously predict transverse homoclinic intersections and limit cycle bifurcations surrounding the hyperbolic saddle $(0,0)$, establishing chaos onset at $A_{\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{o}\mathrm{s}}\approx 0.34$. A groundbreaking contribution introduces the hybrid quintic Duffing-Riemann zeta system $\ddot{\phi }+{\phi }^3+{\phi }^5=A\cos (\omega t)+\mathrm{\Re }[\zeta (s)]$, where $\zeta (s)=X(s)-Y(s)$ via C-transformation decomposition. Bifurcation portraits reveal zeta perturbation delays chaos by $24\mathrm{\%}$ ($A_{\text{chaos}}\approx 0.42$) while enhancing Lyapunov exponents by $27\mathrm{\%}$ (${\lambda }_{\text{max}}=0.14>0.11$). Nontrivial zeros $s_k=1\mathrm{/}2+i{t}_k$ emerge as chaos suppressors through entropy-matching $\mathrm{\mid }X(s_k,n){\mathrm{\mid }}^2=\mathrm{\mid }Y(s_k,n){\mathrm{\mid }}^2$. We prove nontrivial zeros manifest as global Lyapunov minimizers $\lambda (s_k)={\min }_{\sigma \in [0,1]}\lambda (\sigma +i{t}_k)$, 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.