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 \[ \ddotϕ+\frac{1}{q}\dotϕ+ϕ^3+ϕ^5=A\cos(ωt), \] advancing both analytical and numerical chaos theory. Using Melnikov analysis on explicit homoclinic orbits \[ ϕ_0(t) = 1-\tanh(t)-\tanh^2(t) \quad \text{and} \quad ϕ_0(t) = {\rm sech}_{\rm RZ}(t) -{\rm sech}_{\rm RZ}^2(t),\] we rigorously predict transverse homoclinic intersections and limit cycle bifurcations surrounding the hyperbolic saddle $(0,0)$, establishing chaos onset at $A_{\mathrm{chaos}}\approx 0.34$. A groundbreaking contribution introduces the hybrid quintic Duffing-Riemann zeta system $\ddot{\varphi }+{\varphi }^3+{\varphi }^5=A\cos (\omega t)+\Re [\zeta (s)]$, where $\zeta (s)=X(s)-Y(s)$ via C-transformation decomposition. Bifurcation portraits reveal zeta perturbation delays chaos by $24\%$ ($A_{\text{chaos}}\approx 0.42$) while enhancing Lyapunov exponents by $27\%$ (${\lambda }_{\text{max}}=0.14>0.11$). Nontrivial zeros $s_k=1/2+i{t}_k$ 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 $\lambda (s_k)={\min }_{\sigma \in [0,1]}\lambda (\sigma +i{t}_k)$, reformulating the Riemann Hypothesis as a verifiable bifurcation prediction. The unperturbed Hamiltonian $H=\frac{1}{2}{\dot{\varphi }}^2+\frac{1}{4}{\varphi }^4+\frac{1}{6}{\varphi }^6$ and stochastic extensions for biomedical applications are analyzed, positioning number-theoretic chaos control as a novel paradigm bridging nonlinear dynamics and analytic number theory.

English has been improved throughout. A new section "Chaos Analysis in the Hybrid Quintic Duffing-Riemann Zeta System via Decomposition'' has been added with theoretical implications. Special thanks to co-author Pedro Caceres for collaboration and to arXiv administrators for their attention in improving this version