Normal Hyperbolicity in Secondary Hopf Bifurcations

2025/07/14 by Pedro C. C. R. Pereira, Pereira, Pedro C. C. R., Douglas D. Novaes +1 · 1 citation
Computer Science · Mathematics · #34C29 #34C45 #Advanced Differential Equations and Dynamical Systems #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Nonlinear Dynamics and Pattern Formation

paper · pdf · doi:10.48550/arxiv.2507.10721

openalex publication_date 2025/07/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We combine results available in the literature to prove that the torus emerging in a secondary Hopf bifurcation is normally hyperbolic. This result is then applied to establish sufficient conditions for the bifurcation of normally hyperbolic invariant tori in the extended phase space of systems with small time-periodic perturbations via an application of the averaging method.

Citations

• Invariant Tori and Periodic Orbits in the FitzHugh-Nagumo System
• A version of Hilbert's 16th Problem for 3D polynomial vector fields: Counting isolated invariant tori
• A mechanism for detecting normally hyperbolic invariant tori in differential equations
• Zero-Hopf bifurcation in the general Van der Pol-Duffing equation
• Periodic solutions and invariant torus in the Rössler system
• Higher order averaging theory for finding periodic solutions via Brouwer degree
• Transition map and shadowing lemma for normally hyperbolic invariant manifolds

Cited by

• Normally hyperbolic limit tori near monodromic singularities in 3D polynomial vector fields

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• Normal-normal resonances in a double Hopf bifurcation
• Non-autonomous KAM theory for lower dimensional invariant tori (I): Normally elliptic and hyperbolic cases
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• Non-autonomous KAM theory for lower dimensional invariant tori (II): Normally parabolic case
• Unfoldings of singular Hopf bifurcation