[Paper Review] Breakdown of a 2D heteroclinic connection in the Hopf-zero singularity (II). The generic case

Inmaculada Baldomá, O. Castejón+1 more|RECERCAT (Consorci de Serveis Universitaris de Catalunya)|Aug 3, 2016

Advanced Differential Equations and Dynamical Systems11 references3 citations

TL;DR

This paper establishes an explicit asymptotic formula for the exponentially small splitting of the two-dimensional stable and unstable manifolds associated with saddle-focus equilibria in generic analytic unfoldings of the Hopf-zero singularity. Using the inner equation and Stokes constants from singular perturbation theory, it proves the splitting distance is of order $\mathcal{O}(\mu^{-\bar{a}} e^{-a/\sqrt{\mu}})$, confirming a beyond-all-orders phenomenon in dynamical systems.

ABSTRACT

In this paper we prove the breakdown of the two-dimensional stable and unstable manifolds associated to two saddle-focus points which appear in the unfoldings of the Hopf-zero singulariry. The method consists in obtaining an asymptotic formula for the difference between this manifolds which turns to be exponentially small respect to the unfolding parameter. The formula obtained is explicit but depends on the so-called Stokes constants, which arise in the study of original vector field and which corresponds to the so called inner equation in singular perturbation theory.

Motivation & Objective

• To analyze the breakdown of a 2D heteroclinic connection in the generic case of the Hopf-zero singularity, where normal form theory fails to capture the true dynamics due to beyond-all-orders effects.
• To derive an explicit asymptotic formula for the splitting distance between the two-dimensional stable and unstable manifolds of saddle-focus equilibria in generic analytic unfoldings.
• To characterize the exponentially small splitting using the inner equation and Stokes constants, which emerge from the singular perturbation structure of the system.
• To extend previous results on non-generic cases by rigorously treating the generic setting ($q=0$) where the perturbation terms are not artificially scaled.
• To provide a quantitative understanding of the splitting in terms of the unfolding parameters $\mu$ and $\nu$, particularly in the regime $|\nu| < \beta_1\sqrt{\mu}$.

Proposed method

• The method employs the normal form procedure up to order three, followed by rescaling and parameter renaming to express the vector field in a canonical form with explicit dependence on $\mu$, $\nu$, and higher-order terms.
• The analysis relies on the inner equation derived from the original vector field $X^*$, which is independent of parameters and governs the exponentially small splitting behavior.
• Stokes constants are extracted from the solutions of the inner equation, which are essential for computing the asymptotic splitting distance.
• A functional analytic approach is used to estimate the difference between invariant manifolds via the Melnikov-type function and the operator $\mathcal{M}_1^\mathrm{u}$, which captures the nonlinear correction to the splitting.
• The proof uses weighted Banach spaces and estimates in the $\|\cdot\|_{k,\omega}^\mathrm{u}$ norm to control the size of solutions and their differences, ensuring convergence in the singular limit.
• The splitting distance is bounded using a contraction argument on the solution operator, with key estimates relying on the smallness of $\delta$ and the decay properties of the inner solutions.

Experimental results

Research questions

• RQ1 What is the asymptotic behavior of the splitting distance between the two-dimensional stable and unstable manifolds in the generic unfolding of the Hopf-zero singularity?
• RQ2 How do Stokes constants from the inner equation contribute to the exponentially small splitting in the generic case?
• RQ3 Can the splitting distance be expressed as an explicit asymptotic formula in terms of the unfolding parameters $\mu$ and $\nu$?
• RQ4 How does the generic case ($q=0$) differ from the non-generic case ($q>0$) in terms of the splitting mechanism and its quantification?
• RQ5 What role does the inner equation play in capturing the exponentially small splitting in the beyond-all-orders regime?

Key findings

• The splitting distance between the 2D stable and unstable manifolds is asymptotically of order $\mathcal{O}(\mu^{-\bar{a}} e^{-a/\sqrt{\mu}})$ for some positive constants $a$ and $\bar{a}$, confirming the exponentially small nature of the splitting.
• The asymptotic formula depends explicitly on Stokes constants arising from the inner equation, which are intrinsic to the original vector field $X^*$ and not adjustable by perturbation parameters.
• The method successfully extends previous results from the non-generic case ($q>0$) to the generic case ($q=0$), where the perturbation terms are not artificially scaled.
• The splitting is shown to be governed by a Melnikov-type function that captures the nonlinear interaction between the invariant manifolds via the inner equation.
• The proof establishes the existence of a unique solution to the splitting equation using a contraction argument in weighted Banach spaces, with bounds controlled by $\delta$ and $\kappa$.
• The analysis confirms that the splitting is not captured by any finite-order normal form, validating the beyond-all-orders nature of the phenomenon in the generic setting.