[Paper Review] Bifurcation cascades and self-similarity of periodic orbits with analytical scaling constants in Henon-Heiles type potentials

M. Brack|arXiv (Cornell University)|Jun 21, 2000

Quantum chaos and dynamical systems34 references38 citations

TL;DR

This paper investigates isochronous pitchfork bifurcation cascades of straight-line librating orbits in Hénon-Heiles-type potentials, showing analytically that the scaling constant δ for the geometric progression of bifurcation energies is δ = e^{π/\sqrt{2}} ≈ 9.2206, with identical spatial scaling constants α = β = √δ in both directions. The self-similarity and analytical scaling behavior arise from the harmonic saddle structure and transverse curvature at the critical energy e = 1.

ABSTRACT

We investigate the isochronous bifurcations of the straight-line librating orbit in the Henon-Heiles and related potentials. With increasing scaled energy e, they form a cascade of pitchfork bifurcations that cumulate at the critical saddle-point energy e=1. The stable and unstable orbits created at these bifurcations appear in two sequences whose self-similar properties possess an analytical scaling behavior. Different from the standard Feigenbaum scenario in area preserving two-dimensional maps, here the scaling constants \alpha and \beta corresponding to the two spatial directions are identical and equal to the root of the scaling constant \delta that describes the geometric progression of bifurcation energies e_n in the limit n --> infinity. The value of \delta is given analytically in terms of the potential parameters.

Motivation & Objective

• To analyze the cascade of isochronous pitchfork bifurcations in Hénon-Heiles-type potentials as energy approaches the saddle-point energy e = 1.
• To determine whether the self-similar structure of periodic orbits born at successive bifurcations exhibits analytical scaling constants.
• To derive the scaling constant δ analytically in terms of potential parameters, particularly the transverse curvature ω⊥.
• To contrast this scaling behavior with the standard Feigenbaum scenario in area-preserving maps, where δ, α, and β are distinct and universal.

Proposed method

• Solves the scaled Newton equations of motion for the Hénon-Heiles potential in terms of energy e, using numerical integration and Newton-Raphson iteration to locate periodic orbits.
• Analyzes the stability matrix of periodic orbits and tracks the trace tr M(TA) as a function of period TA to identify bifurcation points.
• Derives the asymptotic behavior of the libration period TA near e = 1 using the logarithmic divergence TA ∼ √(2 ln(64/ǫ)) with ǫ = 1 − e.
• Uses the asymptotic periodicity ∆T = 2π/ω⊥ of tr M(TA) and the energy dependence TA ∼ d ln[c/(1 − e)] to derive the scaling constant δ = e^{2π/(ω⊥d)}.
• Confirms analytical results via numerical iteration of bifurcation energies en, yielding δ ≈ 9.2203 with low standard deviation.
• Applies special functions (Lamé functions) to describe small-amplitude oscillations near bifurcations, validating numerical results with high accuracy.

Experimental results

Research questions

• RQ1 What is the analytical form of the scaling constant δ governing the geometric progression of bifurcation energies in Hénon-Heiles-type potentials?
• RQ2 Do the self-similar structures of periodic orbits created at successive bifurcations exhibit identical scaling constants α and β in both spatial directions?
• RQ3 How does the scaling behavior in this system differ from the standard Feigenbaum scenario in two-dimensional area-preserving maps?
• RQ4 Can the scaling constant δ be derived analytically from the potential parameters, particularly ω⊥ and d?
• RQ5 What is the role of the transverse curvature ω⊥ and the Maslov index in determining the asymptotic dynamics near the saddle point?

Key findings

• The scaling constant δ for the geometric progression of bifurcation energies is analytically derived as δ = e^{π/\sqrt{2}} ≈ 9.2206125, with high numerical confirmation (δ ≈ 9.2203 ± 0.02).
• The spatial scaling constants are identical in both directions: α = β = √δ ≈ 3.0365, contrasting with the standard Feigenbaum scenario where α ≠ β.
• The asymptotic period of the librating orbit TA diverges logarithmically as TA ∼ √(2 ln(64/ǫ)) with ǫ = 1 − e, leading to the energy scaling δ = e^{π/\sqrt{2}}.
• The self-similarity of periodic orbit tips is revealed only over two generations due to alternating stable/unstable nature of orbits born at bifurcations, requiring scaling by α² = δ between successive generations.
• The Lyapunov exponent of the new unstable orbits τ (for e > 1) approaches χ = 2πω∥/ω⊥ in the limit e → 1, with σ = χ/Tτ = ω∥.
• The use of Lamé functions Ecm_p(at) and Esm_p(at) accurately describes small-amplitude oscillations near bifurcations, matching numerical results with high precision.