ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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A subcritical planar cubic Hopf bifurcation

Example

Consider $\dot{x} = μ x - y + x (x^{2} + y^{2}), \dot{y} = x + μ y + y (x^{2} + y^{2}) .$ This has the same eigenvalue crossing as the preceding cubic example but the opposite first Lyapunov sign.

Facts & Assumptions

Given: The polynomial vector field in the Example, $ω_{0} = 1$ , and the normalization $q^{*} q = p^{*} q = 1$ .

[F1]

Hopf spectral and multilinear normal-form data: $ℓ_{1} = Re G_{21} / (2 ω_{0})$ with the displayed normalized eigenvectors.

[F2]

A simple nondegenerate Hopf point creates a local periodic branch: a simple crossing with $ℓ_{1} \neq 0$ has its local cycle on $β ℓ_{1} < 0$ ; radial stability requires $β > 0$ .

Verification

1.1givenalgebra

The origin is an equilibrium and $A (μ) = [\begin{array}{cc} μ & - 1 \\ 1 & μ \end{array}]$ has eigenvalues $μ \pm i$ . Hence $β^{'} (0) = 1$ and there is no transverse spectrum.

1.2F1givenalgebra

With $q = p = (1, - i)^{T} / \sqrt{2}$ , $B = 0$ . The cubic vector field is $∥ (x, y) ∥^{2} (x, y)$ , so $C (u, v, w) = 2 ((u \cdot v) w + (u \cdot w) v + (v \cdot w) u)$ . Since $q \cdot q = 0$ and $q \cdot \overset{ˉ}{q} = 1$ , $C (q, q, \overset{ˉ}{q}) = 4 q$ and $G_{21} = 4$ .

2.1F1F2step 1.1step 1.2algebra

Thus $ℓ_{1} = 2 > 0$ , and [F2] gives a small branch only for $μ < 0$ ; its radial multiplier is unstable because $β < 0$ .

3.1step 1.1algebra∎

Directly, polar coordinates give $\dot{R} = μ R + R^{3}$ and $\dot{θ} = 1$ . At $μ < 0$ , $R = \sqrt{- μ}$ is a radial equilibrium with derivative $μ + 3 R^{2} = - 2 μ > 0$ , so the periodic orbit is unstable.

Depends on

Used by

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Dependency tree · two levels

16 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Yuri A. Kuznetsov, Elements of Applied Bifurcation Theory, 2nd ed., Springer (standard reference, not scraped)
J. E. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications, Springer-Verlag, 1976 (standard reference, not scraped)
Yuri A. Kuznetsov, Andronov-Hopf Bifurcation, Scholarpedia 1(10):1858 (2006) (standard reference, not scraped)