Study planar systems near oscillatory instability with practical criteria. Check crossing conditions, frequency estimates, and cycle behavior using reliable nonlinear diagnostics.
Use either directly known trace and determinant values or values derived from your Jacobian.
The first plot shows real parts crossing near the critical parameter. The second plot shows small-amplitude cycle scaling when the normal form predicts a local limit cycle.
Linearization: For a planar system near equilibrium, use the Jacobian matrix \(J\). Its trace is \(\tau = a_{11}+a_{22}\) and determinant is \(\Delta = a_{11}a_{22}-a_{12}a_{21}\).
Eigenvalues: \(\lambda = \frac{\tau}{2} \pm \frac{1}{2}\sqrt{\tau^2 - 4\Delta}\). A Hopf boundary requires \(\tau = 0\) and \(\Delta > 0\), giving a purely imaginary pair.
Transversality: \(\frac{d\tau}{d\mu} \neq 0\). This ensures the real part crosses zero as parameter \(\mu\) changes.
Frequency at onset: \(\omega_0 = \sqrt{\Delta}\) when \(\tau = 0\) and \(\Delta > 0\).
Type classification: The sign of the first Lyapunov coefficient \(l_1\) determines local cycle behavior. \(l_1 < 0\) indicates a supercritical Hopf with a stable small cycle. \(l_1 > 0\) indicates a subcritical Hopf with an unstable small cycle.
Approximate amplitude: In normal-form scaling, the local cycle amplitude behaves like \(r \approx \sqrt{-\frac{\alpha(\mu-\mu_c)}{l_1}}\), where \(\alpha\) is represented here by \(d\tau/d\mu\).
| Case | Trace | Determinant | d(Trace)/dμ | l₁ | Expected Outcome |
|---|---|---|---|---|---|
| Scenario A | 0.00 | 4.00 | 0.80 | -0.12 | Supercritical boundary |
| Scenario B | 0.00 | 2.25 | 0.50 | 0.20 | Subcritical boundary |
| Scenario C | 0.30 | 3.00 | 0.80 | -0.12 | Trace too positive |
| Scenario D | 0.00 | -1.50 | 0.80 | -0.12 | No complex pair |
| Scenario E | 0.00 | 1.44 | 0.00 | -0.12 | Transversality fails |
It tests the classic local Hopf conditions for a planar equilibrium. It checks trace, determinant, transversality, oscillation frequency, and nonlinear cycle classification using the first Lyapunov coefficient.
A positive determinant supports a complex conjugate eigenvalue pair when the trace is near zero. Without that pair, the equilibrium does not pass through a Hopf-type oscillatory crossing.
At a local Hopf boundary, the real part of the complex pair crosses zero. For a 2×2 Jacobian, that crossing is expressed by the trace becoming zero.
It represents the speed and direction of the eigenvalue crossing as the parameter changes. A nonzero value is needed to avoid a tangential or degenerate crossing.
Its sign determines whether the local bifurcation is supercritical or subcritical. Negative values usually create a stable small limit cycle, while positive values usually create an unstable one.
No. It is a local normal-form estimate near the critical parameter. It is most useful for small parameter offsets and for systems already reduced to Hopf normal-form behavior.
Yes, but this page still asks for trace and determinant directly. The Jacobian entries are included for documentation and can be checked against your own symbolic or numerical derivation.
It is designed for local planar analysis or reduced two-dimensional center-manifold models. Higher-dimensional systems often need center-manifold reduction before applying these local Hopf diagnostics.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.