Brusselator Hopf Threshold
Track the local chemical feedback that crosses an oscillatory instability threshold.
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Brusselator Hopf Threshold
The Brusselator is a theoretical model for a type of autocatalytic reaction that exhibits oscillations. This module explores the stability of its fixed point as parameters are varied, specifically focusing on the Hopf bifurcation where the system transitions from a stable steady state to sustained chemical oscillations (a limit cycle).
Worked Example
Let's calculate the stability of the system at the Hopf boundary with the feed parameters set to A = 1 and B = 2.
- Identify the dimensionless reaction model equations: u' = A - (B + 1)u + u²v
v' = B u - u²v - Calculate the unique fixed point (u*, v*): u* = A = 1
v* = B / A = 2 / 1 = 2 - Construct the Jacobian matrix J evaluated at the fixed point: J = [[B - 1, A²], [-B, -A²]] = [[2 - 1, 1²], [-2, -1²]] = [[1, 1], [-2, -1]]
- Calculate the trace (τ) and determinant (Δ) of the Jacobian: τ = B - 1 - A² = 2 - 1 - 1 = 0
Δ = A² = 1² = 1 - Determine the eigenvalues λ = (τ ± √(τ² - 4Δ)) / 2: λ = (0 ± √(0 - 4(1))) / 2 = ± i
- Since the trace is exactly 0 and the determinant is positive, the real part of the maximum eigenvalue is 0. This indicates that the system is exactly at the Hopf bifurcation boundary.
Experiment
Use the interactive model to observe the transition through the Hopf threshold:
- Set Feed A to 1 and Feed B to 1. Observe that the Jacobian trace is -1 (negative), indicating Linear stability is Stable.
- Increase Feed B to 2. Observe that the Jacobian trace becomes exactly 0 and the Linear stability becomes Hopf boundary.
- Further increase Feed B to 3. Observe that the Jacobian trace becomes +1 (positive), and the Linear stability becomes Unstable.
- Change the Probe u to 2 and Probe v to 4 to sample the local vector field derivatives at different locations in the phase space.
Limitations
This module analyzes the local linear stability of the fixed point. While a positive trace indicates linear instability of the fixed point and the birth of a limit cycle (via the Poincaré-Bendixson theorem for this specific 2D system), local linear analysis alone is not a global proof of a stable limit cycle. Furthermore, this is a well-mixed dimensionless model; it does not simulate spatial diffusion or Turing patterns that can occur in spatially distributed versions of the Brusselator.
References
The analysis of nonlinear systems, local stability, and bifurcations is deeply rooted in dynamical systems theory. For more details on the Brusselator and the Hopf bifurcation, please refer to standard literature in this area:
UCSD Physics 221A Lectures on Nonlinear Dynamics
Experience it interactively
Adjust parameters, observe in real time, and build deep intuition with Riano’s interactive Brusselator Hopf Threshold module.
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