Bifurcation Analysis

A numerical study of the changes in the dynamics and stability of a system upon variations in its parameters

Procedure for stability analysis at fixed points

Consider the following system of ordinary differential equations: dx/dt = F(x)

1. Determine the fixed point vector, x^∗, solving F(x^∗) = 0
2. Construct the Jacobian matrix, J(x) = ∂F(x)/∂x
3. Compute eigenvalues of J(x^∗): det |J(x^∗) − λE| = 0
4. Conclude on stability or instability of x^∗ based on the real parts of eigenvalues
   • All eigenvalues have real parts less than zero → x^∗ is stable
   • At least one of the eigenvalues has a real part greater than zero → x^∗ is unstable

Requirements

• SteadyStateDiffEq
• Sundials
• ForwardDiff
• PyPlot

Plotting bifurcation diagram

# Change directory to examples/XXX
include("bifurcation.jl")
using .Bifurcation

Bifurcation.analysis()
Bifurcation.diagram()

Installation

$ git clone https://github.com/himoto/bifurcation.git

Acknowledgements

I would particularly like to thank Dr. Gouhei Tanaka (Graduate School of Engineering, The University of Tokyo) for valuable discussions.

References

• Yao, G., Lee, T. J., Mori, S., Nevins, J. R. & You, L. A bistable Rb-E2F switch underlies the restriction point. Nat. Cell Biol. 10, 476–482 (2008). https://doi.org/10.1038/ncb1711

• Barr, A. R., Heldt, F. S., Zhang, T., Bakal, C. & Novák, B. A Dynamical Framework for the All-or-None G1/S Transition. Cell Syst. 2, 27–37 (2016). https://doi.org/10.1016/j.cels.2016.01.001

• Rata, S. et al. Two Interlinked Bistable Switches Govern Mitotic Control in Mammalian Cells. Curr. Biol. 28, 3824-3832.e6 (2018). https://doi.org/10.1016/j.cub.2018.09.059

License

MIT