Normal form (dynamical systems)

In mathematics, the normal form of a dynamical system is a simplified form that can be useful in determining the system's behavior.

Normal forms are often used for determining local bifurcations in a system. All systems exhibiting a certain type of bifurcation are locally (around the equilibrium) topologically equivalent to the normal form of the bifurcation. For example, the normal form of a saddle-node bifurcation is ${\frac {\mathrm {d} x}{\mathrm {d} t}}=\mu +x^{2}$ where $\mu$ is the bifurcation parameter. The transcritical bifurcation ${\frac {\mathrm {d} x}{\mathrm {d} t}}=r\ln x+x-1$ near $x=1$ can be converted to the normal form ${\frac {\mathrm {d} u}{\mathrm {d} t}}=\mu u-u^{2}+O(u^{3})$ with the transformation $u=x-1,\mu =r+1$ .^[1]

References

↑ Strogatz, Steven. "Nonlinear Dynamics and Chaos". Westview Press, 2001. p. 52.

Normal form (dynamical systems)

References

Further reading