This is a MATLAB function designed to numerically compute and draw bifurcation diagrams. Functionality is currently limited to 1st order 1D functions, but will expand as time goes on.
Last updated 2/16/2021.
Consider a function of the form below, where x is the state variable , r is the parameter.
$$
\frac{dx}{dt} = f(x;r)
$$
Then the bifurcation diagram is the solution to:
$$
f(x_e,r) = 0
$$
The stability of the fixed points thus found is given by
$$
\frac{df}{dx}(x,r)\large |_{x_e} = \begin{cases} < 0 & \text{stable} \ = 0 & \text{half-stable} \ > 0 & \text{unstable} \end{cases}
$$
The main function to use is bifurcation1D(). In order to use it, you must first define the dynamical system as a MATLAB function of the form:
[dx, fprime] = dynSys(x,r)Where the output dx is the value of f(x,r) and fprime is df/dx.
In the current implementation bifurcation1D() autodetects the level of vectorization of the input function and responds accordingly. Best results can be obtained if the dynamic system will respond to a meshgrid of x and r. If not, the system should be vectorized for at least x. If it isn't vectorized at all, results will still be obtained but more slowly.
Once the dynamic system is defined, you may call one of the following commands to generate the bifurcation diagram. xLim and rLim represent limits on x and r respectively, such that -xLim < x < xLim and -rLim < r < rLim. Default values are used if these are not specified.
bifurcation1D(@dynSys)
bifurcation1D(@dynSys,xLim)
bifurcation1D(@dynSys,xLim,rLim)Developed in conjunction with taking ENM512 - Nonlinear Dynamics and Chaos at the University of Pennsylvania in Spring 2021