The Lorenz equation is widely used in the theory of non-linear equations and was encountered in modeling of natural convection

The Lorenz equation is widely used in the theory of non-linear equations and was encountered in modeling of natural convection. While working with these equations Lorenz discovered the phenomena of chaos. The system is described by the following set of differential equations: dx dt = α(y − z) dy dt = −xz + rx − y dz dt = xy − bz where α, r, and b are the parameters. Fix two parameters α = 10 and b = −8/3 and examine the stability of the system with r as a parameter. Write MATLAB code to track the trajectories and prepare phase-plane plots.

245:-Analyze the Frank-Kamenetskii problem for the three standard geometries of slab, cylinder, and sphere. You will need to discretize the operators suitably for the cylinder and sphere. Plot the bifurcation diagram as a function of the thermicity parameter.