I have a system of non-linear differential equations that explains the behaviour of some of the cancer cells.
Looking for help identifying the equilibrium points and eigenvalues of this model in order to determine the type of bifurcation present.
Thanks in advance.
Well it's a good idea to find some of them, first. The first equation implies that y=0 is an equilibrium, so a class of equilibria is of the form (x,0,z). That reduces the problem. From the last equation it then should be possible to solve for z and, from the second, for x.
Then look at the other factor of the first equation; and so on.
Sachin Chandrasekara There are numerous methods for finding the equilibrium points of an ODE system using Python or Matlab. A numerical root-finding algorithm, such as the bisection technique, Newton-Raphson method, or secant method, is one typical way. These methods may be implemented using built-in functions in Python and Matlab, such as scipy.optimize.root. Another approach is to utilize symbolic algebra to solve the system of equations for the equilibrium points analytically, using tools like SymPy in Python or Symbolic Math Toolbox in Matlab.
Once you've determined your system's equilibrium points, you may apply linearization to get the eigenvalues of the Jacobian matrix at those positions. The eigenvalues reflect the stability of the equilibrium points and may be used to calculate them.
It's crucial to note that identifying the equilibrium point and calculating the eigenvalues are only the initial steps in a bifurcation analysis; after that, you may utilize tools like bifurcation diagrams, phase plots, and Poincare sections to analyze the system's behavior.
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