Nonautonomous differential equations

Jupyter Notebook
1

Hello everyone,
I am working on a growth model containing a system of non-autonomous differential equations. The system is as follows:

dx/dt = x^a - (b + c(y) + d(y))x
dy/dt = xe^[c(y) + d(y)] - hy

alpha, a, b, c, and h are parameters that b and c are dependent on y.

would you please do me a favour and help me simulate it using Jupyter notebook?
Thank you very much.

2

You need to be precise on initial and/or boundary values as well, since it changes the methods to use

3

Thank you so much. I’ve defined the initial values, but what do you mean by boundary values? Is it possible to solve a nonautonomous system using ODEINT? Thank you very much.

4

I don’t know anything about ODEINT but if you have an initial value problem, then http://docs.juliadiffeq.org/latest/tutorials/ode_example.html#Example-2:-Solving-Systems-of-Equations-1 is very easy to use.

However, if this is coming from a neoclassical growth moodel, then you have one initial condition (on capital) and one boundary value (for consumption… Either a terminal value or a steady state from the the transversality condition). The typical way to solve those is a shooting method. Pick an initial condition for consumption, solve the system of ODEs as an initial value problem, and then modify the initial condition guess until the boundary value holds approximately.

5

Thank you very much indeed.
Oh yes I understood, because I have a very simple supply-side neoclassical model with environmental constraints, there is no boundary value. I appreciate your help.

6

And for reference, if this is the problem, the following tutorial may be helpful: http://docs.juliadiffeq.org/latest/tutorials/bvp_example.html

7

Thank you very much indeed.