Non-Associative Commutative Algebras for Dynamical Systems

We will consider quadratic dynamical systems, and go to general quasipolynomial differential equations of the form $$\dot{x_i} = x_i\sum_{j=1}^mA_{ij}\prod_{k=1}^nx_k^{B_{jk}}$$

In one dimension we have $$\dot{x}=ax^2+bx+c$$ or in the notation above $$\dot{x}=x(ax+b+c/x)=x\sum_{j=1}^3A_{j}x^{B_j}.$$ With $A_1=a$, $A_2=b$, $A_3=c$ and $B_1=1$, $B_2=0$, $B_3=-1$.



In [2]:

    
import numpy as np
import sympy as sp



In [8]:

    
a1,a2,a3=sp.Symbol("a1"),sp.Symbol("a2"),sp.Symbol("a3")
b1,b2,b3=sp.Symbol("b1"),sp.Symbol("b2"),sp.Symbol("b3")
A=np.array([a1,a2,a3])
B=np.array([b1,b2,b3])



In [16]:

    
M=np.outer(B,A)



In [20]:

    
print M









    



[[a1*b1 a2*b1 a3*b1]
 [a1*b2 a2*b2 a3*b2]
 [a1*b3 a2*b3 a3*b3]]



In [25]:









    Out[25]:





(9, 9)



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