Generalization to Multiple Variable Systems

The generalization of this approach to a system of n ODEs can be illustrated by considering the two-variable dynamical system  

As with the one-variable system, we will carry out a Taylor series expansion of the functions f and g, utilizing the fact that u and v are both very small. For a general function of two variables, h(x, y), the Taylor series expansion about the point (x, y ) will then be 

The resulting system of equations, [23], can be solved using linear algebra techniques. We rewrite this system in matrix form by first defining the column vector w and the matrix J (the Jacobian evaluated at steady state) as 

The solution of a linear system such as this is a linear combination of exponential functions of the form 

We consider first an abstract chemical model invented by a group of investigators in Brussels that has played an important role in the development of nonlinear dynamics. Because of its city of origin, it has become known as the Brusselator, or oscillator from Brussels. The chemical mechanism associated with this abstract model is  

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