By replacing each derivative of order higher than the 1^{st} with a new variable, we can rewrite any set of nonlinear differential equations of arbitrary order by a larger set of 1^{st} order nonlinear differential equations. Therefore, the study of stability can be restricted to the latter kind of equations without loss of generality. Treating the derivative as variables and all else as constants, we can solve for the derivatives by standard algebraic means. A unique solution is expected if the original differential equations is to admit a unique solution. Thus, we can further restrict our attention to equations of the form
where
The existence of a unique solution near a
point
where
The steady state solutions
are often called fixed, singular, or critical points. If we plot the solutions of (1) in the n-dimensional phase plane using t as parameter, they will trace out
trajectories except for the
The stability of a steady state
Taking the time derivatives of (5) gives, with the help of (3),
or, in component form
In linear
stability theory, the
where
Now, the Lipschitz conditions ensure the
existence of n independent
eigenvectors to A so that it can be
diagonalized. The eigenvalues
Let the right eigenvector belonging to the
eigenvalue
Setting
Þ
1.
All
2.
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3.
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5.
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