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Appendix C

Nonlinear Differential Equations and Oscillations

In Section 8.3 we discussed systems of linear differential equations of the form x′ = Ax. These are written in vector notation in which A is the matrix of coefficients and the prime ′ indicates differentiation. Our interest in these equations stems from the fact that the local behavior of solutions of a nonlinear system of differential equations x′ = f(x) about some equilibrium solution is determined by the global behavior of the linearized system x′ = Ax, where A now indicates the Jacobean matrix of f.

Suppose x is an equilibrium point of x′ = f(x), with f(x) = 0. If solutions that begin nearby to x return to this point as t increases, we say that the equilibrium is an attractor or, ...

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