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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