# New PDF release: Applications of an Expansion Theorem to the Development of

By Brown E. W.

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**Example text**

The LC circuit is a typical free oscillator. Nonlinear oscillations are oscillatory solutions to nonlinear diﬀerential equations. This chapter begins with a study of free nonlinear problems in two variables, about which a great deal is known. Most oscillations studied here can be described by functions of the form F (x1 , . . , xN ) that are 2π-periodic in each of N phase variables x1 , . . , xN , which themselves vary with time. A convenient mathematical description of an oscillator is one given in terms of such phase variables, 28 2.

If we can deﬁne a matrix R by the formula log(C) , T so that exp(RT ) = C, then the matrix R= P (t) = Φ(t) exp(−Rt) satisﬁes the identity P (t + T ) = P (t) for all t. This follows because for all t, P (t + T ) = Φ(t + T ) exp(−RT ) exp(−Rt) = Φ(t) exp(−Rt) = P (t). The logarithm of C is well-deﬁned (as shown in [24]), although it might be a matrix of complex numbers. This result is especially helpful in determining the behavior of x(t) as t → ∞. For example, if all of the eigenvalues of R have negative real parts, then x(t) → 0 as t → ∞.

3 Linear Systems with Quasiperiodic Forcing Now consider the equation dx = Ax + g(t), dt where A is a constant diagonalizable matrix and g is a continuous, quasiperiodic function generated by two frequencies, say ω and µ: ∞ g(t) = Cm,n exp[i(mω + nµ)t]. m,n=−∞ We suppose that |Cm,n | ≤ K/(m2 + n2 ) for some constant K. The solution x(t) is given by the formula t exp[(A(t − s)]g(s)ds. 6. Linear Systems with Variable Coeﬃcients: Variation of Constants Formula if these series converge. The series do converge if the eigenvalues of A are not purely imaginary.