F A Convergence Theorem

E Some Techniques in Nonlinear Analysis F A Convergence Theorem 

Matrices are encountered on a number of occasions in the text, particularly in arguments for the linearization about rest points or periodic orbits. On some occasions we encounter matrices of dimension large enough that direct computation of the eigenvalues may not be feasible. Fortunately, the matrices are often of a special form and there are theorems to cover such cases. In this appendix we list some of the useful theorems in the analysis of these special systems, along with appropriate references. 

The next result is also general and, while the conditions are not often met, it is an important tool when it can be applied. The theorem is called the Gerschgorin circle theorem. An excellent general reference on matrices is Lancaster and Tismenetsky [LT], and most of the results here are quoted from that source. Another important source is Berman and Plem-mons [BP], particularly for special results on nonnegative matrices. 

Theorem A.l [LT, p. 371). Let A be an nxn matrix with elements denoted by aij, and let p be defined by pt = Lk dik where X k denotes the sum on k from k = to n with the term k = i omitted. (The diagonal element of the matrix is omitted from the sum.) Then every eigenvalue of A lies in at least one of the disks 

As generally used in stability theory, the diagonal element is negative, and Theorem A.l is used in attempts to show that the radius of the disk (called a Gerschgorin disk) is smaller in absolute value. There are many generalizations that yield finer results at the expense of a more complicated criterion. One of these, useful for our work, involves the concept of an irreducible matrix. 

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Theorem 2 If u x) C G), that is, a solution possesses continuous derivatives in (5 = 0 + F of the first four orders, then the difference scheme converges uniformly with the rate O(h ), that is, it is of second-order accuracy, so that estimate (16) is valid. 

The following theorem guarantees convergence. Let a be the solution to a, =f,(a). Assume that given h>0, there exists a number 0 < p < 1 such that 

The short time solutions obtained in section 8.1.4 (examples 8.1.5 and 8.1.6) require only a few terms in the infinite series at short times to converge. However, at long times the series requires a large number of terms and cannot be used efficiently. The long time solution can be obtained using Heaviside expansion theorem.[l] If we denote the solution obtained in the Laplace domain as F(s)  

The development by Maclaurin s series cannot be used if the function or any of its derivatives becomes infinite or discontinuous when x is equated to zero. For example, the first differential coefficient of f(x) = >Jx, is x which is infinite for x = 0, in other words, the series is no longer convergent. The same thing will be found with the functions log a , cot a , 1/x, a1,x and sec lx. Some of these functions may, however, be developed as a fractional or some other simple function of x, or we may use Taylor s theorem. 

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