Reality of chaos in the double scroll circuit: A computer-assisted proof

Abstract

The authors prove three key inequalities stated in the paper by L.O. Chua, M. Komouro, and T. Matsumoto (see ibid., vol. CAS-33, p. 1072-1118, 1986) by giving verifiable error bounds to the quantities involved in the inequalities with an assistance of a computer. This provides another rigorous proof that the so-called double scroll circuit is chaotic in the sense of Shilnikov. Since a computer is used, everything must be transparent; there should be no black box. To provide a rigorous computer-base free from roundoff errors, it is shown that all the computations are reduced to four logic operations: AND, OR, NOT, and XOR. Based on this, the computer performs internal analysis, which is a method of computing intervals containing the true values. This is important for two reasons: first, the double scroll circuit then becomes, the first real, physical system in which chaos is (i) observed in the laboratory, (ii) confirmed by computer simulation, and (iii) proven mathematically. Second, the author's approach can easily be modified to apply to other problems, providing a powerful tool for a large class of problems. >