| Chaos, which widely exists, can be described as a very typical behavior of many nonlinear systems. Control or applications of it must be under a prerequisite that whether a nonlinear system is chaotic. Besides those criteria using numerical methods, it is now theoretically convinced that the Melnikov method, an analytical method, which estimates the existence of the invariant set of a chaotic motion, is a very important one. Because one can directly use the Melnikov method to make analytical operation, qualitative and quantitative analysis of a system is more convenient. The Melnikov method is used in this paper to determine the threshold value beyond which a specially appointed system will become chaotic. The state of the system can be transformed from chaos to periodicity by adjusting the parameters of the drive signal. A simple algorithm of calculating the Melnikov function via the numerical integral method is employed in this paper. This paper provides the theoretical foundation of how to improve the detection accuracy and the anti-jamming ability of the signal detector. A second-order ordinary differential equation, which has a periodical weak perturbation and homoclinic orbit (or heteroclinic orbit), will rightly represent a kind of dynamical system, for which, using some appropriate mathematical methods, one can obtain the two-dimensional Poincaré mapping. If there are transversal homoclinic points in a Poincaré mapping, there will be a chaos invariant set in the relevant system in the sense of the Smale horseshoes transform. The transversal homoclinic point is characterized by the Melnikov function. To obtain the analytical expression of the Melnikov function, one should get the homoclinic orbit (or the heteroclinic orbit) of the unperturbed nonlinear system, and then calculate the integration of the homoclinic orbit's (or the heteroclinic orbit's) Melnikov function. In general, the residue theorem is used, and the computation of the integration is very complex. One can get the analytical expression of a simple model, but cannot get it from a more complex one. For instance, a constant in the nonlinear item will make the operation of the analytical expression of the homoclinic orbit (or the heteroclinic orbit) impossible; the operation above will work when the exponent of the nonlinear item is large, but the further operation about the analytical expression of the Melnikov function then has been proved impossible. The numerical analysis will be employed, since we are concerned about the approximate solution of the Melnikov function. Our main algorithm of this numerical integral method of the Melnikov function is based on the fact that the time variable t can be written as the functions of the state variable x, then by using the computer, the new integration can be resolved. Thus, one can get curve of threshold beyond which chaos may occur. Then, a qualitative and quantitative analysis via the curve will work. This will avoid computing the analytical expression of the homoclinic orbit (or the heteroclinic orbit) and the Melnikov function. The use of computer will enormously reduce the artificially computational complexity. The fundamental principle and the algorithm of evaluating the chaos threshold by the Melnikov method have been introduced above. The Melnikov method will be used to determine the chaotic threshold of the improved Duffing equation in this paper. The Duffing equation has been given in this paper. When an external additional constant drive is applied, one may get a new chaos threshold using the method discussed above. The effect of and the angular frequency ω of the sinusoidal drive on the performance of the chaotic threshold has been explored. Then a new nonlinear system is founded according to the Duffing equation in which the exponent of the nonlinear item is increased from 3 to 5. The new one has been studied from the following three aspects: 1. The method of numerical integration of the Melnikov function is used to get the curve of threshold. The resul...
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