Considered in this note are numerical tracking algorithms for the accurate following of implicit curves. We start with a fixed point on the curve, and then systematically place on it additional points, one after the o...Considered in this note are numerical tracking algorithms for the accurate following of implicit curves. We start with a fixed point on the curve, and then systematically place on it additional points, one after the other. In this note we first go over the basic procedure of moving forward tangentially from an already placed point then orthogonally returning to the curve. Next, we further consider higher order forward stepping procedures for greater accuracy. We note, however, that higher order methods, desirable for greater accuracy, may harbor latent instabilities. This note suggests ways of holding such instabilities in check, to have stable and highly accurate tracing methods. The note has several supporting numerical examples, including the rounding of a dynamical “snap-through” point.展开更多
In this paper, the periodic initial value problem for the following class of nonlinear Schrodinger equation of high order i partial derivative u/partial derivative t + (-1)(m) partial derivative(m)/partial derivative ...In this paper, the periodic initial value problem for the following class of nonlinear Schrodinger equation of high order i partial derivative u/partial derivative t + (-1)(m) partial derivative(m)/partial derivative x(m) (a(x)partial derivative(m)u/partial derivative x(m)) + beta(x)q(\u\(2))u + f(x, t)u = g(x, t) is considered. A leap-frog finite difference scheme is given, and convergence and stability is proved. Finally, it is shown by a numerical example that numerical result is coincident with theoretical result.展开更多
To enhance the anti-breaking performance of privacy information, this article proposes a new encryption method utilizing the leaping peculiarity of the periodic orbits of chaos systems. This method maps the secret seq...To enhance the anti-breaking performance of privacy information, this article proposes a new encryption method utilizing the leaping peculiarity of the periodic orbits of chaos systems. This method maps the secret sequence to several chaos periodic orbits, and a short sequence obtained by evolving the system parameters of the periodic orbits in another nonlinear system will be the key to reconstruct these periodic orbits. In the decryption end, the shadowing method of chaos trajectory based on the modified Newton-Raphson algorithm is adopted to restore these system parameters. Through deciding which orbit each pair coordinate falls on, the original digital sequence can be decrypted.展开更多
文摘Considered in this note are numerical tracking algorithms for the accurate following of implicit curves. We start with a fixed point on the curve, and then systematically place on it additional points, one after the other. In this note we first go over the basic procedure of moving forward tangentially from an already placed point then orthogonally returning to the curve. Next, we further consider higher order forward stepping procedures for greater accuracy. We note, however, that higher order methods, desirable for greater accuracy, may harbor latent instabilities. This note suggests ways of holding such instabilities in check, to have stable and highly accurate tracing methods. The note has several supporting numerical examples, including the rounding of a dynamical “snap-through” point.
文摘In this paper, the periodic initial value problem for the following class of nonlinear Schrodinger equation of high order i partial derivative u/partial derivative t + (-1)(m) partial derivative(m)/partial derivative x(m) (a(x)partial derivative(m)u/partial derivative x(m)) + beta(x)q(\u\(2))u + f(x, t)u = g(x, t) is considered. A leap-frog finite difference scheme is given, and convergence and stability is proved. Finally, it is shown by a numerical example that numerical result is coincident with theoretical result.
基金This project was supported by the National Natural Science Foundation of Shaan'Xi Province, China (2003F40).
文摘To enhance the anti-breaking performance of privacy information, this article proposes a new encryption method utilizing the leaping peculiarity of the periodic orbits of chaos systems. This method maps the secret sequence to several chaos periodic orbits, and a short sequence obtained by evolving the system parameters of the periodic orbits in another nonlinear system will be the key to reconstruct these periodic orbits. In the decryption end, the shadowing method of chaos trajectory based on the modified Newton-Raphson algorithm is adopted to restore these system parameters. Through deciding which orbit each pair coordinate falls on, the original digital sequence can be decrypted.
