Lyapunov exponents and chaos investigation

This dissertation describes different methods of chaos investigation paying special attention to one of the most popular methods: Lyapunov exponents.; First a problem of Lyapunov directions and exponents of a system of ordinary differential equations with known general solutions is presented and equations describing both Lyapunov directions and exponents are derived. To verify these equations a number of simple theorems describing Lyapunov exponents of systems of linear ordinary differential equations with constant coefficients are proven.; Ordinary differential equations describing Lyapunov directions and exponents are derived and a new algorithm based on these equations is created. The new algorithm does not use reorthogonalisation (orthogonalisation is “built in” into equations describing Lyapunov directions).; A number of miscellaneous results is included. A diagonalizing transformation that can be used to compute Lyapunov exponents is presented. A number of estimates and alternative methods that can be used to compute Lyapunov exponents are also shown. Effects of differential equations stabilization are described.; A number of examples is presented through this dissertation to illustrate derived theorems and algorithms. The most common test system is a system of ordinary differential equations with constant coefficients. Also Lorenz system, Mathieu equation and Duffing's equation are used as an illustration. The most interesting application of derived methods presented in this dissertation is a three body problem (satellite motion with a Moon influence). This example shows that there are regions of phase space where orbits are chaotic.