Differential System＇s Nonlinear Behaviour of Real Nonlinear Dynamical Systems

Differential System's Nonlinear Behaviour of Real Nonlinear Dynamical Systems

Chaos attractor behaviour is usually preserved if the four basic arithmetic operations, i.e. addition, subtraction, multiplication, division, or their compound, are applied. First-order differential systems of one-dimensional real discrete dynamical systems and nonautonomous real continuous-time dynamical systems are also dynamical systems and their Lyapunov exponents are kept, if they are twice differentiable. These two conclusions are shown here by the definitions of dynamical system and Lyapunov exponent. Numerical simulations support our analytical results. The conclusions can apply to higher order differential systems if their corresponding order differentials exist.