A novel Lie-group theory and complexity of nonlinear dynamical systems

The complexity of a dynamical system x ̇ = f ( x , t ) is originated from the nonlinear dependence of f on x . We develop a full Lorentz type Lie-group for the augmented dynamical system in the Minkowski space. Depending on a signum function Sign ≔ sign ( ‖ f ‖ 2 ‖ x ‖ 2 - 2 ( f · x ) 2 ) = - sign ( cos 2 θ ) , where θ is the intersection angle between x and f , we can derive a group-preserving scheme based on SO o ( n , 1 ) , which has two branches: compact one and non-compact one. The barcode to show the complexity of nonlinear dynamical systems is obtained by plotting the value of Sign with respect to time. The Rössler equation possesses the most elementary chaotic mechanisms of stretching and folding, which is useful to validate the present theory of nonlinear dynamical system. Analyzing the barcode, we can point out the bifurcation of subharmonic motions and the chaotic range in the parameter space. The bifurcation diagram of the percentage of the first set of dis-connectivity A 1 - ≔ { sign ( cos θ ) = + 1 and sign ( cos 2 θ ) = + 1 } with respect to the amplitude of harmonic loading leads to a finer structure of devil staircase for the ship rolling oscillator, and the cascade of subharmonic motions to chaos for the Duffing oscillator.