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

• A novel Lie-group theory was developed for nonlinear dynamical system.
• The complexity can be revealed by a barcode.
• We can construct bifurcation diagram in terms of the percentage of first set of dis-connectivity.
• The present method is effective to analyze chaos of nonlinear dynamical system in parameter space.

The complexity of a dynamical system ẋ=f(x,t) is originated from the nonlinear dependence of ff on xx. 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(cos2θ), where θθ is the intersection angle between xx and ff, we can derive a group-preserving scheme based on SOo(n,1)SOo(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 A1-≔{sign(cosθ)=+1andsign(cos2θ)=+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.