A Lie-group DSO(n)DSO(n) method for nonlinear dynamical systems

It is known that a nonzero vector x∈Rn can be decomposed into a direction multiplied by a length, i.e., x=‖x‖n. For a nonlinear dynamical system ẋ=f(x,t) we can derive a Jordan dynamics for n, and a generalized Hamiltonian dynamics for x with a diagonal symmetric and a skew-symmetric coefficient matrix in a quasilinear system: ẋ=[S+W]x. The new system endows a Lie-symmetry DSO(n)DSO(n). Then we derive a closed-form formula x(t)=G(t)x(0),G(t)∈DSO(n) for a small time step with t≤ht≤h, where hh is a small stepsize. Three numerical examples are given to validate the accuracy and efficiency of the DSO(n)DSO(n) method.