A Lie-group method for nonlinear dynamical systems

It is known that a nonzero vector x ∈ R n can be decomposed into a direction multiplied by a length, i.e., x = ‖ x ‖ n . For a nonlinear dynamical system x ̇ = 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: x ̇ = [ S + W ] x . The new system endows a Lie-symmetry D S O ( n ) . Then we derive a closed-form formula x ( t ) = G ( t ) x ( 0 ) , G ( t ) ∈ D S O ( n ) for a small time step with t ≤ h , where h is a small stepsize. Three numerical examples are given to validate the accuracy and efficiency of the D S O ( n ) method.