Carleman Embedding of the Driven Pendulum

Carleman embedding is a linearization technique that allows writing a finite system of nonlinear (NL) first-order initial-value ordinary differential equations (ODEs) as an infinite linear system of first-order ODEs.  Most challenging computational plasma problems can be cast as such a first-order NL dynamical system.  Hence, it is worthwhile to explore the Carleman linearization (CL) technique.  Additionally, quantum circuits can be represented as a linear system, and there is much recent work in the development of quantum linear systems algorithms (QLSAs).  Therefore, CL may provide a tool to solve nonlinear problems on a quantum computer [1].  Efficient QLSAs will have a computational complexity that is logarithmic with system dimension.  Here, we examine the driven pendulum problem as a test for applying CL to NL Hamiltonian dynamics.  The pendulum problem is seemingly elementary and is also a classic example of Hamiltonian chaos.  It is fundamental for the understanding of island overlap and other aspects of Hamiltonian dynamics. We show that the exact solution can be expressed as an infinite linear system by a change of variables.  We truncate the linear system and explore numerical convergence properties.  Using the eigenvalue solution of the truncated CL system, we can time advance, effectively taking a very large timestep on the order of the nonlinear oscillation period. 

[1] "Linear embedding of nonlinear dynamical systems and prospects for efficient quantum algorithms," A. Engel, G. Smith, S.E. Parker, Phys. Plasmas 28, 062305 (2021).