Abstract:
In the field of nonlinear science, Hao Berlin is the most deserving scholar in China to receive a prestigious award, but it is truly regrettable that he is no longer with us. Using this opportunity of NSC 2026, I would like to pay my tribute to him.
Renormalization-group methods provide us with a tool to characterize the transition to chaos and to understand the universal critical properties of the dynamics. An approximate renormalization transformation is explicitly constructed for the last destroyed invariant torus with a golden mean winding number in the Taylor-Chirikov standard map. The renormalization map exhibits a trivial stable fixed point corresponding to the pure rotation and a non-trivial saddle point describing the transition to chaos. It is derived that the products of the amplitude and frequency associated with the critical orbit are constant for the significant components with frequencies being Fibonacci numbers. A renormalization approach for the electronic spectrum of Fibonacci chains is also briefly discussed.