  A theoretic framework for dynamics is obtained by transferring dynamics from
state space to its dual space. As a result, the linear structure where dynamics
are analytically decomposed to subcomponents and invariant subspaces
decomposition based on local Koopman spectral theory are revealed. However,
nonlinear dynamics are distinguished from the linear by local exponential
dynamics and infinite dimension, where the latter is due to nonlinear
interaction and characterized by recursively proliferated Koopman eigenspaces.
The new framework provides foundations for dynamic analysis techniques such as
global stability analysis (GSA) and dynamic mode decomposition (DMD) technique.
Additionally, linear structure via Mercer eigenfunction decomposition derives
the well-known proper-orthogonal decomposition (POD). A Hopf bifurcation
process of flow past fixed cylinder is decomposed numerically by the DMD
technique. The equivalence of Koopman decomposition to the GSA is verified at
the primary instability stage. The Fourier modes, the least stable Floquet
modes, and their high-order derived modes around the limit cycle solution are
found to be the superposition of countably infinite number of Koopman modes
when the flows reach periodic. The nonlinear modulation effects on the mean
flow is the saturation of the superimposed monotonic Koopman modes. The
nonlinear resonance phenomenon is attributed to the alignment of infinite
number of Koopman spectrums. The analysis of above nonlinear dynamic process
relies on the properties of continuity of Koopman spectrums and
state-invariance of Koopman modes discussed in part 1. The coherent structures
are found related to the state-invariant modes.
