A theoretical framework for Koopman analyses of fluid flows, part 1: local Koopman spectrum and properties

Nonlinear dynamic systems play an important role in scientific research and engineering applications because most systems are inherently nonlinear. Unfortunately, nonlinearity makes a system much more complicated. On one hand, a nonlinear dynamic system can admit various complex solutions such as stable or unstable nodes/spirals/limit cycles or even multiple of them. Sometime there can be strange attractor, or even chaos solution (Strogatz, 2018). On the other hand, nonlinearity can create more complicated modes by self-interacting of flow structures. The modulation of modes as well as the dynamics are all attributed to nonlinearity, for example, the facts that base flow and the growth rate change in a Hopf-bifurcation process were attributed to nonlinearity (Landau & Lifshitz, 1959). Different from the well-studied linear dynamics, a nonlinear process remains a challenge in many situations.

Recently, there are emerging efforts to analyze the dynamics by a spatial-temporal decomposition in a data-driven manner. Researchers apply various decomposition techniques on the outcome of the systems to analyze their dynamic properties, especially in the fluid regime. These techniques are efficient in reconstructing dynamics in low-dimensional modeling. For instance, proper orthogonal decomposition (POD) was found to capture coherent structures of turbulent flow (Holmes et al., 1996) and rebuild dynamics efficiently (Lucia et al., 2004). Dynamic mode decomposition (DMD) based on eigendecomposition is also efficient to capture dynamics for many fluid problems (Schmid, 2010). The above decomposition techniques provide exact solutions for linear systems, while approximates well for certain nonlinear systems. Other than the mentioned data-driven manner using a set of dynamics-induced spatial bases, other fixed spatial bases derived from Sturm-Liouville theory (Marchenko, 1977), such as Fourier series, Chebyshev series, Bessel series or other trigonometry or polynomial series, are found efficient in many applications (Kim et al., 1987; Orszag, 1971; Andrews & Andrews, 1992; Mathelin et al., 2005; Jaganathan et al., 2008).

Besides the above decomposition techniques, the spectral decomposition of the linear, infinite-dimensional Koopman operator defined on a nonlinear dynamic system provides a promising decomposition for the nonlinear system (Mezić, 2005; Rowley et al., 2009). Koopman operator has a long history for nonlinear dynamics analysis. Koopman (1931) first introduced a linear operator, which is known today as Koopman operator, to ‘transform’ a nonlinear system to be evoluted by a linear operator. Neumann (1932b) found the fixed point (if it exists) of the Koopman operator defined on gas molecules dynamic system was an efficient device to determine if the system is ergodic. For instance, the equilibrium of thermodynamics (the zeroth law of thermodynamics) can be defined at the molecular level by finding such a fixed point of molecular dynamics despite the constantly moving gas molecules (Neumann, 1932a). Later, the spectral theory was developed for operators, since linear mapping on the infinite-dimensional linear space naturally raises spectral problem (Reed & Simon, 1972; Dunford & Schwartz, 1963; Dowson, 1978). Therefore, the recent works of spectral Koopman decomposition have received a lot of attention.

However, spectral theory of infinite-dimensional operator is more complicated than the finite-dimensional counterpart, the matrix spectral theory. For example, it is known bounded operators have bounded spectrums, while spectrums for unbounded operators remain undetermined (Reed & Simon, 1972). Therefore, current applications (Mezić, 2005) focused on periodic or ergodic systems, which define the unitary operator, one type of bounded operators. Unfortunately, extending operator spectral problem to a general nonlinear system is difficult from current authors and a few others’ experiences. For instance, inconsistency was discovered when approximating Koopman spectral decomposition using numerical DMD technique (Page & Kerswell, 2019). Or, to address the spectral problem of operator, artificial families of discontinuous unimodular Koopman operator were constructed for a nonlinear system (Mohr, 2014).

To address the mentioned difficulties, a local Koopman spectral problem on nonlinear dynamic systems is defined and studied in this work. This is the first step current authors have taken to understand the complicated behavior of nonlinear dynamics. They are the key to understand the properties of nonlinear dynamics, which will be explored extensively in part 2 of this paper, where a theoretical framework for both linear and nonlinear dynamics will be developed.

In part 1, Koopman spectrums of various linear systems are introduced in §2. Starting from a linear time-variant (LTI) system to a general linear time-variant system, Koopman spectrums are found to change from constants to time-dependent values. Then state-dependent Koopman spectrums for nonlinear dynamic systems are introduced in §3, generalizing the local spectrum discovered in LTV systems to nonlinear autonomous systems. Several important properties of Koopman spectrums of nonlinear dynamics are introduced. They are the proliferation rule, the hierarchy structure, and the continuity property. In §4, Koopman modes of the full-state variable are discussed for different linear systems, while modes for nonlinear systems will be discussed in part 2. DMD, the data-driven technique to realize Koopman decomposition, is examined in §5 for its applicability to various dynamic systems, such as LTI, LTV, or asymptotic nonlinear systems. Finally, two numerical examples are presented in §6. The primary and secondary instabilities of flow passing a fixed cylinder are studied to support the discovery of local Koopman spectrums.

Considering a discrete dynamic system evolving on a manifold $\mathcal{M}$ such that,

$\boldsymbol{x}_{k}\in\mathcal{M}$. $f$ is a map from $\mathcal{M}$ to itself, which evolves the dynamics one step forward, and $k$ is an integer time step.

An observable $g$ is a function defined on the manifold $\mathcal{M}$, $g:\mathcal{M}\rightarrow\mathbb{R}$. The use of observable to study the dynamics introduces flexibility for many applications, but more importantly, to transfer dynamics to its functional space whose importance will be made clear in part 2. Besides, the trajectory of $\boldsymbol{x}(t)$ may not always be available or of interest for some applications. For example, Lasota & Mackey (2013) studied the probability density function (PDF) instead of the sensitive and chaotic trajectory to understand the dynamics of nonlinear deterministic systems.

The Koopman operator $U$ is an infinite-dimensional linear map defined on the observable. It evolves the observable one-step forward by

The Koopman operator is a linear operator, since

Linear map acting on a linear space, though both are infinite-dimensional, raises the spectral problem. In fact, the Koopman operator admits a unique decomposition of singular and regular parts (Reed & Simon, 1972; Mezić, 2005).

where $U_{s}$, $U_{r}$ are a singular and a regular operator respectively defined on the same manifold $\mathcal{M}$. $U_{s}$ has pure discrete spectrums while $U_{r}$ has continuous spectrums. For simplicity this paper consider only $U=U_{s}$. Spectral problem of Koopman operator reads

where $\phi_{i}(\boldsymbol{x}):\mathcal{M}\rightarrow\mathbb{R}$ is the Koopman eigenfunction, and $\rho_{i}\in\boldsymbol{C}$ is the Koopman spectrum.

It will be clear later, that the spectral problem of a linear system is part of the Koopman spectral problem defined on that linear system. And the linear spectrums of the linearized system provides a subset of Koopman spectrums of the original nonlinear systems. Therefore, discrete Koopman spectrums are common, since fluid systems with bounded domain usually have discrete spectrums. On the other hand, continuous spectrums usually appear in the unbounded fluid system. For instance, Bénard convection between infinite horizontal planes and the plane Poiseuille flow have continuous spectrum (Grosch & Salwen, 1978; Drazin et al., 1982). Therefore, the continuous Koopman spectrums are not uncommon.

The Koopman operator can be defined on continuous differential systems (32) as well. For simplicity, definition of the discretized system (1) is presented since the continuous system can always be reasonably discretized. Later, the reader will realize the continuous and discretized systems will share the same eigenfunctions and there is a logarithmic relation between their spectrums.

The infinite dimensional eigenfunctions $\phi_{i}(\boldsymbol{x})$ provide a set of bases for functional analysis. An observable $g(\boldsymbol{x})$ defined on $\mathcal{M}$ can be expanded by above bases

If $\phi(\boldsymbol{x})$s are complete bases of mapping $\mathcal{M}\rightarrow\mathbb{R}$, residue $r(\boldsymbol{x})$ is zero. The evolution of observable $g(\boldsymbol{x})$ is obtained by applying the Koopman operator

Moreover,

by absorbing the constant $\phi_{i}(\boldsymbol{x}_{0})$ to $a_{i}$ and assuming $\phi_{i}(\boldsymbol{x})$s are complete.

Koopman spectrums for a general system are usually hard to obtain. However, for linear systems, they are compatible with the linear spectrum theory and can be conveniently constructed. The following sections will introduce Koopman spectrums for various linear systems.

For a nonlinear autonomous dynamic system

global Koopman spectrum (independent of $\boldsymbol{x}$) defined in Eqn. (5) describes the global dynamic characteristics. As mentioned before, global spectrums are widely used to determine if an system is ergodic. If all fixed points of the defined Koopman operator $U$ are constant functions (Lasota & Mackey, 2013, see, chap. 4.2), then the system is ergodic. Therefore, for an ergodic system $\lambda=1$ (the fix point) is the spectrum of Koopman operator, and the corresponding eigenfunction is everywhere constant.

However, difficulties persist when applying above global Koopman spectrums to dynamics analysis. Firstly, it is hard to prove such Koopman spectrums exist for a general nonlinear system. Secondly, even if they exist, there lacks an algorithm to compute them. More importantly, some important spectrums characterizing the dynamics are not included in the global Koopman spectrums. For example, in a Hopf bifurcation process, the unstable spectrums (from linear stability analysis) around the unstable equilibrium are not. Otherwise, we could construct a particular initial condition, where only components associated with the growing spectrums are nontrivial. From this particular initial condition, the dynamics exponentially grow without bound, which contradicts to the physical system.

The above difficulties can be solved by defining local Koopman spectrums for nonlinear dynamics.

Here $D(\boldsymbol{x})$ is an open domain covering $\boldsymbol{x}$. $\lambda$ is the local Koopman spectrum and $\phi(\boldsymbol{x})$ is the corresponding eigenfunction. The local spectrum is backward compatible with the global spectrum, since $D(\boldsymbol{x})$ can be the whole manifold $\mathcal{M}$, for instance, LTI systems admit global spectrums.

In many applications, the observable is a vector valued function defined on system state, such that

Koopman decomposition for this observable is obtained by expanding each component of $\boldsymbol{g}$ by the Koopman eigenfunction $\phi_{i}(\boldsymbol{x})$, see (50)

Here $\boldsymbol{v}_{i}$ is the Koopman mode. Each Koopman mode is a collection of component which has the same dynamics, that is, the same growth rate and frequency. They provide rich information of the given dynamic system, especially when the full-state observable $\boldsymbol{x}$ is investigated. In the following discussion, Koopman modes for the full-state observable $\boldsymbol{x}$ are considered.

As mentioned earlier, a remarkable feature of Koopman modes is that they are state-independent for LTI systems, periodic LTV systems. It is also state-independent for the nonlinear autonomous systems under suitable conditions.

Koopman decomposition can be approximated in a data-driven manner (Rowley et al., 2009; Schmid, 2010). In this method a set of data $\{\boldsymbol{x}_{1},\cdots,\boldsymbol{x}_{m+1}\}$ is first collected. The DMD algorithm is then applied to get the Koopman spectrums and modes.

DMD considers the Koopman operator acting on the system, with a finite-dimensional approximation $K$, such that time-resolved state variables collected from the system will satisfy the following realtion

Here $K\in\mathbb{R}^{n\times n}$ is the discretized Koopman operator which has $n\times n$ dimensions.

In a recent paper (Zhang & Wei, 2019), the character dynamic information of operator $K$ is effectively extracted by solving a generalized eigenvalue problem (GEV)

where $Y=\left[\boldsymbol{x}_{2}\,\cdots\,\boldsymbol{x}_{m+1}\right]$ and $X=\left[\boldsymbol{x}_{1}\,\cdots\,\boldsymbol{x}_{m}\right]$. This algorithm has the advantage of circumventing the singularity issue occurred in other DMD algorithms, which is valuable for applications such as stability analysis. The GEV provides the Koopman spectrum $\lambda$ and Koopman mode $X\boldsymbol{v}$ since

DMD algorithm is a data-driven technique and can only capture dynamics contained in the data. For example, if only periodic snapshots are given, the Fourier spectrum will be resolved and no spectrums for the perturbation will be obtained.

DMD algorithm can effectively approximate the spectrums and modes of a system which has constant spectrums. It may fail to capture spectrums for a general LTV system or strong nonlinear systems. In fact, the DMD algorithm provides a time averaged approximation of the local Koopman spectrums. Let us use DMD decomposition and local Koopman decomposition (see Eqn. 52) to approximate the same data

DMD spectrum $\rho_{i}$ and Koopman exponents ($\lambda_{i}(t_{k})=\log{\rho_{ik}}$) is related by

Eqn. (66) used the knowledge that Koopman modes are state invariant.

Generally, DMD numerical algorithm works for the following nonlinear cases.

1. 1.

   The transient spectrum: If the observation interval is much smaller than the characteristic time $T$ of the dynamics system, the DMD algorithm may be useful to compute the transient spectrums and modes.

2. 2.

   The infinite spectrum: If the observation duration is large enough such that $\tau\rightarrow\infty$ in Eqn. (29) is approximately satisfied, $\lambda(\infty,t_{0})$ defined in Eqn. (29) will be approximated. If the system is further UES (or UAS), these infinite spectrums are time-independent and can be computed correctly by the DMD algorithm.

3. 3.

   The limit cycle system: As stated earlier, periodic systems have constant Koopman spectrums which are independent on observation time and observation duration. DMD algorithm will correctly reveal their Koopman modes and spectrums.

In this section, the DMD algorithm is applied to two examples to obtain Koopman decomposition, where the hierarchy structure of the Koopman spectrums and the proliferation rule will be examined. The two examples are associated with the primary and secondary instability of wake past a cylinder.

The flow passing a fixed cylinder is chosen for its simple configuration, geometry and rich dynamics. The Reynolds number $Re=\frac{UD}{\nu}$ influences the instability of wake, where $U$ is the incoming flow velocity, $D$ is the diameter of the cylinder, and $\nu$ is the dynamic viscosity of the fluids. For low Reynolds number, such that $Re<Re_{c1}\approx 6$ (Jackson, 1987), viscosity dominates. The flow is laminar, steady and does not separate from the cylinder. Above $Re_{c1}$, the flow separates from the cylinder surface and rejoins in the wake, creating a recirculating zone with two counter-rotating vortices. The flow is still symmetry, steady and laminar. Experiments show at around $Re_{c2}\approx 50$ (Tritton, 1959; Roshko, 1954), the wake breaks symmetry. The original two steady vortices shed alternatively off the cylinder, resulting the well-known Kármán vortex. Further increasing $Re$, it is observed at some point after $Re_{c3}\approx 190$ (Williamson, 1988) there is a sharp drop in the lift, drag as well as shedding frequency indicating a transition occured. At this Reynolds number, the original two-dimensional Kármán vortex begins to wobble in the spanwise direction, and eventually develops the three-dimensional waves. The above phenomena are summarized in figure 5.

These bifurcation phenomena are attributed to the linear instability mechanism. In the range of $Re_{c2}<Re<Re_{c3}$, the linearized homogeneous Navier-Stokes equation around the steady base flow will provide a pair of unstable complex conjugate normal modes. When $Re$ exceeds the critical $Re_{c3}$, a three-dimensional secondary wave is formed and superimposed on the primary wave (Landahl, 1972). Orszag & Kells (1980) expanded the three-dimensional perturbation around the essential two-dimensional periodic base flow and obtained the Floquet system for the perturbation. Since the well-known instability mechanism and relatively simple flow dynamics, therefore, it is chosen as the study object.

Data for DMD analysis was obtained by numerically solving the governing equation. The incompressible Navier-Stokes equation was solved by the projection method (Brown et al., 2001). A second-order central difference scheme was used for spatial discretization of the viscous term. Adams-Bashford scheme was employed for the convection terms and the Crank-Nicolson scheme was used for time advancement of viscous terms, and the sharp immersed boundary method (IBM) (Mittal et al., 2008) was implemented to handle the solids in the fluid. A non-conformal Cartesian mesh was used for the simulation. The staggered Cartesian grid was used to avoid the pressure-velocity decoupling. The numerical algorithm was the same as the one reported and tested in papers (Yang et al., 2010; Xu et al., 2016).

This paper studied the local Koopman spectral problem for nonlinear dynamic systems. The local Koopman spectral problem for various linear systems is first studied and found to be compatible with the linear spectral theory. For an LTI system, its linear spectrums constitute part of the Koopman spectrums. For an LTV system, the spectrums of the fundamental matrix are a subset of local Koopman spectrums. While for a periodic LTV system, it contains the Floquet spectrums or their augmentations with Fourier frequencies. Correspondingly, eigenvectors of these linear systems are also Koopman modes. For a nonlinear dynamic system, the local spectral problem is defined on the time-parameterized semigroup Koopman operator acting on it.

Several properties of Koopman spectrums are revealed. The recursive proliferation of eigenspaces into infinite dimensionl was found because of the nonlinearity of the systems. The hierarchy structure of Koopman spectrums of a nonlinear system is revealed, that is, the eigenspace is decomposed into the base and perturbation. The continuity of local Koopman spectrums is studied and found to be conditional continuous for nonlinear systems. The continuity property extends the knowledge of local dynamics to the global manifold. As a result, the Koopman modes are found to be state independent.

Primary and secondary instability of flow passing a fixed cylinder are numerically analyzed using the DMD algorithm. The triad-chain or the lattice distribution of spectrums confirmed the proliferation rule and hierarchy structure. The obtained Koopman modes capture the main flow structures.

The authors gratefully appreciate the support from the Army Research Lab (ARL) through Micro Autonomous Systems and Technology (MAST) Collaborative Technology Alliance (CTA) under grant number W911NF-08-2-0004.

The authors report no conflict of interest.