## 13 Conclusions  
The primary results of this work are summarized in the following points:  
1. Modes in system I-O responses accurately capture essential system dynamics for each I-O pair of a mechanistic (multicompartmental, linear or nonlinear) model, defining _minimal_ system dimensionality (size and complexity) based on that I-O pair. In other words, the granularity of the model (or extent of its multicompartmentalization) is experiment-dependent, and the minimum dimension--the smallest number of possible compartments--is equal to the number of its distinct _modes_ visible in its output response--for any particular specification of input and output (I-O) locations.
2. More extensively, modal analysis methodology can establish more than model dimensionality from particular datasets. It also can help in designing new experiments to extract \"hidden\" structures and select among multiple _candidate_ models (model distinguishability analysis) or simplify (reduce) models to their dynamical essentials.
3. In particular, the testing of all I-O pairs of a hypothesized candidate multicompartmental structure can extract hidden modes and thus provide complete structure.
4. Finding modes (and thus compartments) visible in output data signals by _graphical inspection_ speeds up the model-building process and visually exposes new experiment designs for discovering compartment connectivity.
5. By analogy, finding the modes in dynamic system data is akin to finding the _principal components_ (principal component analysis--PCA), or _singular values_ (singular-value decomposition--SVD), of a large dataset (e.g., bioinformatics data matrix), thereby simplifying it to the minimum dimension (size and complexity) for analysis.
6. For linear systems, diagonalized modal solutions (linearized normal modes LNMs) have an _invariance_ property, i.e., a motion initiated on one specific LNM--representable as a _two-dimensional planar surface in phase space_--has no effect on the remaining LNMs because they are decoupled. This property permits the modal concept to be readily extended to NL systems. A nonlinear normal mode (NNM) is a _two-dimensional invariant (nonplanar) manifold surface in phase space_, tangent to the planar surfaces of their _linearized_ ODE LNMs at the equilibrium point of their NL ODE system.
7. NL modal analysis is thus about generating _minimal reduced-order models_ that accurately capture the dynamics of higher-order models in mechanics, with the help of NNMs.
8. For biological systems, the analogous application is reducing complex biosystem models using _conservation_ relations, combined with _dynamical approximations_ (quasi-steady-state approximations, QSSA) common in biochemical reaction modeling. This means NNM analysis is equivalent to QSSA analysis, with invariant manifolds of the biosystem in the substrate-product phase space of each reaction submodel.
9. Compartments are state variables, not spaces or containers of state variables.
10. Linear model simplification also can be helped by examining the poles and zeros of I-O transfer functions for cancellations.
11. Prospectively, this work introduces and explains the analogies between LNM and NNM analyses in mechanics and the QSSA in a broad class of regulated systems in biochemistry and systems biology. Hopefully, it will serve to promote the study of other NL system classes, like those that exhibit more complex behaviors.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** Not applicable.  
**Conflicts of Interest:** The author declares no conflict of interest.