## 10 Nonlinear Modes in Nonlinear Models  
Nonlinear (NL) ODE models also have \"transfer functions\" (TFs), i.e., input-output solution response functions. Unlike linear models, however, their dynamic responses are dependent on inputs--input shapes or magnitudes. So we cannot write them only in terms of model parameters, like we do for linear model TFs. Nevertheless, the principle result above carries over to NL models. \"Modes\" in NL systems can cancel each other (like poles and zeros)--at least approximately--or be vanishingly small in magnitude, or present for vanishingly small transient times--thereby rendering complex structures less visible in data characterized by specific inputs and outputs. Similarly simplified NL models, also with input-dependent dynamics, are characteristic of enzyme kinetic models. We tie the linear mode concept together here with a novel view of _quasi-steady state approximations_ (QSSA) in nonlinear biochemical kinetics models (described in Section 11)--viewed as _nonlinear modes_ with the _same kinds of geometric properties_ as linear modes. The goal is to better understand NL systems and facilitate NL model building and analysis. The field of mechanics has a developed theory of nonlinear modes associated with certain oscillatory phenomena in mechanical systems. Biological systems share many of the same dynamical properties, not recognized as such in the field. We make the connection precise here. As noted earlier, linear modes are a _minimal set of \(n\)-dimensional invariants_ of the dynamical system and, for any set of ICs or input(s), the modes accurately capture essential system dynamics. Among other things, this means linear ODEs of an order higher than \(n\) can be accurately reduced (simplified) to order \(n\), and this simplified model can be represented by a set of \(n\)_two-dimensional modes_ in a state vector space. Mode shape is represented by eigenvector component \(v_{j}\), associated with a frequency (eigenvalue) component \(\lambda_{j}\). The intrinsic parameters of the measurement model \(\mathbf{y}(t)=C\mathbf{q}(t)=C\mathbf{e}^{Kt}\mathbf{q}(0)\) for the homogeneous linear system are then the eigenvalues and eigenvectors of \(K\), a set of \(2n\) structural invariants of \(K\). Importantly, \(\lambda_{j}\) can be complex as well as real. In mechanics, modal analysis is typically concerned with vibrations and resonances in mechanical structures [23]. When their models are linear or NL and linearized about their equilibrium solutions--a common practice in analyzing stability properties--some \(\lambda_{j}\) are complex. This means there exist oscillatory components in their dynamical behavior. In this context, linear modes are called **linear normal modes** ([[https://en.wikipedia.org/wiki/Normal_mode](https://en.wikipedia.org/wiki/Normal_mode)]([https://en.wikipedia.org/wiki/Normal_mode](https://en.wikipedia.org/wiki/Normal_mode)), accessed on 13 July 2023), or LNMs, and the governing equations of motion can be decoupled into LNMs, e.g., by diagonalization of their system matrix. In decoupled form, i.e., with a basis chosen to decouple the state variables, a linear system vibrates as if it consisted of independent oscillators governed by the (complex) modes, a property not shared with NL systems. Linear modal solutions also obey the superposition principle, i.e., free and forced oscillations are conveniently expressed as linear combinations of individual LNM motions in phase space. Importantly, modal solutions 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. By analogy, for NL models of the form \(dx/dt=f(x,u)\), a **nonlinear normal mode** (NNM) is defined similarly as a two-dimensional invariant manifold in phase space.