SLOW INVARIANT MANIFOLDS OF
SLOW-FAST DYNAMICAL SYSTEMS

Following the works of Jones Jones (1994) and Kaper Kaper (1999) some fundamental concepts and definitions for systems of ordinary differential equations with two time scales, i.e., for singularly perturbed dynamical systems are briefly recalled. Thus, we consider systems of differential equations of the form:

where $\vec{x}\in\mathbb{R}^{m}$, $\vec{y}\in\mathbb{R}^{p}$, $\varepsilon\in\mathbb{R}^{+}$, and the prime denotes differentiation with respect to the independent variable $t$. The functions $\vec{f}$ and $\vec{g}$ are assumed to be $C^{\infty}$ functions (In certain applications these functions will be supposed to be $C^{r}$, $r\geqslant 1$) of $\vec{x}$, $\vec{y}$ and $\varepsilon$ in $U\times I$, where $U$ is an open subset of $\mathbb{R}^{m}\times\mathbb{R}^{p}$ and $I$ is an open interval containing $\varepsilon=0$.

In the case when $\varepsilon\ll 1$, i.e., is a small positive number, the variable $\vec{x}$ is called fast variable, and $\vec{y}$ is called slow variable. Using Landau’s notation: $O(\varepsilon^{l})$ represents a real polynomial in $\varepsilon$ of $l$ degree, with $l\in\mathbb{Z}$, it is used to consider that generally $\vec{x}$ evolves at an $O\left(1\right)$ rate; while $\vec{y}$ evolves at an $O\left(\varepsilon\right)$ slow rate.

Reformulating the system (1) in terms of the rescaled variable $\tau=\varepsilon t$, we obtain:

The dot $\left(\cdot\right)$ represents the derivative with respect to the new independent variable $\tau$. The independent variables $t$ and $\tau$ are referred to the fast and slow times, respectively, and (1) and (2) are called the fast and slow systems, respectively. These systems are equivalent whenever $\varepsilon\neq 0$, and they are labeled singular perturbation problems when $\varepsilon\ll 1$, i.e., is a small positive parameter. The label “singular” stems in part from the discontinuous limiting behavior in the system (1) as $\varepsilon\to 0$. In such case, the system (1) reduces to an $m$-dimensional system called reduced fast system, with the variable $\vec{y}$ as a constant parameter:

System (2) leads to the following differential-algebraic system called reduced slow system which dimension decreases from $m+p$ to $p$:

By exploiting the decomposition into fast and slow reduced systems (3) and (4), the geometric approach reduced the full singularly perturbed system to separate lower-dimensional regular perturbation problems in the fast and slow regimes, respectively.

Fenichel geometric theory for general systems (1), i.e., a theorem providing conditions under which normally hyperbolic invariant manifolds in system (1) persist when the perturbation is turned on, i.e., when $0<\varepsilon\ll 1$ is briefly recalled in this subsection. This theorem concerns only compact manifolds with boundary.

Earliest geometric approaches to singularly perturbed systems have been developed by Cole Cole (1968), O’Malley O’Malley (1974, 1991), Fenichel Fenichel (1971, 1974, 1977, 1979) for the determination of the slow manifold equation. Generally, Fenichel theory enables to turn the problem for explicitly finding functions $\vec{y}=\vec{Y}\left({\vec{x},\varepsilon}\right)$ whose graphs are locally invariant slow manifolds $M_{\varepsilon}$ of system (1) into regular perturbation problem (O’Malley, 1974, p. 112). Invariance of the manifold $M_{\varepsilon}$ implies that $\vec{Y}\left({\vec{x},\varepsilon}\right)$ satisfies:

According to Guckenheimer et al. (Guckhenheimer & Holmes, 1983, p.131), this (partial) differential equation for $\vec{Y}\left({\vec{x},\varepsilon}\right)$ cannot be solved exactly. So, its solution can be approximated arbitrarily closely as a Taylor series at $\left({\vec{x},\varepsilon}\right)=(\vec{0},0)$. Then, the following perturbation expansion is plugged:

into (5) to solve order by order for $\vec{Y}\left({\vec{x},\varepsilon}\right)$. The Taylor series expansion O’Malley (1991) for $\vec{f}(\vec{x},{\vec{Y}\left({\vec{x},\varepsilon}\right),\varepsilon})$ and $\vec{g}(\vec{x},{\vec{Y}\left({\vec{x},\varepsilon}\right),\varepsilon})$ up to terms of order two in $\varepsilon$ reads:

•    At order $\varepsilon^{0}$, Eq. (5) gives:

which defines $\vec{Y}_{0}\left(\vec{x}\right)$ due to the invertibility of $D_{\vec{y}}\vec{f}$ and the Implicit Function Theorem.

•    The next order $\varepsilon^{1}$ provides:

which yields $\vec{Y}_{1}\left(\vec{x}\right)$ and so forth.

So, regular perturbation theory makes it possible to build an approximation of locally invariant slow manifolds $M_{\varepsilon}$. Thus, in the framework of the Geometric Singular Perturbation Method, three conditions are needed to characterize the slow manifold associated with singularly perturbed system: existence, local invariance and determination. Existence and local invariance of the slow manifold are stated according to Fenichel theorem for compact manifolds with boundary while asymptotic expansions provide its equation up to the order of the expansion.

At the end of the 1980s, Rossetto Rossetto (1986, 1987) proposed the Successive Approximations Method to approximate the slow invariant manifold of singularly perturbed dynamical systems. He considered that the singular approximation, i.e., the zero-order approximation in $\varepsilon$ of the slow invariant manifold, i.e., the critical manifold $M_{0}$ defined by $\vec{f}\left({\vec{x},\vec{y},\varepsilon}\right)=\vec{0}$ is overflowing invariant with respect to (1) when $\varepsilon=0$. Invariance of the manifold $M_{0}$ implies that $\vec{f}\left({\vec{x},\vec{y},\varepsilon}\right)$ satisfies:

According to (1), this leads to:

By application of the Implicit Function Theorem, we have:

implies that:

Left multiplication of (15) by $(D_{\vec{y}}\vec{f})^{-1}$ gives:

Taking into account (16) & (17), we find the following equation:

which is absolutely identical to (5). Thus, identity between Geometric Singular Perturbation Method and Successive Approximations Method is stated.

In 2005, an iterative model reduction method called Zero-Derivative Principle (ZDP) was presented by Gear et al. Gear et al. (2005). Four years later, Zagaris et al. Zagaris et al. (2009) applied the ZDP to singularly perturbed dynamical systems (1) to approximate their slow invariant manifold. They considered that if a function is locally invariant under (1) the zero set of its time derivative provides an even more accurate approximation of the slow invariant manifold (SIM) of (1). In other words, it means that the $k^{th}$ approximation of the slow invariant manifold of (1) is given by the following equation:

According to Benoît et al. Benoît et al. (2015): “estimates [of the SIM] can be obtained in terms of approximations of the slow manifold to order $\varepsilon^{k}$ when looking at the zeros of the $k^{th}$ derivative of the fast components of the original singularly perturbed system.” First of all, let’s notice that such result has been already stated and published by Rossetto Rossetto (1986, 1987) and has not been quoted in any reference. In his works, Rossetto Rossetto (1986, 1987) presented his Successive Approximations Method and stated that:

“The $m^{th}$ order approximation of slow motion is solution of the dynamical system $\vec{f}^{(m)}=\vec{0}$.”

Thus, according to the ZDP, if $\vec{x}^{\prime}=\vec{f}\left({\vec{x},\vec{y},\varepsilon}\right)=\vec{0}$ provides the zero-order approximation in $\varepsilon$ of the slow invariant manifold of (1), $\vec{x}^{\prime\prime}=\vec{0}$ will define its first-order approximation in $\varepsilon$. So, according to (1), we have the following equation:

which reads:

This equation (22) is absolutely identical to Eq. (15). This proves that both Successive Approximations Method and Zero-Derivative Principle are identical. Moreover, since we have already stated above that both Geometric Singular Perturbation Method and Successive Approximations Method are identical, it follows by transitivity that these three methods are absolutely identical and so belong to the first category called: Singular Perturbation-Based Methods.

In 1992, Maas and Pope introduced the Intrinsic Low-Dimensional Manifold (ILDM) method Maas & Pope (1992). Within the framework of application of the Tikhonov’s theorem Tikhonov (1952), this method uses the fact that in the vicinity of the slow manifold the eigenmode associated with the fast eigenvalue is evanescent. According to Mass and Pope Maas & Pope (1992):

“If a subspace in the composition space can be found where the system is “in equilibrium” with respect to its smallest $m$ eigenvectors, this subspace defines a low-dimensional manifold that is characterized by the fact that movements along it are associated with slow time scales and can be used to simplify chemical kinetics.”

Thus, IDLM method provides an approximation of the slow invariant manifold of the slow-fast dynamical system (23) by considering that its velocity vector field is coplanar to the slow eigenvectors of the Jacobian of $\overrightarrow{\Im}(\vec{X})$.

At the end of the nineties, Rossetto et al. Rossetto et al. (1998) presented their Tangent Linear System Approximation Method. The application of this method requires that the slow-fast dynamical system (23) satisfies the following assumptions:

• (H₁)

  The components $f_{i}$, of the velocity vector field $\overrightarrow{\Im}(\vec{X})$ defined in $E$ are continuous, $C^{\infty}$ functions in $E$ and with values included in $\mathbb{R}$.

• (H₂)

  The slow-fast dynamical system (23) satisfies the nonlinear part condition, i.e., that the influence of the nonlinear part of the Taylor series of the velocity vector field $\overrightarrow{\Im}(\vec{X})$ of this system is overshadowed by the fast dynamics of the linear part.

To the slow-fast dynamical system (23) is associated a tangent linear system defined as follows:

where $\delta\vec{X}=\vec{X}-\vec{X}_{0}$, $\vec{X}_{0}=\vec{X}(t_{0})$ and $J(\vec{X}_{0})=\left.D_{\vec{X}}\overrightarrow{\Im}(\vec{X})\right|_{\vec{X}_{0}}$ is the Jacobian matrix. Thus, the solution of the tangent linear system (25) reads:

So,

where $n$ is the dimension of the eigenspace, $a_{i}$ represents coefficients depending explicitly on the co-ordinates of space and implicitly on time and $\vec{Y}_{\lambda_{i}}$ the eigenvectors associated with the Jacobian matrix of the tangent linear system. Thus, according to the TLSA method we have the following proposition:

Hence, the Tangent Linear System Approximation method consists in the projection of the velocity vector field $\vec{V}$ on the eigenbasis associated with the eigenvalues of the functional Jacobian of the tangent linear system. Indeed, by taking account of (25) and (27) we have according to (28):

Thus, Proposition 1 leads to:

The equation (30) constitutes in dimension two (resp. dimension three) a condition called collinearity (resp. coplanarity) condition which provides the analytical equation of the slow manifold of the slow-fast dynamical system (23). An alternative proposed by Rossetto et al. Rossetto et al. (1998) uses the “fast” eigenvector on the left associated with the “fast” eigenvalue of the transposed functional Jacobian of the tangent linear system. In this case the velocity vector field $\vec{V}$ is then orthogonal with the “fast” eigenvector on the left. This constitutes a condition called orthogonality condition which provides the analytical equation of the slow manifold of the slow-fast dynamical system (23) (See Rossetto et al. Rossetto et al. (1998) and Ginoux et al. Ginoux & Rossetto (2006a); Ginoux et al. (2008)).

For a two-dimensional slow-fast dynamical system (23) the projection of the velocity vector field $\vec{V}$ on the eigenbasis reads thus:

where $\alpha$ and $\beta$ represent coefficients depending explicitly on the co-ordinates of space and implicitly on time and where $\vec{Y}_{\lambda_{1}}$ represents the “fast” eigenvector and $\vec{Y}_{\lambda_{2}}$ the “slow” eigenvector. The existence of an evanescent mode in the vicinity of the slow manifold implies according to Tikhonov’s theorem Tikhonov (1952): $\alpha\ll 1$. We deduce:

So, the following collinearity condition between the velocity vector field $\vec{V}$ of a two-dimensional slow-fast dynamical system (23) and the “slow” eigenvector $\vec{Y}_{\lambda_{2}}$ provides an approximation of its slow manifold:

For a three-dimensional slow-fast dynamical system (23) the projection of the velocity vector field $\vec{V}$ on the eigenbasis reads thus:

where $\alpha$, $\beta$ and $\delta$ represent coefficients depending explicitly on the co-ordinates of space and implicitly on time and where $\vec{Y}_{\lambda_{1}}$ represents the “fast” eigenvector and $\vec{Y}_{\lambda_{2}},\vec{Y}_{\lambda_{3}}$ the “slow” eigenvectors. The existence of an evanescent mode in the vicinity of the slow manifold implies according to Tikhonov’s theorem Tikhonov (1952): $\alpha\ll 1$. We deduce:

So, the following coplanarity condition between the velocity vector field $\vec{V}$ of a three-dimensional slow-fast dynamical system (23) and the “slow” eigenvectors $\vec{Y}_{\lambda_{2}}$ and $\vec{Y}_{\lambda_{3}}$ provides an approximation of its slow manifold:

This brief presentation enables to state that both Intrinsic Low-Dimensional Manifold method and Tangent Linear System Approximation method are absolutely identical. Let’s notice that these methods can be also applied to singularly perturbed dynamical systems (1). However, they presented a major drawback since they required the computation of eigenvalues and eigenvectors of the Jacobian which could only be done numerically for dynamical systems of dimension greater than two. Moreover, according to the nature of the “slow” eigenvalues (real or complex conjugated) the plot of their slow manifold analytical equation was difficult even impossible. So, to solve this problem it was necessary to make the slow manifold analytical equation independent of the “slow” eigenvalues. That’s was the aim of the Flow Curvature Method.

Fifteen years ago, Ginoux et al. Ginoux & Rossetto (2006a) published the first article in which the Flow Curvature Method was presented. They explained that by multiplying the slow manifold analytical equation of a two dimensional dynamical system by a “conjugated” equation, that of a three dimensional dynamical system by two “conjugated” equations make the slow manifold analytical equation independent of the “slow” eigenvalues of the tangent linear system.

For a two-dimensional slow-fast dynamical system (23), let’s multiply equation (31) by its “conjugated” equation, i.e., by an equation in which the eigenvalue $\lambda_{2}$ is replaced by the eigenvalue $\lambda_{1}$. Let’s notice that the “conjugated” equation of the equation (31) corresponds to the collinearity condition between the velocity vector field $\vec{V}$ and the fast eigenvector $\vec{Y}_{\lambda_{1}}$. The product of the equation (31) by its “conjugated” equation reads:

It has been proved in Ginoux et al. Ginoux & Rossetto (2006a) that:

where $\vec{\gamma}=\dot{\vec{V}}=\ddot{\vec{X}}$ represents the acceleration vector field, i.e., the time derivative of the velocity vector field (23). But, in the framework of Differential Geometry, curvature of trajectory curve integral of slow-fast dynamical system (23) is defined by:

where $\mathcal{R}$ represents the radius of curvature. Thus, it followed from this result that the slow manifold of a two-dimensional slow-fast dynamical system (23) can be directly approximated from the collinearity condition between the velocity vector field $\vec{V}=\dot{\vec{X}}$ (23) and its acceleration vector field $\vec{\gamma}=\dot{\vec{V}}=\ddot{\vec{X}}$. More than ten years ago, Ginoux et al. Ginoux et al. (2008) proved that this collinearity condition can be written as:

This led Ginoux et al. Ginoux et al. (2008) to the following proposition:

For a three-dimensional slow-fast dynamical system (23), equation (32) must be multiplied by two “conjugated” equations obtained by circular shifts of the eigenvalues. The product of equation (32) by its “conjugated” equation reads:

It has been proved in Ginoux et al. Ginoux & Rossetto (2006a) that:

where $\dot{\vec{\gamma}}$ represents the over-acceleration vector field. But, in the framework of Differential Geometry, torsion of trajectory curve integral of slow-fast dynamical system (23) is defined by:

where $\mathcal{T}$ represents the radius of torsion. Thus, it followed from this result that the slow manifold of a three-dimensional slow-fast dynamical system (23) can be directly approximated from the coplanarity condition between the velocity vector field $\vec{V}=\dot{\vec{X}}$ (23), its acceleration vector field $\vec{\gamma}$ and its over-acceleration $\dot{\vec{\gamma}}$. More than ten years ago, Ginoux et al. Ginoux et al. (2008) proved that this coplanarity condition can be written as:

This led Ginoux et al. Ginoux et al. (2008) to the following proposition:

Then, Ginoux et al. Ginoux et al. (2008) and Ginoux Ginoux (2009) generalized these results to any $n$-dimensional slow-fast dynamical system (23) as well as any $n$-dimensional singularly perturbed dynamical system (1). This led to the following general proposition which encompasses proposition 2 & 3:

Let’s notice that within the framework of Differential Geometry, $n$-dimensional smooth curves, i.e., smooth curves in Euclidean $n-$space are defined by a regular parametric representation in terms of arc length also called natural representation or unit speed parametrization. According to Herman Gluck Gluck (1966) local metrics properties of curvatures may be directly deduced from curves parametrized in terms of time and so natural representation is not necessary. This fundamental result allows applying the Grahm-Schmidt orthogonalization process, with a collection of $i=1,\ldots,n$ vectors $\vec{u}_{i}(t)$ depending explicitly on time (not on arc length) and forming an orthogonal basis, to deduce the expression of curvatures of trajectory curves integral of dynamical systems (1) or (23) in Euclidean $n-$space.

Then, to prove that the slow manifold (41) is locally invariant by the flow of (23), we use Darboux invariance theorem. According to Schlomiuk Schlomiuk (1993) and Llibre et al. Llibre & Medrado (2007) it seems that Gaston Darboux Darboux (1878) has been the first to define the concept of invariant manifold which can be presented as follows:

More than ten years ago, Ginoux Ginoux (2009) established for any $n$-dimensional singularly perturbed dynamical systems (1) the identity between Fenichel’s invariance (5) and Darboux invariance (42) (see also Ginoux Ginoux (2014)).

Thus, it follows from what precedes that both Tangent Linear System Approximation Method and Flow Curvature Method are absolutely identical.

In 1994, Brøns and Bar-Eli Brøns & Bar-Eli (1994) presented their Inflection Line Method for planar or two-dimensional slow-fast dynamical system (23) which can be written as follows:

According to Brøns and Bar-Eli Brøns & Bar-Eli (1994):

“The curvature $\kappa$ of a smooth curve measures the rate of turn of the tangent vector with respect to arc length $s$. The inflection line $l_{i}$ associated with (49) is the locus of points where $\kappa_{t}=0$. The inflection line typically consists of one or more smooth curves. For our purposes, it suffices to consider systems (49) in a region where $f(x,y,\varepsilon)\neq 0$. Then time can be eliminated to give the equivalent equation for the trajectories,

$$\dfrac{dy}{dx}=\dfrac{g\left(x,y,\varepsilon\right)}{f\left(x,y,\varepsilon\right)}.$$

(50)

For curves in the ($x,y$)-plane of the form $y=y(x)$, the curvature can be computed as

$$\kappa=\dfrac{y^{\prime\prime}(x)}{\left(1+y^{\prime}(x)\right)^{3/2}},$$

(51)

where the positive root in the denominator is used. Thus, the inflection line for (50) is determined by $y^{\prime\prime}(x)=0$, and an equation for it can be found by differentiating (50).”

For such two-dimensional dynamical systems (49), the issue of the parametrization highlighted by Gluck Gluck (1966) just above can be easily solved by dividing the two components of the velocity vector field (49) as done by Brøns and Bar-Eli Brøns & Bar-Eli (1994). Nevertheless, such simplification is no more possible in dimension higher than two. Let’s notice that for three-dimensional dynamical systems the use of curvature instead of torsion has already been carried out by Ginoux et al. Ginoux & Rossetto (2006b) and by Gilmore et al. Gilmore et al. (2010). However, the use of curvature or more precisely, the first curvature require to solve a system of two nonlinear equations since in dimension three a curve is defined as the intersection of two surfaces. Nowadays, this kind of problem cannot be solved analytically in the general case. Thus, such considerations precludes any generalization of the Inflection Line Method proposed by Brøns and Bar-Eli Brøns & Bar-Eli (1994) at least from an analytical point of view.

Now let’s prove that the Inflection Line Method is just a particular case of the more general Flow Curvature Method. According to Brøns and Bar-Eli Brøns & Bar-Eli (1994) “the inflection line for (50) is determined by $y^{\prime\prime}(x)=0$, and an equation for it can be found by differentiating (50).” So, let’s differentiate equation (50):

So, the condition $y^{\prime\prime}(x)=0$ given by Eq. (52) is absolutely identical to the condition $\det(\dot{\vec{X}},\ddot{\vec{X}})=0$ where $\vec{X}=(x,y)^{t}$ provided by (36). As a consequence, it is thus proved that the Inflection Line Method is just a particular case of the more general Flow Curvature Method.

To resume, we have proved in this section that the Flow Curvature Method encompasses the Intrinsic Low-Dimensional Manifold Method, the Tangent Linear System Approximation Method and the Inflection Line Method which all belong to the second category called Curvature-Based Methods.

Now, a last issue arises as follows: for $n$-dimensional singularly perturbed dynamical systems (1), does the Flow Curvature Method provide the same approximation of their slow invariant manifold as the Geometric Singular Perturbation Method?