LOCAL ORTHOGONAL RECTIFICATION: DERIVING NATURAL COORDINATES TO STUDY FLOWS RELATIVE TO MANIFOLDS

. We recently derived a method, local orthogonal rectiﬁcation (LOR), that provides a natural and useful geometric frame for analyzing dynamics rel- ative to a base curve in the phase plane for two-dimensional systems of ODEs (Letson and Rubin, SIAM J. Appl. Dyn. Syst. , 2018). This work extends LOR to apply to any embedded base manifold in a system of ODEs of arbitrary dimension and establishes a corresponding system of LOR equations for analyzing dynamics within the LOR frame, which maps naturally back to the original phase space. The LOR equations encode geometric properties of the underlying ﬂow and remain valid, in general, beyond a local neighborhood of the embedded manifold. In addition to developing a general theory for LOR that makes use of a given normal frame, we show how to construct a normal frame that conveniently simpliﬁes the computations involved in LOR. Finally, we illustrate the utility of LOR by showing that a blow-up transformation on the LOR equations provides a useful decomposition for studying trajectories’ behavior relative to the embedded base manifold and by using LOR to identify canard behavior near a fold of a critical manifold in a two-timescale system.

1. Introduction and motivation. Often the dynamics of an ordinary differential equation (ODE) defies rectangular coordinate schemes; that is, the geometry induced by a flow may be difficult to represent in Cartesian coordinates. In fact, a common early step in analysis is to exchange Cartesian coordinates for a geometry better suited for the problem [11,2,10,15,16,12,17]. In this work, we present a technique that allows us to use any embedded manifold (not necessarily invariant) to generate a natural coordinate frame for a dynamical system. We call this technique Local Orthogonal Rectification (LOR). We previously derived the planar version of LOR and illustrated its utility in a variety of scenarios; for example, we used the LOR framework to precisely define transiently attracting or repellling rivers in the two-dimensional phase plane, to develop a convenient method to locate rivers, and to implement this method to study transient dynamics in some example systems [9]. The focus of the present work is on providing a rigorous, much more general derivation for the LOR approach and its properties that applies in arbitrary dimensions and on illustrating its use in the non-planar setting.
Consider an ODE and initial conditioṅ where f ∈ C r (Ω, R n ) for n, r ≥ 1 and Ω is an open subset of R n , which induces a flow Φ : Ω × R → Ω. For simplicity, we introduce the notation ∂ i = ∂/∂η i . Suppose that M is a codimension-k C s -regular manifold embedded in Ω. Specifically, suppose there exist an indexing set A and an atlas of charts {(U α , σ α )} α∈A where M = ∪ α∈A σ α (U α ), such that for all α, β ∈ A: 1. each U α ⊆ R n−k is open with corresponding σ α ∈ C s (U α , Ω) a homeomorphism on its image; 2. if σ α (U α ) ∩ σ β (U β ) = ∅ then the map κ α,β : 4. M can be equipped with a local normal frame; that is, there are mappings N j σ α ∈ C 1 (U α , R n ) such that where ·, · denotes the standard Euclidean inner product, j 1 , j 2 ∈ {1, . . . , k}, and δ j1,j2 is the Kronecker delta. We call these four conditions the LOR assumptions. These assumptions guarantee that the tangent space to M at any point p ∈ M, denoted by T p M, is an n − k dimensional space.
To simplify indexing, we will use the convention that any index related to tangential objects will be denoted by i, or when necessary by i 1 , i 2 , . . ., and any index related to normal objects will be denoted by j or by j 1 , j 2 , . . .. Stated simply, {N j σ α (η)} k j=1 forms an orthonormal basis of (T σα(η) M) ⊥ on U α . The existence of such a basis is straightforward to establish locally, and by refining our domains U α we can guarantee that such mappings exist.
With our notation and assumptions in hand, we now motivate the underlying geometric idea for constructing the LOR frame. Suppose that we are interested in studying the dynamics near a point x 0 ∈ Ω that lies close to our embedded manifold. Furthermore, suppose that x 0 can be written in the form To write equation (2), we have assumed that x 0 can be decomposed into a point on M, namely σ α (η 0 ), and a vector in the orthogonal complement of T σα(η0) M, namely k j=1 ξ 0,j N j σ α (η 0 ). We will establish that such a decomposition is generic, sufficiently close to M. We define Ψ α : U α × R k → R n by Ψ(η, (ξ 1 , · · · , ξ k )) = σ α (η) + k j=1 ξ j N j σ α (η) and denote ξ = (ξ 1 , · · · , ξ k ), so that (2) can be more succinctly expressed as x 0 = Ψ(η 0 , ξ 0 ). Now, denote by φ(t) the trajectory of (1) such that φ(0) = x 0 . We want to continue tracking φ(t) in our decomposition. To do so, we seek smooth η : (−δ, δ) → U α , ξ : (−δ, δ) → R k such that φ(t) = Ψ(η(t), ξ(t)) for t ∈ (−δ, δ). We will establish that this continuation can be achieved, providing a convenient new set of coordinates (η, ξ), which we will call the LOR frame. We will also derive a system of ODEs that govern the evolution of η(t), ξ(t), the LOR equations, by using the ODE satisfied by φ(t). Figure 1. The geometric setup for Local Orthogonal Rectification. We consider an inital condition x 0 near a given manifold, and decompose the trajectory through x 0 , denoted by φ, into a curve on the manifold and a curve in the normal bundle to the manifold.
We call (η, ξ) the LOR coordinates for the base manifold M. By choosing M to be dynamically relevant (i.e., a structure that can be observed to play a role in organizing the flow), we will be able to study the dynamics near M using the corresponding LOR frame. The geometric nature of the LOR frame can offer striking insights into the local behavior of the flow, representing a powerful approach for the theoretical study of dynamical systems.
The remainder of the paper is organized as follows. In Section 2, we state our main result, which stipulates the existence of the LOR frame and the form of the LOR equations. Next, in Section 3, we set up the notation and definitions needed to prove this result, while the proof itself appears in Section 4. In Section 5, we consider LOR in a variety of settings. We treat the special cases where M is codimension-1 or codimension-(n − 1), present an algorithm for constructing a particularly useful normal frame assuming that M is equipped with a Frenet-type atlas, and introduce a blow-up approach to studying how trajectories in Ω evolve relative to M. We conclude in Section 6 with a computational example to explicitly illustrate the implementation of LOR, in this case to easily identify trajectories organizing canard behavior near a fold of a critical manifold in a two-timescale system.

Statement of the main result.
In what follows, we will pick a specific chart of M and drop our indexing subscript; that is, we will study the LOR frame on the chart (U, σ) of M. Once our theory is established locally, we will prove that transfer across charts is well-defined, a result that is critical to the utility of LOR.
To state our main result, we define T f : . . .
Note that T f (η, ξ) projects the vector field at Ψ(η, ξ) into the tangent space of M and N f (η, ξ) projects the vector field at Ψ(η, ξ) into the normal space of M. With this additional notation, we can state our main result.
The following section will supply the explicit forms of S σ (η, ξ) and K σ (η, ξ) and will establish some key results that underlie the LOR Equations Lemma. We call S σ (η, ξ) and K σ (η, ξ) the tangent and normal exchange operator, respectively, for reasons that will become apparent in subsection 3.2 and in Section 4, where we give the proof of the LOR Equations Lemma.
3. Constructing the LOR frame. To start the construction, we will present a set of tools for the local analysis of embedded manifolds and some of their properties, establish our notation and review relevant concepts for readers. Subsequently, in subsection 3.3, we establish an important tracking result.
3.1. First and second fundamental forms. When studying surfaces embedded in R 3 , the first and second fundamental forms, often denoted I, II respectively, play a crucial role in representing local properties or shape. We will review the generalization of these concepts to an (n − k)-manifold embedded in R n . Remark 1. There is a substantial body of work regarding the differential geometry of embedded manifolds including generalization of the concept of a second fundamental form [7]. While there may be a more elegant presentation of the following material, we set up just what we need for the problem at hand. Definition 3.1. Given a chart (U, σ) of the C r codimension-k manifold M with r ≥ 2 and a C s normal frame {N j σ(η)} k j=1 with s ≥ 1, define the mappings I, II j : U → R n−k×n−k entrywise by where i 1 , i 2 ∈ {1, · · · , n − k}, j ∈ {1, · · · , k}. We call I(η), {II j (η)} k j=1 the first and second fundamental forms of σ, respectively.
In the instance where n = 3 and k = 1, such that M is a surface embedded in R 3 , our representations of the first and second fundamental forms reduce to the standard definition. For any manifold embedded in R n , the arclength of a curve along the manifold is defined using the first fundamental form; given a C 1 curve where the final equality follows from transposition. Therefore, the arclength functional may be expressed as Note that the second fundamental forms depend on the choice of normal frame and hence are not (generically) intrinsic features of M. However, I(η) is preserved under diffeomorphic reparameterization. The following result presents the properties of these matrices that are relevant for our analysis. and Proof. For notational convenience we will suppress η-dependence. From the symmetry of the inner product, we note that I is self-adjoint. Furthermore, the regularity of M (LOR assumption 3) implies that {∂ i σ} n−k i=1 forms a linearly independent set. Thus, I is a Gram matrix and hence is positive definite. Now consider Equation (6) implies that (II j ) i1,j2 = − ∂ i1 ∂ i2 σ, N j and, as second derivatives commute, II j is self-adjoint. The equalities in (5) follow immediately from matrix multiplication.
3.2. The tangent and normal exchange operators. We will use the first and second forms to define a linear operator that is fundamental to the LOR frame. We call this operator the tangent exchange operator because it takes tangent vectors in T σ(η) U and exchanges them for other tangent vectors in T σ(η) M, as will become clear later, in Section 4. Definition 3.3. Given a chart (U, σ) of the C r codimension-k manifold M with r ≥ 2 and a C s normal frame {N j σ(η)} k j=1 with s ≥ 1, define the mapping S σ : As noted in Section 2, we call S σ (η, ξ) the tangent exchange operator of σ.
For our analysis, we will require the following result, which shows that given an embedding into R n , S σ is invariant under change of normal frame (i.e., S σ is extrinsically invariant).
To conclude this subsection, we present a definition for another linear map, which has no obvious analogue in the theory of surfaces.
3.3. LOR tracking lemma. The following lemma provides the backbone of our main result.
The non-constructive wording "for ξ 0 sufficiently small in norm" in the LOR Tracking Lemma can actually be made exact.
Definition 3.6. Given a chart (U, σ) of the C r codimension-k manifold M with r ≥ 2 and a C s normal frame {N j σ(η)} k j=1 with s ≥ 1, for each η ∈ U, let P η be the maximal, by inclusion, path-connected subset of R k containing the origin such that det S σ (η, ξ) > 0 ∀ξ ∈ P η .
Define the maximal parameter domain P by The definition of the maximal parameter domain provides the following rewording of the LOR Tracking Lemma.
Reversing the proof gives the opposite inclusion. Note that Ψ is defined on an approriate domain of (η, ξ) independent of LOR tracking, so Ψ(P) is uniquely defined as desired, and the claim regarding Ψ(P 0 ) follows analogously.

4.
Proof of the main result. With our definitions and preliminary results laid out, we can prove our main result.
Proof of the LOR Equations Lemma. Suppose that x 0 ∈ Ω can be expressed as Suppressing time dependence, we computė Note , as S σ is invertible near (η 0 , ξ 0 ). The equation governingξ follows analogously by left acting on equation (14) by the expression given in (12). (15) highlights the exchange of tangent vectors via the action of S σ that motivates our choice to name this the exchange operator. Note that the standard shape operator from differential geometry also exchanges tangent vectors, but these operators are not identical.

Remark 2. Equation
We have once again used non-constructive language to constrain ξ 0 . We can strengthen this language by using P and "reverse" the statement of the main result.
Given a vector field f and a chart (U, σ) of a smooth, regular, codimension-k manifold, we denote the LOR vector field by Interestingly, the operator L σ is linear; that is, as f, g only appear in T f, T g, N f, N g, which are vectors of inner products. Therefore the LOR dynamics respects additive perturbations. We hope to take advantage of this fact in future work. It is natural to wonder whether the LOR flow is topologically conjugate to the flow induced by (1). It can be computationally difficult to find a codomain on which Ψ is a homeomorphism. However, the flow induced by (1) is a submersion of the LOR flow via Ψ.
This is simply the definition of topological conjugacy of flows with the injectivity of the homeomorphism relaxed.

Extensions and applications.
In this section, we highlight several scenarios where the computations involving LOR simplify in useful ways. We also discuss certain auxiliary ideas, namely a measure of near-invariance for a surface in a flow that can be defined based on LOR and a blow-up transformation that can be applied to the LOR equations, that enhance the utility of LOR. The near-invariance measure plays a useful role in Section 6.

Hypersurfaces and Frenet curves.
In the cases where k = 1 and k = n−1, which correspond to M being a curve and a hypersurface, respectively, there is a canonical choice of normal frame that allows us to simplify the LOR dynamics. First, suppose that k = n − 1 and M is a codimension-1 manifold. In this case, T p M is a n − 1 dimensional space, with a one-dimensional orthogonal complement in R n . Therefore there are exactly 2 unit vectors that could serve as a normal frame. We choose the normal vector in accordance with the right hand rule; specifically, we define where x 1 ∧ · · · ∧ x n−1 is the outer product of n − 1 vectors, which is a generalization of the cross product. We easily attain the following result.
Using this particular normal frame allows us to greatly simplify the LOR equations; we find that in this frame, ). With significantly more index wrangling, we find that We call system (18) the Frenet LOR equations. This example illustrates that when σ is a Frenet curve, K σ reduces to the matrix of curvatures. If we take σ(η) to be a trajectory of (1), then N f (η, ξ) = 0 andη| ξ=0 = 1.
By dividing through byη to eliminate time from system (18), we find This set of equations (19) is well-suited to study transition maps: suppose we choose σ to be a dynamically relevant trajectory, and we want to study how the normal space of σ at η = 0 is mapped to the normal space at η = 1. Usually, one has to deal with approximating the time it takes for trajectories to travel from one section Σ 1 to another section Σ 2 , defined to represent this mapping. In (19), however, we have eliminated time dependence; thus, we can simply integrate trajectories from η = 0 to η = 1. Therefore it is very natural to represent Poincaré maps in the LOR frame.

5.2.
A measure for near-invariance. In this section, we present a natural generalization of a technique used for curves in planar systems in [9]; namely, we construct a quantitative measure that quantifies how close a surface is to being invariant under a flow. We will focus on the simplest case k = 1, when M is a hypersurface embedded in Ω. Recall from the preceding section that the ξ dynamics in the LOR frame for a codimension-one chart is given byξ = N f (η, ξ). Note that σ(U) is locally invariant if and only ifξ| ξ=0 = 0; that is, N f (η, 0) = 0 for all η ∈ U. Geometrically, N f (η, 0) = 0 if and only if f • σ(η) ∈ T σ(η) M; thus, the normal vector N σ(η) is orthogonal to our vector field along σ. Heuristically, we note that σ(U) is closer to being invariant if |N f (η, 0)| is small, as LOR trajectories cannot rapidly escape U × {0}.
There are several natural candidates for constructing a measure of this "nearinvariance" property. We choose one that we find particularly informative.
Definition 5.1. Given a chart (U, σ) of the C r codimension-1 manifold M with r ≥ 2 and a C s normal vector N σ(η) with s ≥ 1, we define µ σ : U → R + by with the convention that inf ∅ = −∞. We call µ σ the near-invariance measure of σ. Furthermore, define Ξ σ : U → R by We call Ξ σ the correction to σ.
Generically,σ is a regular chart itself; that is, it parametrizes a codimension-1 manifoldM, in which case we callσ the corrected chart of σ. Interestingly, the corrected chart is often more nearly-invariant than the original chart, with µσ(η) < µ σ (η) for η ∈Û. We make use of this property in our computational example in Section 6 to identify canard behavior in the normal form system for a folded-saddle node bifurcation [6,14].

5.3.
Constructing a normal frame to a manifoldà la Frenet. In the theory presented thus far, we have simply assumed that the user can provide a normal frame. In this section, we present a rigorous algorithm for constructing these frames.
In the construction of the Frenet frame for the k = 1 case, we used higher derivatives of the curve to "fill-up" T p M ⊥ and then applied the Gram-Schmidt process to orthonormalize these derivatives and thus construct our moving frame. Here, we apply the same ideas, albeit with more notation, to generalize this construction to higher codimensions.
Clearly we can perform the Gram-Schmidt process on {∂ αj σ} to construct an orthonormal basis. The first m vectors will be linear combinations of {∂ ej σ} and hence form an orthonormal basis for T σ(η) M ; thus, we denote them T i σ(η), i = 1, . . . , m. The final n − m orthonormal vectors produced will be orthogonal to T σ(η) M and can be denoted N j σ(η) without any abuse of notation; they form a normal frame per LOR assumption 4. Therefore, we can produce normal frames to manifolds in an algorithmic way, provided that we make a fairly generic assumption about the manifolds' higher derivatives.
One advantage of the Frenet frame in the preceding section is that it allows us to easily compute II i , K σ using inherent properties of the LOR base curve. We shall next establish that we can compute II i , K σ from the properties of our generalized Frenet frame as well.
The Gram-Schmidt process, being linear, can be represented as a matrix multiplication. That is, there is an n × n matrix G : U → R n×n such that . . .
and G is C 1 , everywhere invertible, and lower triangular. Note that G is of the form where I is the m × m identity matrix, then by definition To finish our generalization of the Frenet frame, we need to understand how the derivatives of T i σ, N j σ can be expressed as linear combinations of T i σ, N j σ, as in (17). To this end, we define the following. Definition 5.3. Suppose that (U, σ) is a Frenet chart with index set {α j } n j=1 . Then {∂ αj σ(η)} n j=1 forms a basis for R n and therefore there are matrices A i : U → R n×n such that . . .
We call these matrices the algebraic closures of the index set.
Remark 3. At first glance, the entries of A i (η) appear to be arbitrary and thus could be difficult to compute. However, by filling out our normal frame with higher derivatives of σ, we can greatly simplify A i . Note that ∂ ei ∂ αs σ(η) = ∂ αs+ei σ(η), and therefore if α s + e i ∈ {α j } n j=1 , then the s th row of A i (η) is a canonical basis vector; that is, if α s + e i = α j then A i (η) s = e j , hence A i can be fairly sparse. In fact, the remarkable simplicity of the Frenet-Serrat equation is caused by the sparseness of the relevant algebraic closure.
Proof. Taking derivatives of both sides of (23), using the definition of the algebraic closures, and applying (23) yields the result.
Proposition 2 allows us to compute II i , K σ from first principles, after we define the following operators.
These operators have no clear analogue in the theory of surfaces; however, they bear some resemblance to the Christoffel symbols. The third fundamental forms are useful insofar as they determine K σ .
Lemma 5.5. Given a chart (U, σ) of the C r codimension-k manifold M with r ≥ 2 and a C s normal frame {N j σ(η)} k j=1 with s ≥ 1, Now we can derive formulae for II j , III j using properties of our Frenet-type frame. Let Proof. From the definition of II i , III i and Lemma 3.2 we can compute where the blank m × m matrix block is irrelevant, and v(η) is defined in (25). Comparing this expression to (24) and using the definitions (21), (22) yields the result, after some simple block matrix computation.

Blowing up an invariant manifold to study local transient dynamics.
Suppose that (U, σ) is a chart of a locally invariant codimension-k manifold M that satisfies our LOR assumptions. We will study the LOR flow generated by (3) to determine the dynamics near σ(U) and will introduce what we term LOR blow-up coordinates to serve in this process. First we let ξ = rξ where ξ = 1 such that r 2 = ξ, ξ . Intuitively, r represents the Hausdorff distance of Ψ(η, ξ) from the manifold M andξ represents the "angle" of ξ relative to the manifold.
It is straightforward, using the fact that ξ ,ξ = 1 and hence ξ ,ξ = 0, to compute that for r > 0,ṙ = ξ ,ξ , System (27) becomes potentially problematic in the r → 0 + limit, which is precisely what we are interested in, in order to study dynamics local to M. Note that if lim r→0 +ξ/r is well-defined then we can continuously extend the dynamics of (27) to r = 0. In our new coordinate scheme, the LOR equations (3) give uṡ By definition, K σ (η, rξ) = rK σ (η,ξ); hence, as r → 0 + , the right hand side of (28) limits to 0. Therefore, we can compute the L'Hôpital type limit so that from (28), (29), and ξ = rξ, we havė Note that in the above representation, at leading order, theη equation depends only upon η and theξ equation depends only on η,ξ. Therefore, we have decoupled the dynamics along the manifold (the η dynamics) from the angular dynamics relative to the manifold (theξ dynamics) from the contraction/expansion dynamics (the r dynamics) for r sufficiently small. This non-trivial decoupling is possible because we have chosen to represent our dynamics in their natural frame.
We call the (η,ξ, r) coordinates the LOR blow-up coordinates, as the corresponding phase space has the form U × S k−1 × [0, ε), where S k−1 = {x ∈ R k | x = 1} is the k-sphere and 0 < ε 1. This coordinate representation is effectively equivalent to performing geometric desingularization on the entirety of M.
Hence the LOR equations are also fast-slow. Transforming to LOR blow-up coordinates and expanding in ε yields, from equations (30) and (29), where B(η) = D ξ N f (η, 0; 0). Note that since Sσ, T f, N f are being evaluated on ε = 0, we use the existence ofσ only to justify our derivation; the equations in (34) that we have obtained depend only on the critical manifold M 0 . Thus, system (34) gives a natural framework to study dynamics relative to a slow manifold M ε that relates the dynamics relative to M ε to the dynamics relative to M 0 . 6. A computational example. In this final section, we present a long-form computational example of the LOR frame in action. We choose a well-known, fast-slow system for our analysis, the normal form for a folded saddle-node canard point [13], which is given by  ẋ ẏ z for parameter µ > 0 as well as 0 < ε 1. In the ε → 0 + limit, the critical manifold M 0 , parameterized by σ(η 1 , η 2 ) = (−η 2 2 , η 1 , η 2 ), becomes a surface of critical points. Let U L = {(η 1 , η 2 )|η 2 < 0}, U R = {(η 1 , η 2 )|η 2 > 0}, U F = {(η 1 , η 2 )|η 2 = 0}, then σ(U L ) is a sheet of normally hyperbolic, stable fixed points in the ε = 0 system, while σ(U R ) is a sheet of normally hyperbolic, unstable fixed points. The set σ(U F ), called the fold, is a line of nilpotent fixed points. Fenichel theory guarantees, for ε > 0 but sufficiently small, the existence of charts σ L,ε , σ R,ε such that σ i,ε (U i ) is a locally-invariant manifold, σ(η) − σ i,ε (η) = O(ε) for η ∈ U i , and σ i,ε (U i ) has the same stability properties as σ(U i ) for i ∈ {L, R}.
The most intriguing feature of (35) is the existence of trajectories beginning in σ L,ε (U L ) that cross into σ R,ε (U R ); these trajectories are referred to as canards and play an organizing role in various forms of fast-slow dynamics including mixed-mode oscillations.
The standard technique for identifying canards is to perform geometric desingularization in a neighborhood of the origin of system (35) and track carefully chosen trajectories across the multiple "charts" of the desingularized variables. Here, we present an approach based on LOR techniques.
Two features of this result bear mentioning. First, we have demonstrated that any trajectories that escape from a neighborhood of the stable critical manifold must do so through the fold. Second, recalling that Ξ − (η) = O(ε), we have verified a weak version Fenichel's Theorem for system (35). The organizational role of the correction is demonstated in the left half of Fig. 2.
By performing an additional transformation of our system, we can extract canardtype solutions, which pass across the fold {η 2 = 0}, in a very natural way. We have demonstrated that the correction manifold plays a strong role in organizing the flow near the critical manifold, so it is natural to use the correction as a base manifold for another LOR transformation. That is, we definê which is a manifold embedded in our original space that satisfies all of our LOR assumptions, such that we can compute the LOR dynamics relative toσ. We suppress the computations for brevity; however, they follow exactly along the lines of the previous computations, and can be executed extremely quickly by symbolic computation software (e.g., Mathematica, Maple, Julia; see https://github.com/ LORlab2020/Mathematica for Mathematica code that can be used for LOR). Note that in this transformation,η is geometrically equivalent to η; that is,η 1 still corresponds to x in the original system, just asη 2 corresponds to y, but we need theη notation to represent the change from ξ toξ.
The lowest order terms of the resultant system are given bẏ Note that {ξ = 0} is invariant to O(ε 2 ), which is an improvement to the previous system; trajectories starting on {ξ = 0} must spend an O(ε −2 ) amount of time near {ξ = 0} before escaping. We remain interested in trajectories beginning in {η 2 < 0} that cross the fold curve. If we numerically plot theη dynamics on {ξ = 0}, we notice that there is a single trajectory that plays an outsized role in organizing theη phase plane.
It is easily verified that γ(t) = (εt, εµt/2, 0) solves (40) to O(ε 2 ), and this is, in fact, the "central" trajectory of the approximate dynamics; see Fig. 3. We also note that γ is the only trajectory that remains well-defined as it crosses the fold, through a L'Hopital type limit. Finally, we note that γ(t) is constrained to lie in a η 2η 1 x z y γ(t) Figure 3. (Left) The dynamics on {ξ = 0} of system (40) for µ = 3. Note how the approximate trajectories organize around the orange curve, called γ in the text. (Right) A plot of the rivers of (35), three of which cross the fold of the critical manifold. The orange curve, γ(z), is indistinguishable from the identified approximate canard solution Ψ 2 (γ(t), 0). plane, as its trace inη is a line; stated equivalently, γ(t) has everywhere vanishing curvature.
In earlier work [9], we suggest that rivers in planar systems, which are centrally organizing orbits with no apparent source, are trajectories for which curvature and torsion vanish simultaneouly. Previous work has established the zero curvature set plays an organizational role in planar systems [1,4,5], we use the LOR frame to study where the zero curvature set is invariant. One advantage of this definition, is that it generalizes easily to arbirary dimension. In R n we define a river to be a Frenet trajectory Γ(t) whose Frenet curvature κ n−2 (t) vanishes to second order; that is, if there exists t * such that κ n−2 (t * ) =κ n−2 (t * ) = 0 then Γ(t) is a river, and we call Γ(t * ) a confluence [9].
Our analysis suggests that γ(t) is an organizing trajectory that lies in a plane and thus has identically zero Frenet curvature. This result suggests a strong connection between canards and rivers. In fact, if we study the zero curvature locus of the original system (35), we find that there are seven trajectories that have identically zero curvature and hence are rivers by our definition. The most interesting of these is given by and is invariant, such that it can be reparameterized as a solution. Furthermore, Γ(z) lies O(ε) close to our approximate canard Ψ 2 (γ(t), 0). For z < 0, Γ lies near the stable branch of the critical manifold, at z = 0 it passes near the fold, and for z > 0 it remains O(ε) close to the unstable critical manifold; see Fig. 3. We intend to formalize the connection between rivers and canards in future work. Additionally, it would be interesting to investigate how the LOR frame, specialized to the context of fast-slow systems, relates to other geometrically-motivated manifold approximation schemes that apply in that setting, such as computational singular perturbation (CSP) methods [8,18] or the zero derivative principle [1], and whether LOR provides a way to extend ideas like CSP beyond the fast-slow context. For the time being, we hope that the computations and sketch of ideas in this section serve as an amuse-bouche to demonstrate how the LOR frame can highlight nontrivial features of phase space.

7.
Conclusions. LOR provides a general approach to deriving a natural coordinate frame, based on a geometric representation, that is well suited to study dynamics relative to any manifold embedded in the flow of an ODE of arbitrary dimensions. A variety of methods have been previously developed to describe the flow in the vicinity of a periodic orbit or other attractor [11,2,10,15,16,12,17]. LOR is advantageous because, to our knowledge, it is the unique approach with a full range of desirable properties: it is not based on linearization, it is not limited to scenarios involving periodic orbits, it generally extends well beyond the local neighborhood of the base manifold used to define it, and it applies naturally in arbitrary dimensions. LOR leads to LOR equations describing the evolution of trajectories in the LOR frame, which encode the geometry of the flow. These equations do not depend on the choice of normal frame used in their derivation, which allows a specific frame to be chosen to simplify associated calculations. An additional blow-up transformation of the LOR equations provides an especially convenient decomposition for studying transient aspects of the behavior of trajectories near structures in a phase space. Alternatively, we have seen that LOR can be used to identify canard behavior that organizes flow along a slow manifold. Given the derivation and illustration of these ideas and their properties in this work, we hope that LOR will serve as a useful tool for future studies of dynamical systems.