Model Reduction of Nondensely Defined PiecewiseSmooth Systems in Banach Spaces
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Abstract
In this paper, a model reduction technique is introduced for piecewisesmooth (PWS) vector fields, whose trajectories fall into a Banach space, but the domain of definition of the vector fields is a nondense subset of the Banach space. The vector fields depend on a parameter that can assume different discrete values in two parts of the phase space and a continuous family of values on the boundary that separates the two parts of the phase space. In essence, the parameter parametrizes the possible vector fields on the boundary. The problem is to find one or more values of the parameter so that the solution of the PWS system on the boundary satisfies certain requirements. In this paper, we require continuous solutions. Motivated by the properties of applications, we assume that when the parameter is forced to switch between the two discrete values, trajectories become discontinuous. Discontinuous trajectories exist in systems whose domain of definition is nondense. It is shown that under our assumptions the trajectories of such PWS systems have unique forwardtime continuation when the parameter of the system switches. A finitedimensional reducedorder model is constructed, which accounts for the discontinuous trajectories. It is shown that this model retains uniqueness of solutions and other properties of the original PWS system. The model reduction technique is illustrated on a nonlinear bowed string model.
Keywords
Piecewisesmooth Model reduction Invariant manifolds Nondense domainMathematics Subject Classification
37L25 35B65 35Q70 47D061 Introduction
The purpose of model reduction is to extract the essence of a complex model, disregarding details that are irrelevant to a specific application. Depending on the question asked from the model, different kinds of model reduction are required. In many cases, only qualitative predictions are needed, where loworder analytically solvable models, such as normal forms used in bifurcation theory (Kuznetsov 2004), are useful. In other cases, the reducedorder model has to be solved numerically with a specified accuracy using constrained computational resources (Benner et al. 2017). Similar to model reduction, any numerical scheme that solves a continuum problem, such as finite elements, spectral collocation or finite differences, turns an infinitedimensional continuoustime problem into a finitedimensional problem. A numerical scheme, however, tends to emphasize quantitative accuracy, which might miss some qualitative features, such as differentiability of solutions. In this paper, we focus on the qualitative properties of solutions of piecewisesmooth (PWS) systems, with applications to numerical schemes and reducedorder models in mind.
For smooth systems, there are rigorous ways to obtain reducedorder models. Center manifold reduction (Carr 1981) about an invariant set, such as an equilibrium or periodic orbit, captures the slowest dynamics and can be used to study bifurcations, regardless of the dimensionality of the system (Kuznetsov 2004). In multiple timescale systems (Kuehn 2015) attracting slow manifolds that contain dynamics much slower than the rest of the system can be used to obtain reducedorder models.
This paper discusses model reduction for infinitedimensional systems that are piecewise smooth. The theory of PWS systems is summarized in Filippov (1998), which contains the basic definitions and results on existence of solutions in finite dimensions. There are numerous applications of PWS systems, where discontinuities are essential to the model or where rapid variations of the vector field over small regions of the phase space naturally lead to discontinuous approximations. Some applications in finite dimensions include neuron models with resetting (Coombes et al. 2012; Izhikevich 2003), DC–DC converters (di Bernardo et al. 1998), network dynamics (Danca 2002; DeLellis et al. 2015), friction oscillators (Oestreich et al. 1997; Szalai and Osinga 2008), gene regulatory networks (Glass and Kauffman 1973; Mestl et al. 1995) and so on. We consider the special case of differential equations that are discontinuous along a codimensionone hypersurface of their phase space, called the switching manifold. We assume that the phase space is a Banach space and that the domain of definition of the differential equation is not dense.
In contrast to smooth systems, center manifolds or slow manifolds that continue through switching manifolds do not exist for PWS systems. In general, the dynamics of singularly perturbed PWS systems cannot be reduced to an invariant manifold, because smallscale instabilities persist as the perturbation vanishes (Sieber and Kowalczyk 2010). For a special class of PWS systems, slow manifolds with similar properties to smooth systems exist (Fridman 2002; Cardin et al. 2013, 2015). It is also possible to find equivalents of invariant manifolds which allow model reduction by considering the dynamics on the invariant manifold. Invariant cones can be found in systems with equilibria on the switching manifold (Weiss et al. 2012, 2015). Invariant polygons may also appear when an unstable focustype periodic orbit interacts with discontinuities of the vector field (Szalai and Osinga 2008), which leads to periodic or chaotic dynamics (Szalai and Osinga 2009).
In infinite dimensions, the theory of PWS systems is focused on sliding mode control (Orlov and Utkin 1987) and PWS delay equations (Sieber 2006; Londoño et al. 2012). Sliding mode control applies a discontinuous control signal to a plant, in order to restrict the system onto an engineered hypersurface with a prescribed dynamics. The main objective of sliding mode control is to establish conditions that guarantee the prescribed dynamics. The results in this area concern systems that are densely defined on reflexive Banach spaces (Levaggi 2002a, b), which suggests that these systems are similar to finitedimensional PWS systems.
In this paper, we relax the assumption of a dense domain of definition and not surprisingly we find different dynamics to what has been studied before. For this class of systems, we are able to prove uniqueness of solutions and also construct a reducedorder model. One consequence of the nondense domain is the existence of discontinuous solutions, which is just the inverse of the Hille–Yosida theorem (Pazy 1983): trajectories of a linear autonomous system (as described by a semigroup) are strongly (or weakly) continuous if and only if the infinitesimal generator is closed and densely defined. The relevant mathematics describing our class of systems is the nonautonomous generalization of integrated semigroups (Neubrander 1988; da Prato and Sinestrari 1992). To illustrate that our class of systems are necessary to describe physical phenomena, we refer to McIntyre and Woodhouse (1979). In McIntyre and Woodhouse (1979), the authors have noticed that the measured impulse response function of a string has a discontinuity in the velocity component, which is manifest of the nondense domain and that the initial condition is outside of the closure of the domain. This is shown later in the paper for the relevant mathematical model. Crucially, accounting for the discontinuity of the impulse response explains the observed asymmetric hysteresis of the stick–slip motion that causes ‘flattening’ of notes when the string is bowed in a certain way. The discontinuity of the impulse response is exactly the property that allows us to carry out model reduction.
The outline of the paper is as follows. We first carry out model reduction on a simple linear example to illustrate each step of the process, but without a rigorous justification of the steps. In Sect. 3, we review basic classes of PWS systems and highlight some cases where solutions may be nonunique. Section 4 describes the model reduction process in a general setting and shows that uniqueness of solutions and some other properties carry over to the reducedorder model. Section 5 describes a nonlinear example, the classical example of the bowed string, which highlights the significance that nonlinearity plays in the reduction process and uncovers some possibly surprising results that were not known about friction oscillators.
2 The Reduction Procedure Through an Example
The phase portrait of the skeleton model (2.7) can be seen in Fig. 1. We focus on the dynamics at stick, which occurs on the switching manifold, highlighted by the horizontal redshaded plane. The dashed red lines correspond to discontinuities in \(\lambda \). The solid green line on the horizontal redshaded plane is the stick solution, where the friction force \(\lambda \) grows with a constant rate with respect to t and y until it reaches the limit \(\lambda =\pm \,1\) and slip ensues.
The phase portrait of the reducedorder model (2.19) can be seen in Fig. 3. In the simulation, we have used the result of Appendix A to extend the valid time interval to an appropriate length. In comparison with the skeleton model (2.7) shown in Fig. 1 the dynamics at stick becomes more complicated. The dynamics in slip is the same, because \(\lambda \) is constant and decoupled from the rest of the variables due to the choice of immersion (2.6). The stick dynamics is now described by the differential Eq. (2.18), and therefore there is no discontinuity of \(\lambda \). Due to the higher dimensional dynamics that arise from the inclusion of \(\kappa \) as a dynamic variable and delayed values of \(\lambda \), the dynamics depicted in Fig. 3 is only a projection. Regardless of the differences, the phase portrait in Fig. 3 appears as a smoothed version of the same dynamics in Fig. 1, even though no smoothing or regularization was applied. Furthermore, to solve the reducedorder model (2.19) we did not need an arbitrary closure, such as Filippov’s to define the dynamics at stick, instead the solution followed straight from the initial problem (2.1) and (2.2).
In Sect. 4, we explore a generalization of Eqs. (2.1) and (2.2). We consider models whose solutions may be normdiscontinuous as illustrated by the linear string model. Before we embark on the general theory, we recall basic definitions and properties of PWS models.
3 FiniteDimensional PWS Models
In this section, we summarize two commonly used closures of PWS systems. As it turns out, these common PWS systems are special forms of the skeleton model to be defined in Sect. 4.2. An introduction to the state of the art can be found in Glendinning and Jeffrey (2017); however, the book of Filippov (1998) contains the most general definitions of PWS systems. Below, we review the cases defined in (Filippov 1998, Chapter 2, §4), which are used most commonly in applications. We avoid cases where the vector field is a setvalued function (Filippov 1998, Chapter 2, §5,§6). We also limit the description to the bimodal case, where the discontinuity occurs along a single implicitly defined manifold in the phase space.
Note 3.1
In addition to various notation for derivatives, in what follows D is also used to denote the Frechet derivative of a function; a subscript of D, such as \(D_{k}\) denotes the partial derivative with respect to the kth argument of a function and a superscript such as \(D_{k}^{j}\) denotes the jth derivative with respect to the kth argument.
We have assumed two possibilities, (3.3) or (3.4), for the domain of definition of \(\varvec{f}\). The case of (3.3) is the minimum necessary to make Eqs. (3.1) and (3.2) consistent. In many applications, such as mechanics, the larger domain of definition (3.4) is naturally given, which is useful to define the solutions of (3.1) and (3.2) on \(\varSigma \) as we show later in this section.
The system (3.1) and (3.2) has a unique solution on an interval of nonzero length for initial condition \(\varvec{y}(0)\in G\) if \(h(\varvec{y}(0))\ne 0\), because \(\varvec{f}\) is a smooth vector field (Coddington and Levinson 1955). However, for \(h(\varvec{y}(0))=0\) the vector field is not defined and one needs to reason how trajectories continue once they reach \(\varSigma \).
3.1 Filippov’s Closure
Definition 3.2
Assume that (3.8) holds. We call the vector field (3.9), where \(\lambda \) is given by (3.11), Fillipov’s closure.
It can be shown that \(\lambda \in (\,1,1)\) when condition (3.8) holds (Filippov 1998). Fillipov’s closure is illustrated in Fig. 4c, which shows that the vector field given by (3.9) and (3.11) is chosen from all convex combinations of \(\varvec{f}(\varvec{y},\pm \,1)\) so that \(\varvec{r}(\varvec{y})+\varvec{b}(\varvec{y})\lambda \) is tangential to \(\varSigma \). A trajectory at its first point of contact with \(\varSigma \) is continuous, but not continuously differentiable, because \(\lambda \) becomes discontinuous due to (3.11).
Trajectories may not have unique continuation when they are tangent to \(\varSigma \). When both vector fields \(\varvec{f}(\varvec{y},\pm \,1)\) are tangential to \(\varSigma \) at a point, \(\lambda \) is not uniquely defined by (3.11) as both the numerator and denominator of (3.11) vanish. Consequently, the forwardtime solution of (3.1), (3.2) and (3.11) is not unique. This case is illustrated in Fig. 4d, which shows a set of possible directions that a solution can follow. A particular case of this double tangency is the Teixeira singularity, where an open set of initial conditions generate trajectories that go through the double tangency. The Teixeira singularity (Teixeira 1982) was studied extensively (Colombo and Jeffrey 2011; Filippov 1998; Kristiansen and Hogan 2015; Szalai and Jeffrey 2014) in various contexts.
3.2 Utkin’s Closure
Definition 3.3
Assume that (3.8) holds. We call the vector field (3.9), where \(\lambda \) is given by the solutions of Eq. (3.13), Utkin’s closure.
The root of (3.13) may not be unique, which renders the solution of (3.1) and (3.2) nonunique. We also note that (3.13) can have a solution even when (3.7) holds in the crossing region.
A simple case of Utkin’s closure is illustrated in Fig. 4a. The green curve connecting \(\varvec{f}(\varvec{y},1)\) to \(\varvec{f}(\varvec{y},\,1)\) represents the possible values of the vector field on \(\varSigma \). There is one intersection of this family of vectors with the tangent plane of \(\varSigma \), represented by the thick red arrow, which satisfies Eq. (3.13). Figure 4b shows that there can be multiple intersections of \(\varvec{f}(\varvec{y},[\,1,1])\) with the tangent plane of \(\varSigma \), that then yields multiple solutions. Note that in the case of Fig. 4b, Filippov’s closure yields a unique solution. The contrary, when Utkin’s closure predicts a unique solution and Filippov’s closure predicts a family of solutions, is also possible. For example, when the convex hull represented by the green dashed line in Fig. 4d is deformed slightly, the possible number of solutions can be reduced to three. Out of these three solutions, there is only one with \(\lambda \in (\,1,1)\).
4 Model Reduction
Remark 4.1
The notation of Eq. (4.1) facilitates that \(\lambda \) is an unknown, which needs to be found when \(h(\varvec{x})=0\). Therefore, \(\lambda \) may not be a function of \(\varvec{x}\), but it may become part of the phase space. The solution for \(\lambda \), when \(h(\varvec{x})=0\) is defined in Sect. 4.3. This is a similar setting to Sect. 3, except that the phase space is now infinite dimensional, and therefore a different kind of solution is required for \(\lambda \).
4.1 The Invariant Manifold
 (A1)
 Existence of an invariant manifold. We assume that there exists a function \(\varvec{W}\in C^{p}(\mathbb {R}^{n}\times [\,1,1],\varvec{X})\), \(p\ge 2\) and a vector field \(\varvec{f}\in C^{p}(G\times [\,1,1],\mathbb {R}^{n})\), which satisfies the invariance conditionwhere G is a compact and connected subset of \(\mathbb {R}^{n}\). The invariant manifold is given by$$\begin{aligned} \varvec{F}(\varvec{W}(\varvec{y},\lambda ),\lambda )=D_{1}\varvec{W}(\varvec{y},\lambda )\varvec{f}(\varvec{y},\lambda ), \end{aligned}$$(4.3)and the dynamics of (4.2) on \(\mathcal {M}_{\lambda }\) is described by$$\begin{aligned} \mathcal {M}_{\lambda }=\left\{ \varvec{W}(\varvec{y},\lambda ):\varvec{y}\in G,\lambda \in [\,1,1]\right\} \end{aligned}$$(4.4)\(\varvec{W}\) is called the immersion of \(\mathcal {M}_{\lambda }\).$$\begin{aligned} \dot{\varvec{y}}=\varvec{f}(\varvec{y},\lambda ). \end{aligned}$$(4.5)
 (A2)
 We assume that for every \(\lambda \in [\,1,1]\), \(\varvec{F}(\cdot ,\lambda )\) is Frechet differentiable on \(\mathcal {M}_{\lambda }\). This derivative is denoted byWe also assume that the domain of definition of \(\varvec{A}_{1}\), i.e., \(\varvec{\mathcal {D}}(\varvec{A}_{1}(\varvec{y},\lambda ))=\left\{ \varvec{x}\in \varvec{X}:\varvec{A}_{1}(\varvec{y},\lambda )\varvec{x}\in \varvec{X}\right\} \), is independent of \(\varvec{y}\) and \(\lambda \), and we define \(\varvec{Z}=\overline{\varvec{\mathcal {D}}(\varvec{A}_{1}(\varvec{y},\lambda ))}\) (in general, \(\varvec{\mathcal {D}}(\varvec{A}_{1}(\varvec{y},\lambda ))\ne \varvec{\mathcal {D}}_{\lambda }(\varvec{F})\)).$$\begin{aligned} \varvec{A}_{1}(\varvec{y},\lambda )=D_{1}\varvec{F}(\varvec{W}(\varvec{y},\lambda ),\lambda ). \end{aligned}$$
 (A3)
 Unique continuous solutions. We assume that the abstract Cauchy problemwith initial conditions \(\varvec{y}(s)\in G\), \(\varvec{z}(s)\in \varvec{Z}\), \(s\in \mathbb {R}\) and with \(\lambda \in C^{1}([s,\infty ),\mathbb {R})\) has a unique solution \(\left( \varvec{y},\varvec{z}\right) \in C([s,\infty ),G\times \varvec{Z})\), even though we only have \(D_{2}\varvec{W}(\varvec{y}(t),\lambda (t))\in \varvec{X}\). We also assume that the \(\varvec{Z}\) component of the solution can be written as$$\begin{aligned} \left. \begin{array}{rl} \dot{\varvec{y}} &{} =\varvec{f}(\varvec{y},\lambda )\\ \dot{\varvec{z}} &{} =\varvec{A}_{1}(\varvec{y},\lambda )\varvec{z}D_{2}\varvec{W}(\varvec{y},\lambda )\dot{\lambda } \end{array}\right\} \end{aligned}$$(4.6)where \(\varvec{K}\) is bounded for \(\tau \ge s\) and continuous in both variables for \(\tau >s\). The underlying conditions of existence of unique solutions can be found in da Prato and Sinestrari (1992). For discussion, see Remarks 4.3 and 4.4.$$\begin{aligned} \varvec{z}(t)=\varvec{U}(t,s)\varvec{z}(s)\int _{s}^{t}\varvec{K}(t,\tau )\dot{\lambda }(\tau )\mathrm {d}\tau , \end{aligned}$$(4.7)
 (A4)
 \(\mathcal {M}_{\lambda }\) is attracting and normally hyperbolic. We assume that there exist two families of projections \(\varPi ^{c}(\varvec{y},\lambda )\) and \(\varPi ^{s}(\varvec{y},\lambda )\), strongly continuous in \(\varvec{y}\) and \(\lambda \) such that$$\begin{aligned} \varPi ^{c}(\varvec{y},\lambda )+\varPi ^{s}(\varvec{y},\lambda )&=\varvec{I},\nonumber \\ \varPi ^{c}(\varvec{y},\lambda )D_{1}\varvec{W}(\varvec{y},\lambda )&=D_{1}\varvec{W}(\varvec{y},\lambda ),\end{aligned}$$(4.8)Consider the nonautonomous ordinary differential equation \(\dot{\varvec{\eta }}=D_{1}\varvec{f}(\varvec{y},\lambda )\varvec{\eta }\), whose solutions with initial condition \(\varvec{\eta }_{0}\) at \(t=s\) are denoted by \(\varvec{\eta }(t,s,\varvec{\eta }_{0})\). We assume that there exist real numbers \(\sigma _c >0, \sigma _s <  \sigma _c\) ,\(M_{c}>0\) and \(M_{s}>0\) such that$$\begin{aligned} \varvec{U}(t,s)\varPi ^{s}(\varvec{y}(s),\lambda (s))\varvec{z}&=\varPi ^{s}(\varvec{y}(t),\lambda (t))\varvec{U}(t,s)\varvec{z},\quad \forall \varvec{z}\in \varvec{Z},\quad t\ge s. \end{aligned}$$(4.9)$$\begin{aligned}&\forall (ts)\in \mathbb {R},\,\varvec{\eta }_{0}\in \mathbb {R}^{n} :\left\ \varvec{\eta }(t,s,\varvec{\eta }_{0})\right\ \le M_{c}\left\ \varvec{\eta }_{0}\right\ \mathrm {e}^{\sigma _{c}\left ts\right },\\&\forall s\le t,\,\varPi ^{s}(\varvec{y}(s),\lambda (s))\varvec{z}=\varvec{z} :\left\ \varvec{U}(t,s)\varvec{z}\right\ \le M_{s}\left\ \varvec{z}\right\ \mathrm {e}^{\sigma _{s}(ts)}. \end{aligned}$$
 (A5)
 We assume that for \(t\ge s\) there exists \(0<M<\infty \) and \(\sigma <0\) such that$$\begin{aligned} \left\ \varvec{K}(t,s)\right\ \le M\mathrm {e}^{\sigma (ts)}. \end{aligned}$$(4.10)
Remark 4.2
For systems with an equilibrium it is natural to consider spectral submanifolds (Haller and Ponsioen 2016), that are the smoothest invariant manifolds tangent to an invariant linear subspace of the variational problem about the equilibrium. The uniqueness and existence of such manifolds are established in Cabré et al. (2003). In order to be meaningful, these manifolds need to contain the slowest dynamics within the system to capture longtime behavior. This requirement is outlined in points R1 and R2 of Haller and Ponsioen (2017).
Remark 4.3
Remark 4.4
Remark 4.5
The uniqueness or persistence of \(\mathcal {M}_{\lambda }\) is not addressed by the assumptions. For persistence of \(\mathcal {M}_{\lambda }\) under a perturbation, additional smoothness conditions on the solutions of (4.2) have to hold, which can be found in Bates et al. (1998).
Remark 4.6
The condition (4.10) implies that the convolution in (4.7) remains bounded when \(\dot{\lambda }\) is bounded. This will be useful later when the reducedorder model is constructed.
Figure 5 shows the invariant manifold \(\mathcal {M}_{\lambda }\) and its intersection with the switching manifold \(\varSigma \). The Banach space \(\varvec{X}\) is represented by two coordinates \(\varvec{x}_{1}\) and \(\varvec{x}_{2}\). The two parts of the invariant manifold (\(\mathcal {M}_{1}\) and \(\mathcal {M}_{1}\)) do not join up in Fig. 5a. If trajectories cross \(\varSigma \) instantaneously, they are discontinuous. When discontinuity is not allowed, the crossing cannot be instantaneous. In certain cases, however, the disconnected nature of \(\mathcal {M}_{\lambda }\) may be overlooked. For example, when the switching function solely depends on the parameter \(\varvec{y}\) of the immersion \(\varvec{W}\), i.e., \(\varvec{y}=\varvec{x}_{1}\) in Fig. 5. The case when the dynamics is restricted to \(\mathcal {M}_{\lambda }\) is discussed in Sect. 4.2.
Figure 5b shows the extended phase space and how solutions of (4.1) behave about \(\mathcal {M}_{\lambda }\), when instantaneous crossing is not allowed. In Fig. 5b, \(\mathcal {M}_{\lambda }\) is a connected manifold. When a solution of (4.1) arrives at \(\varSigma \), the value of \(\lambda \) must change, so that a trajectory can enter \(\varSigma \). \(\mathcal {M}_{\lambda }\) is only invariant for constant \(\lambda \), and therefore a trajectory (denoted by dotted lines) will not continue on \(\mathcal {M}_{\lambda }\) while also in \(\varSigma \). Once a trajectory has left \(\varSigma \), it will be attracted to \(\mathcal {M}_{\lambda }\) as per Assumption (A4).
In the following sections, we discuss how the departure of a trajectory from \(\mathcal {M}_{\lambda }\) can be captured and whether or not capturing this dynamics makes a qualitative difference in the predictions of the model. In Sect. 2, we have already seen that including a correction that captures the departure from \(\mathcal {M}_{\lambda }\) makes a difference and trajectories can no longer cross \(\varSigma \) instantaneously.
4.2 The Skeleton Model
Definition 4.7
Equation (4.13) is called the skeleton model of (4.1) on the invariant manifold \(\mathcal {M}_{\lambda }\).
Definition 4.7 alludes to what follows next. We will use Eq. (4.13) to build upon and not consider it as an end result. Equation (4.13) is inaccurate when \(\lambda \) varies and that causes solutions to become nonunique, even if they were unique in the full problem (4.1). Nevertheless, we highlight some properties of the skeleton model that carry over to the reducedorder model.
Theorem 4.8
 1.
\(\mathcal {T}\) is not tangent to \(\varSigma _{0}^{\pm }\), i.e., \(D_{1}h_{0}\left( \varvec{y}^{\star },\lambda ^{\star }\right) \varvec{f}\left( \varvec{y}^{\star },\lambda ^{\star }\right) \ne 0\)
 2.\(\mathcal {T}\) is tangent to \(\varSigma _{0}^{\pm }\) and the order of the tangency is less than the smoothness order (\(C^{p}\)) of \(h_{0}\). In other words, there exists \(0<\ell \le p\) such that$$\begin{aligned} \frac{\mathrm {d}^{\ell }}{\mathrm {d}t^{\ell }}h_{0}(\varvec{y}(t),\lambda ^{\star })\vert _{t=0}\ne 0. \end{aligned}$$(4.22)
Proof
The proof can be found in Appendix B. \(\square \)
Remark 4.9
Theorem 4.8 excludes the case \(D_{2}h_{0}(\varvec{y},\lambda )>0\). For \(D_{2}h_{0}(\varvec{y},\lambda )>0\), transverse trajectories (case 1 of Theorem 4.8) cannot cross \(\varSigma _{0}^{\pm }\). Tangential trajectories with even \(\ell \) may have multiple continuation, which is the case of the Teixeira singularity (Colombo and Jeffrey 2011). Tangential trajectories with odd \(\ell \) cannot cross \(\varSigma _{0}^{\pm }\), similar to transverse trajectories. To investigate the case of \(D_{2}h_{0}(\varvec{y},\lambda )>0\) in detail, a definition of how trajectories move along \(\varSigma _{0}^{\pm }\) (with \(\lambda =\pm \,1\)) is also required, which falls outside of the scope of this paper.
4.3 Dynamics About Manifold \(\mathcal {M}_{\lambda }\) Due to Switching
This section describes a correction to the skeleton model (4.13) that resolves the dynamics in the neighborhood of \(\mathcal {M}_{\lambda }\) up to linear order. The correction is necessary, because the \(\dot{\lambda }=0\) assumption does not hold: Eq. (4.20) states that \(\lambda \) varies on \(\varSigma _{0}\). The correction that is introduced here captures trajectories that depart from \(\mathcal {M}_{\lambda }\) when \(h=0\) (see dashed line in Fig. 5b).
Lemma 4.10
Proof
The proof can be found in Appendix C. \(\square \)
Remark 4.11
The quantity \(d^{\pm }(\varvec{y},\varvec{z},\lambda )\) in (4.31) measures the discontinuity of the convolution kernel \(\varvec{K}\) at \(t=s\). A discontinuous \(\varvec{K}\) is possible, because \(D_{2}\varvec{W}(\varvec{y},\lambda )\in \varvec{X}\backslash \varvec{Z}\), and the continuity Assumption (A3) does not apply at \(t=s\). Such a discontinuity allowed us to find a differential equation for \(\lambda \) in Sect. 2.
Definition 4.12
We call the quantity \(d^{\pm }(\varvec{y},\varvec{z},\lambda )\) in Eq. (4.31) the normal discontinuity gap.
Theorem 4.13
 1.$$\begin{aligned} d^{\pm }(\varvec{y},\varvec{z},\lambda )>0, \end{aligned}$$(4.36)
 2.
\({Dh}(\varvec{W}(\varvec{y},\lambda )+\varvec{z})\cdot \varvec{U}(t,s)\varvec{z}\) is continuously differentiable with respect to t for \(t\ge s\) and
 3.one of the vector fields, (4.34) or (4.35) is not tangent to \(\varSigma ^{\pm }\), that is,$$\begin{aligned} {Dh}(\varvec{W}(\varvec{y},\lambda )+\varvec{z})\cdot \left( D_{1}\varvec{W}(\varvec{y},\lambda )\varvec{f}(\varvec{y},\lambda )+\varvec{A}_{1}(\varvec{y},\lambda )\varvec{z}\right) \ne 0. \end{aligned}$$(4.37)
Remark 4.14
The linear correction about the invariant manifold is carried out here without an assessment whether trajectories of the corrected model (4.28) and the full model (4.1) are qualitatively the same. If \(\left\ \varvec{z}\right\ \ll 1\), the linear correction is accurate. Because on \({{\mathcal {M}}}_{\lambda }\), we have \(\varvec{z}=\varvec{0}\), when a trajectory enters \(\varSigma \), the rate of change of \(\varvec{z}\) is determined by \(\dot{\lambda }\). The magnitude of \(\dot{\lambda }\) depends on the \(\varvec{f}\) and \(d^{\pm }\). Smaller \(d^{\pm }\) makes \(\lambda \) faster. The value of \(d^{\pm }\) is not necessarily a small parameter, and therefore the deviation from \({{\mathcal {M}}}_{\lambda }\) can stay small. For the linear string \(d^{\pm }=\frac{1}{2}\). In the literature of regularized PWS systems (Jeffrey 2015; Kristiansen and Hogan 2015), to stay close to the skeleton model, fast \(\lambda \) is assumed.
Remark 4.15
If \(d^{\pm }(\varvec{y},\varvec{z},\lambda )=0\), the dynamics about \(\mathcal {M}_{\lambda }\) as captured by variable \(\varvec{z}\) can only have a secondorder effect on h due to the nonlinearity of h. Therefore, (4.30) is independent of \(\dot{\lambda }\) and \(\frac{\mathrm {d}}{\mathrm {d}t^{+}}h=0\) cannot be solved for \(\dot{\lambda }\). When \(d^{\pm }(\varvec{y},\varvec{z},\lambda )=0\), the corrected model (4.28) needs a closure, such as Filippov’s or Utkin’s. \(d^{\pm }(\varvec{y},\varvec{z},\lambda )=0\) occurs when \(\varvec{U}\) is strongly continuous on the whole of \(\varvec{X}\), i.e., \(\varvec{Z}=\varvec{X}\). This case for linear systems is explored in Orlov (1995) and Levaggi (2002a, b).
Remark 4.16
The transversality condition (4.37) is the equivalent of case 1 of Theorem 4.8. The equivalent of case 2 of Theorem 4.8 is not proven here, but a similar argument can be made while carefully accounting for the infinitedimensional nature of the problem.
Remark 4.17
4.4 TimeScale Separation
We already have some indication that switching has a great influence on the normal dynamics. For example, ignoring the normal dynamics as in the skeleton model (4.13) leads to a different uniqueness condition than for the corrected model (4.28). In this section, we restrict the analysis to the simplest case where there is a separation of time scales. We assume a parameter \(0\le \varepsilon \le 1\) and denote the dependence on \(\varepsilon \) by a subscript, that is \(\varvec{F}_{\varepsilon }\). Here, the \(\varepsilon =0\) limit is represented by the skeleton model (4.13) and \(\varepsilon =1\) refers to the corrected model (4.28). Naturally, the immersion \(\varvec{W}_{\varepsilon }(\varvec{y},\lambda )\) of the invariant manifold also depends on \(\varepsilon \), which implicitly assumes that \({{\mathcal {M}}}_{\lambda }\) persists for \(0\le \varepsilon \le 1\). Whenever we write \(\varvec{F}_{0}\) or \(\varvec{W}_{0}\), we mean the \(\varepsilon =0\) limit.
 (\(\overline{\mathbf{A3 }}\))
 Assumptions (A3) holds when (4.6) is replaced by (4.42) for all \(\varepsilon \in [0,1]\). The unique solution of (4.42) can be written as$$\begin{aligned} \varvec{z}(t)=\varvec{U}_{\varepsilon }(t,s)\varvec{z}(s)\int _{s}^{t}\varvec{K}_{\varepsilon }(t,\tau )\dot{\lambda }(\tau )\mathrm {d}\tau . \end{aligned}$$
 (\(\overline{\mathbf{A4 }}\))

Assumption (A4) holds when (4.6) is replaced by (4.42) and \(\sigma _{s}<\varepsilon \sigma _{c}\).
 (\(\overline{\mathbf{A5 }}\))
 The perturbation \(D_{2}\varvec{W}_{0}(\varvec{y},\lambda )\) acts in the invariant normal bundle of \(\mathcal {M}_{\lambda }\), that is,$$\begin{aligned} \varPi ^{c}(\varvec{y},\lambda )D_{2}\varvec{W}_{0}(\varvec{y},\lambda )=\varvec{0}. \end{aligned}$$(4.44)
Remark 4.18
Theorem 4.19
Proof
Remark 4.20
Remark 4.21
Similar to Remark 4.5, normal hyperbolicity does not imply the persistence of \(\mathcal {M}_{\lambda }^{\mathrm{crit}}\) under variations in \(\varepsilon \). The theorem of Bates, Lu and Zeng Bates et al. (1998) suggests that the evolution operator \(\varvec{U}\) needs to be differentiable (among other conditions) for \(\mathcal {M}_{\lambda }^{\mathrm{crit}}\) to persist for small \(\varepsilon >0\). Note that the nonlinear string example in Sect. 5 generates such a differentiable \(\varvec{U}\) on \(\varvec{Z}\).
Remark 4.22
When both regions of \(\mathcal {M}_{\lambda }\), that is \(\mathcal {M}_{\lambda }\cap \varSigma \) and \(\mathcal {M}_{\lambda }\backslash \left( \mathcal {M}_{\lambda }\cap \varSigma \right) \), persist for \(\varepsilon >0\), they most likely become discontinuous at the boundaries \(\varSigma ^{\pm }\), and hence as a whole, \(\mathcal {M}_{\lambda }\) does not persist. This is because the vector fields are discontinuous. Therefore, for \(\varepsilon >0\), trajectories that followed one part of \(\mathcal {M}_{\lambda }\) must jump to the other part of \(\mathcal {M}_{\lambda }\), which induces fast transients that we are unable to characterize under general settings.
4.5 Qualitative Approximation of Normal Dynamics and the ReducedOrder Model
A key difference between the skeleton model (4.13) and the corrected model (4.28) is that they have unique solutions under different conditions. This difference is caused by the fact that the skeleton model does not take into account the normal discontinuity gap \(d^{\pm }\). To rectify the omission of \(d^{\pm }\), the skeleton model is extended by a scalar variable, which represents the dynamics of the convolution kernel \(\varvec{K}\) in Eq. (4.7). We call this extension the reducedorder model. It is then shown that the reducedorder model reproduces uniqueness of solutions and the existence of a critical manifold under equivalent conditions to those of Theorems 4.13 and 4.19.
 (A6)

\(h(\varvec{x})\) is linear, therefore \(h(\varvec{x})=h(\varvec{0})+{Dh}\cdot \varvec{x}\), where Dh is a constant linear functional.
We can now check that the reducedorder model (4.57) has the same key properties as the corrected model (4.28). In what follows, we outline the equivalents of Theorems 4.13 and 4.19 for the reducedorder model (4.57).
Proposition 4.24
A trajectory \(\mathcal {T}\) of the reducedorder model (4.57) with an end point at \((\varvec{y}^{\star },\kappa ^{\star },\lambda ^{\star })\in \varSigma _{\varepsilon }^{\pm }\) has a unique continuation through \((\varvec{y}^{\star },\kappa ^{\star },\lambda ^{\star })\) if
 1.
\(d^{\pm }(\varvec{y}^{\star },0,\lambda ^{\star })>0\) as defined by Eq. (4.31) and
 2.when trajectory \(\mathcal {T}\) is not tangent to \(\varSigma ^{\pm }\) or trajectory \(\mathcal {T}\) is tangent to \(\varSigma _{\varepsilon }^{\pm }\) and the of order of the tangency is not greater than the smoothness order (\(C^{p}\)) of \(h_{\varepsilon }\), that is, there exists \(0<\ell \le p\) such that$$\begin{aligned} \frac{\mathrm {d}^{\ell }}{\mathrm {d}t^{\ell }}h_{\varepsilon }(\varvec{y}(t),\kappa (t),\lambda ^{\star })\vert _{\varvec{y}=\varvec{y}^{\star },\kappa =\kappa ^{\star }}\ne 0. \end{aligned}$$
Proof
The proof is the same as for Theorem 4.8 if we replace \(\left( \varvec{y},\kappa \right) \rightarrow \varvec{y}\) and \(d^{\pm }\rightarrow D_{2}h_{0}(\varvec{y},\lambda )\). \(\square \)
Proposition 4.25
Proof
Remark 4.26
We know that \(\sigma <0\), because of Assumption (A5) or (\(\overline{\hbox {A5}}\)). If Proposition 4.24 also holds, \({{\mathcal {M}}}_{\lambda }^{\mathrm{crit}}\) is attracting when \(d^{}<0\). This is the same condition under which solutions of the skeleton model (4.13) are unique due to Theorem 4.8.
Next, we investigate in what sense the reducedorder model (4.57) is similar to the corrected model (4.41) with timescale separation. It turns out that on \(\varSigma _{\varepsilon }\) the critical manifold is likely to be attracting or repelling under the same conditions. The precise statement is in the following proposition.
Proposition 4.27
Assume that \(d^{\pm }(\varvec{y},\varvec{0},\lambda )>0\) and \({Dh}\cdot \varvec{A}_{0}^{1}(\varvec{y},\lambda )D_{2}\varvec{W}(\varvec{y},\lambda )>0\) along a smooth curve \(\gamma =\left\{ \left( \varvec{y}(\alpha ),\lambda (\alpha )\right) \in \varSigma _{0}:\alpha \in (\,\delta ,\delta )\right\} \) with \(\left( \varvec{y},\lambda \right) \in C^{1}\left( (\,\delta ,\delta ),\varSigma _{0}\right) \) and \(\delta >0\). For \(\varepsilon =0\), the stability of equilibria along \(\gamma \) changes through a zero root (saddlenode bifurcation) at the same value(s) of \(\alpha \in (\,\delta ,\delta )\) for both systems (4.59) and (4.45).
Proof
Remark 4.28
In the next section, for the example of the nonlinear string, \(d^{}\) is a small parameter, which measures how well the equilibrium shape of the string is approximated by a truncated Fourier series. The error gets smaller with increasing number of terms in the truncated series, and therefore \(d^{}\) also gets smaller. Without damping or nonlinearity, \(d^{}\) entirely vanishes, as was the case in Szalai (2014). When both \(d^{}\) and \(\varepsilon \) vanish, we arrive at a system that is subject to Utkin’s closure in Sect. 3.2. If \(d^{}\) vanishes, but we have \(d^{+}>0\) then for \(\varepsilon >0\) the trajectories are still unique, but there is no critical manifold in \(\varSigma \) that is being perturbed.
Remark 4.29
A more rigorous analysis would inspect the dynamics in the perturbed vector bundle corresponding to the near zero root of (4.47) for \(\varepsilon =1\). If this dynamics has a Lyapunov exponent \(\sigma _{0}\) such that \(\sigma _{s}<\left \sigma _{0}\right \) as in Assumption (A4), then this perturbed vector bundle could be attached to \(\mathcal {M}_{\lambda }\), which would become a normally hyperbolic invariant manifold of the corrected model (4.28) in \(\varSigma \).
5 A Bowed Nonlinear String Model Reduced to Single Degree of Freedom
 1.
Calculate the equilibrium of (5.5) as a function of \(\lambda \), which is denoted by \(\varvec{x}^{\star }\).
 2.
Find the smoothest twodimensional spectral submanifold (Haller and Ponsioen 2016) \(\mathcal {M}_{\lambda }\) about \(\varvec{x}^{\star }\), corresponding to the pair of complex conjugate eigenvalues with the least negative real part. The immersion of the manifold is denoted by \(\varvec{W}:{{\mathbb {R}}}^{2}\times [\,1,1]\rightarrow \varvec{X}\). Assume a function \(\varvec{y}^{\star }:{{\mathbb {R}}}^{2}\times [\,1,1]\rightarrow {{\mathbb {R}}}^{2}\), which shifts the parametrization of \(\mathcal {M}_{\lambda }\), such that \(\varvec{W}(\varvec{y},\lambda )=\varvec{W}_{ fix }(\varvec{y}+\varvec{y}^{\star }(\varvec{y},\lambda ),\lambda )\), where \(\varvec{W}_{ fix }\) is just one parametrization of \(\mathcal {M}_{\lambda }\). \(\varvec{y}^{\star }\) is an unknown and will be calculated in step 4.
 3.
Introduce an artificial parameter \(\varepsilon \) that slows down the dynamics on \(\mathcal {M}_{\lambda }\) to standstill at \(\varepsilon =0\) and has no effect at \(\varepsilon =1\). Then for \(\varepsilon =0\), calculate the invariant normal bundle of \(\mathcal {M}_{\lambda }\), which is formed by the subspace orthogonal to the kernel of the adjoint \(\varvec{A}_{0}^{\star }(\varvec{y},\lambda )\) at each point on \(\mathcal {M}_{\lambda }\).
 4.
Choose a coordinate shift \(\varvec{y}^{\star }\) so that \(D_{2}\varvec{W}\) falls into the invariant normal bundle of \(\mathcal {M}_{\lambda }\) at \(\varepsilon =0\), that is, \(\varvec{W}\) satisfies assumption (\(\overline{\hbox {A4}}\)). This now fully specifies the immersion \(\varvec{W}\).
 5.
Obtain the skeleton model by substituting the immersion \(\varvec{W}\) into (5.5).
 6.
Calculate the normal discontinuity gap from the dynamics in the invariant normal bundle of \(\mathcal {M}_{\lambda }\). Also determine \(\sigma \), the rate of convergence of the trajectory in the normal bundle with initial condition \(D_{2}\varvec{W}\).
Proposition 5.1
5.1 The Invariant Manifold and Its Parametrization
We identify the invariant manifold \(\mathcal {M}_{\lambda }\) with the spectral submanifold (Haller and Ponsioen 2016) of the string’s equilibrium corresponding to its first natural frequency. When \(\lambda \) is constant, the string has an equilibrium. We choose the smoothest invariant manifold about the equilibrium corresponding to the first natural frequency of the string, which is a unique twodimensional linear subspace. We note that the theory of Cabré et al. (2003) does not apply, because damping makes backwardtime solutions nonunique.
Lemma 5.2
Proof
5.2 Linearized Dynamics About the Invariant Manifold
The linearized dynamics about \(\mathcal {M}_{\lambda }\) is characterized by the Frechet derivative \(\varvec{A}_{1}(\varvec{y},\lambda )\) of Eq. (5.5), which is calculated here.
Lemma 5.3
Proof
The domain of definition of \(\varvec{A}_{1}(\varvec{y},\lambda )\) must now include that \(\varvec{x}_{1}^{\prime }(\xi ^{\star })\varvec{x}_{1}^{\prime }(\xi ^{\star }+)=0\), because the equilibrium is already included in the definition of \(\varvec{W}\). However, \(\varvec{x}_{1}^{\prime }(\xi ^{\star })\varvec{x}_{1}^{\prime }(\xi ^{\star }+)=0\) is already satisfied if \(\varvec{x}_{1}^{\prime \prime }\in L^{\infty }([0,1],\mathbb {R})\), and therefore we arrive at (5.21). To determine the closure \(\overline{\varvec{\mathcal {D}}},\) we first note that \(\varvec{x}_{1}^{\prime }\) must be Lipschitz continuous, and therefore \(\varvec{x}_{1}\) is Lipschitz continuously differentiable. The closure of such functions in the Lipschitz norm is the continuously differentiable functions \(C^{1}([0,1],\mathbb {R})\). Since \(\mathfrak {D}\) is the square root of \(\mathfrak {D}^{2}\), they have the same domain of definition, and therefore \(\varvec{x}_{2}\) is Lipschitz continuously differentiable in \(L^{\infty }\). The closure for this set in the \(L^{\infty }\) norm is the set of continuous functions \(C^{0}([0,1],\mathbb {R})\). Summing up this argument, we have found (5.22). Lipschitz functions are not dense in \(C^{1}\), and continuous functions are not dense in the set of bounded functions either, and therefore \(\varvec{Z}\ne \varvec{X}\). \(\square \)
5.3 Invariant Normal Bundle and TimeScale Separation
There is no small parameter in Eq. (5.5) that controls the spectral gap between the tangential and normal dynamics about \(\mathcal {M}_{\lambda }\). We therefore introduce such a scaling by artificially constructing \(\varvec{A}_{\varepsilon }(\varvec{y},\lambda )\) such that for \(\varepsilon =1\), we recover the original dynamics and for \(\varepsilon =0\) the tangential dynamics becomes \(\dot{\varvec{y}}=\varvec{0}\) when time is rescaled. This allows us to calculate the invariant normal bundle of \(\mathcal {M}_{\lambda }\) at \(\varepsilon =0\) and determine the parametrization \(\mathcal {M}_{\lambda }\) (i.e., the unknown coordinate shift \(\varvec{y}^{\star }(\varvec{y},\lambda )\) in the immersion of \(\mathcal {M}_{\lambda }\)) such that \(D_{2}\varvec{W}(\varvec{y},\lambda )\) is strictly in the invariant normal bundle of \(\mathcal {M}_{\lambda }\).
Lemma 5.4
Proof
Remark 5.5
For \(\varGamma =0\), we have \(\varvec{C}=\varvec{0}\) and also \(D_{2}y_{2}^{\star }=0\). This implies that for the linear string, the bundle projection is simply \(\varvec{Q}\). The normal bundle is independent of \(\varepsilon \), and there is no need to introduce timescale separation. Instead of \(\varepsilon \), \(\varGamma \) can be used to track the deformation of the invariant normal bundle, which persists for a sufficiently small \(\varGamma >0\) due to the properties of \(\mathcal {M}_{\lambda }\) (Bates et al. 1998).
5.4 The Vector Field \(\varvec{f}(\varvec{y},\lambda )\) on the Invariant Manifold
Lemma 5.6
Proof
5.5 The Normal Discontinuity Gap \(d^{\pm }\) and Decay Rate \(\sigma \)
The normal discontinuity gap \(d^{\pm }\) measures the discontinuity of the correction about the invariant manifold with initial conditions \(D_{2}\varvec{W}(\varvec{y},\lambda )\) at \(t=0\) and determines the uniqueness of solutions according to Theorem 4.13. We calculate \(d^{\pm }\) for the \(\varepsilon \rightarrow 0\) limit, when the dynamics in the normal bundle of \(\mathcal {M}_{\lambda }\) becomes autonomous. Therefore, it is sufficient to evaluate the limit \(\lim _{t\downarrow 0}{Dh}\cdot \mathrm {e}^{\varvec{A}_{0}(\varvec{y},\lambda )t}D_{2}\varvec{W}(\varvec{y},\lambda )\).
Lemma 5.7
Proof
The decay rate (5.39) is found by reading off the smallest exponent from formula (5.40). \(\square \)
Remark 5.8
The normal discontinuity gap \(d^{\pm }\) is a local property of the string; it depends on material properties and the tension in the string. However, \(d^{\pm }\) is independent of the boundary conditions and the position where the string is forced.
5.6 Spectrum of the Normal Dynamics on \(\varSigma \)
We use Theorem 4.19 to find out whether there exists an attracting critical manifold.
Lemma 5.9
Proof
The plot of this rightmost root is shown in Fig. 9 in orange (solid lines), which indicates that the critical manifold is partly attracting (negative root), partly repelling (positive root). This might be surprising because the system dissipates energy as a whole. However, the constraint \(h=0\) and nonlinearity couple the dynamics in the tangent and normal bundles and energy is exchanged between them causing instability. In contrast, there is no such coupling in the linear string (\(\varGamma =0\)), the green root near the origin in Fig. 8 remains at the origin, and therefore the normal dynamics is neutrally stable.
Remark 5.10
Continuing from Remark 5.5, we find that for \(\varGamma =0\) and for all \(\varepsilon \in [0,1]\) the characteristic function (5.42) is valid due to \(\varvec{A}_{1}\) being constant. Let us denote the zero root of the characteristic function (5.42) for \(\varGamma =0\) by \(\sigma ^{0}\). The invariant vector bundle corresponding to \(\sigma ^{0}\) is then isomorphic to \(\mathcal {M}_{\lambda }\times \mathbb {R}\). For \(\varGamma >0\), the invariant vector bundle of \(\sigma ^{0}\) is continuously perturbed. The perturbation turns \(\sigma ^{0}\) into a small Lyapunov exponent of the now nonautonomous dynamics within the invariant vector bundle. \(\varGamma >0\) can be chosen such that \(\sigma _{s}<\varepsilon \left \sigma ^{0}\right \), that is, the invariant vector bundle is an attracting normally hyperbolic invariant manifold in \(\left( \mathcal {M}_{\lambda }\times \varvec{X}\right) \cap \varSigma \), that is the phase space of the corrected model in \(\varSigma \). The dynamics in the invariant vector bundle is represented by the reducedorder model on \(\mathcal {M}_{\lambda }\times \mathbb {R}\).
5.7 Equivalent ReducedOrder Model on \(\Sigma _{\varepsilon }\)
5.8 Dynamics of the Skeleton Model on \(\Sigma _{0}\)
A different picture emerges when \(D_{2}y_{2}^{\star }(\overline{y}_{1},\lambda )\) is considered. Figure 10b shows a typical phase portrait of (5.52). Parts of trajectories are denoted by dashed lines when \(D_{2}y_{2}^{\star }(\overline{y}_{1},\lambda )<0\). The arrows indicate the correct forward direction of time. Black lines indicate when \(D_{2}y_{2}^{\star }(\overline{y}_{1},\lambda )=0\), and therefore Eq. (5.52) is singular and the direction of time changes in Eq. (5.53). The black lines also form a set of nullclines of Eq. (5.53), because at these points \(\dot{\overline{y}}_{1}=0\). Another nullcline is shown in green, where \(\dot{\lambda }=0\). At the intersection of the green and black lines, Eq. (5.53) has an equilibrium, which is a node. The weak stable manifold of this equilibrium is close to the green nullcline of \(\dot{\lambda }=0\).
5.9 Dynamics of the ReducedOrder Model on \(\varSigma _{\varepsilon }\)
In this section, we investigate the reducedorder model (4.57), which is the extension of the skeleton model by a single variable representing the dynamics in the normal bundle of \(\mathcal {M}_{\lambda }\). The dynamics on \(\varSigma _{\varepsilon }\) is given by Eq. (5.51) with parameters derived in Sect. 5.5. Proposition 4.27 shows that the reducedorder model captures the stability of \(\mathcal {M}_{\lambda }\) for \(\varepsilon =0\) well. Figure 9 confirms this: In the illustrated part of the phase space, the stability of the critical manifold of the corrected model and the reducedorder model is the same. The critical manifold is repelling where \(d^{}>0\) as per Proposition 4.25. The skeleton model does not capture the repelled trajectories and also displays singular dynamics for \(d^{}>0\) as shown in Fig. 10b. Here, we illustrate that the positive value of \(d^{\pm }\) for the reducedorder model resolves the singularities that occur in the skeleton model according to Proposition 4.24.
We first choose a small parameter value \(\varepsilon =10^{7}\) to show the qualitative differences between Eqs. (5.52) and (5.51). Figure 11a shows the twodimensional projection of the phase portrait ignoring variable \(\kappa \). When trajectories start in the shaded part with \(\lambda =1\), where the critical manifold is repelling, they quickly pass to \(\lambda =\,1\) without much change in \(y_{1}\), while \(\kappa \) exponentially explodes. Trajectories starting with \(\lambda =\,1\), in the region where the critical manifold is attracting, follow the manifold while being attracted to the stable node of (5.52) at the intersection of the green and black lines. At the node, the stability of the critical manifold changes and trajectories are again repelled with growing magnitude of \(\kappa \). This is illustrated in Fig. 11b. After passing the node, trajectories tend to either \(\lambda =\pm \,1\). It is then likely that trajectories will start a violent oscillation between \(\lambda =\pm \,1\), because they interact with the two repelling parts of the critical manifold. This dynamics has some resemblance to Fig. 10b except that there is no need to rescale time, since there is no division by \(d^{}\).
Increasing \(\varepsilon \) leads to less violent oscillations between \(\lambda =\pm \,1\), which eventually continues without reaching \(\lambda =\pm \,1\). Such a case is shown in Fig. 11d, where the oscillation is reduced to a single loop about the line where the critical manifold becomes repelling. For \(\varepsilon =1\), the dynamics becomes relatively slow for all variables and resembles that of typical friction oscillators with welldefined stick and slip phases. This phase portrait is shown in Fig. 11c. For \(\varepsilon \) sufficiently large the time scale of the normal dynamics (\(\kappa \) variable) becomes much longer than the dynamics of the rest of the variables, and therefore during a stick phase \(\kappa \) does not change much, which also means that the instability of the critical manifold loses its influence on the dynamics. Indeed, the leading characteristic root of \(\Delta (s)\) is a small perturbation of the zero root, and hence it is easily dominated by other time scales. In fact by removing nonlinearity (\(\varGamma =0\)), this root remains zero, and hence \(\kappa \) simply becomes an integral of other quantities without a dynamics of it own. In our example at \(\varepsilon =1\), \(\kappa \) is almost without its own dynamics. The justification why \(\varepsilon \) can be increased to \(\varepsilon =1\) can be found in Remark 5.10.
The conclusion from the analysis is that simply applying reduction to an invariant manifold is not sufficient, one needs to take into account at least a qualitative approximation of the normal dynamics. This is because the skeleton model (4.13) overemphasizes instabilities and turns them into singularities. The main component that makes the reducedorder model (4.57) well behaved is that \(d^{\pm }\) is positive in all parts of the phase space. For the nonlinear string example, \(d^{\pm }\) is the velocity jump of the contact point due to a unit jump in \(\lambda \), i.e., the contact force. Therefore in light of Newton’s second law, it is understandable why \(d^{\pm }>0\). In case we had found \(d^{\pm }=0\), the reducedorder model (4.57), including an approximation of the normal dynamics about \(\mathcal {M}_{\lambda }\), would not be necessary, the skeleton model would be sufficient.
6 Conclusion
In this paper, we have investigated PWS systems on Banach spaces with nondense domain of definition. Specific application areas that satisfy this assumption are the elastodynamics equations (Kausel 2006), delay equations (Diekmann et al. 1995) or agedependent population dynamics (Metz and Diekmann 1986). Such systems are different from other classes of PWS systems, because they can have unique solutions under general conditions. We were also able to construct a finitedimensional reducedorder model that inherits key properties of an infinitedimensional model. Nondense domain of definition can arise if the phase space is nonreflexive, for example, when the phase space consists of continuous, bounded or Lipschitz continuous functions. In some cases, boundary conditions can make the domain nondense (Neubrander 1988).
The key quantity that decides uniqueness of solutions is the normal discontinuity gap, which is due to discontinuous trajectories that systems with nondense domains have. For the linear and nonlinear string, the normal discontinuity gap represents the velocity jump of a contact point in response to a unit jump in force. The presence of the normal discontinuity gap allows the dynamics inside the switching manifold to become smooth. As a result, two new discontinuity boundaries arise, where trajectories can enter or leave the switching manifold. If the normal discontinuity gap is positive, trajectories cross the new discontinuity boundaries under general conditions.
Despite uniqueness of solutions, invariant manifolds that extend over the switching manifold do not exist. We have assumed the existence of an invariant manifold when the switching parameter of the vector field is constant. This invariant manifold does not persist when the switching parameter varies, but we have found that pieces of this manifold do persist, while discontinuities between the persisting pieces develop along the two new discontinuity boundaries. We have also shown that switching can make the invariant manifold repelling. However, in the example of the nonlinear string the invariant manifold is repelling only in a single direction, which can be captured by a scalar variable. We have constructed a reducedorder model that captures this instability. It remains to be shown under what conditions there is a spectral gap between the reduced model and the rest of the dynamics, so that the reducedorder model captures all the essential dynamics. We have only shown that the invariant manifold becomes repelling within the reducedorder model and within the infinitedimensional system under the same conditions through a real root (see Proposition 4.27).
While the theory presented is incomplete, we hope that the results in this paper will find applications in simulating PWS continuum systems. Using the reducedorder model instead of the skeleton model eliminates singularities and allows for a unique solution. This allows wellconditioned numerical schemes that lead to robust solutions unlike what is currently possible (Kane et al. 1999). While it is not proven that the reducedorder model fully captures all dynamics, we expect that this will be shown in the future either in general or under further conditions.
We have demonstrated the model reduction procedure on a bowed nonlinear string model. In this example, we have found that the skeleton model has nonphysical singularities, where the friction force between the bow and the string remains at its maximal limit. The skeleton model also has a singularity that resembles a folded node of singularly perturbed systems (Wechselberger 2005; Kristiansen 2017). After correcting the skeleton model with the dynamics that arises in the eliminated parts of the system due to switching, the pictures becomes clearer. It turns out that the correction is a largely decaying motion with the possibility of an instability along a onedimensional subspace. When this possible instability is taken into account, the model becomes free of singularities and the dynamics resembles what a friction oscillator would exhibit when the friction force is regularized (Sotomayor and Teixeira 1998).
Notes
Acknowledgements
The author would like to thank Alan R. Champneys and S. John Hogan for the feedback on the manuscript. He would also like to thank the anonymous reviewers who have helped with the clarity of the text and the accuracy of calculations. The author is especially thankful to Galit Szalai, who has proofread the final draft. Funding was provided by Engineering and Physical Sciences Research Council (GB) (Grant No. EP/K003836/1).
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