The Infinitesimal Phase Response Curves of Oscillators in Piecewise Smooth Dynamical Systems

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1 arxiv: v1 [math.ds] 11 Mar 2016 The Infinitesimal Phase Response Curves of Oscillators in Piecewise Smooth Dynamical Systems Youngmin Park Department of Mathematics, Applied Mathematics, and Statistics. Case Western Reserve University, Cleveland, Ohio, 44106, USA. Present address: University of Pittsburgh Kendrick M. Shaw Department of Biology Case Western Reserve University Cleveland, Ohio, 44106, USA. Present address: Department of anesthesia, critical care, and pain medicine, Massachusetts General Hospital, Boston, MA Hillel J. Chiel Department of Biology Case Western Reserve University Cleveland, Ohio, 44106, USA Peter J. Thomas Department of Mathematics, Applied Mathematics, and Statistics. Case Western Reserve University, Cleveland, Ohio, 44106, USA December 6,

2 Abstract We derive a formula for the infinitesimal phase response curve (iprc) of a limit cycle occurring in a piecewise smooth dynamical system satisfying a transverse flow condition. Discontinuous jumps in the iprc can occur at the boundaries separating subdomains. When the subdomain dynamics are linear, we obtain an explicit expression for the iprc. We present examples from cell biology (Glass networks) and neuroscience (central pattern generator models). 1 Introduction 1.1 Overview A stable limit cycle is a closed, isolated periodic orbit in a nonlinear dynamical system that attracts nearby trajectories [1]. Limit cycles arise in models of biological motor control systems [2, 3], excitable membranes [4, 5], sensory systems [6], neuropathologies such as Parkinsonian tremor [7] and epilepsy [8]. Chemical oscillations arise when the differential equations describing mass action kinetics admit a limit cycle [9]. Limit cycle dynamics arise not only in biological but also in engineered systems. For instance, phase locked loops play a role in radio and electronic communications devices [10], and control of oscillations is an important problem in mechanical and electrical engineering [11]. Formally, a nonlinear autonomous n-dimensional ordinary differential equation, d x(t) = F (x(t)), (1) dt has a T -periodic limit cycle if it admits a periodic solution γ with minimal period γ(t) = γ(t + T ), t R, (2) and an open neighborhood of γ (the basin of attraction, B.A.) within which all initial conditions give solutions converging, as t, to the set Γ = {γ(s) : s [0, T )}. (3) In many situations the multidimensional dynamics of a stable limit cycle oscillator can be accurately captured in a one dimensional phase model, 2

3 representing the fraction of progress around the limit cycle. The effect of weak inputs on the oscillator can be represented in terms of their effect on the timing of the limit cycle alone, the linear approximation to which is the infinitesimal phase response curve (iprc). The iprc has become a fundamental tool for understanding entrainment and synchronization phenomena in weakly driven and weakly coupled oscillator systems, respectively [12, 13]. The iprc gives the relative shift in timing per unit stimulus, as a function of the phase at which the stimulus occurs, in the limit of small stimulus size (Figure 1; Appendix C). The iprc is known in closed form in a handful of special cases: near a supercritical Andronov-Hopf bifurcation [5], near a saddle-node-on-invariantcircle (SNIC) bifurcation [14], and for certain piecewise linear oscillator models [15, 16, 17]. The form of the iprc near a homoclinic bifurcation is not known in general, cf. [14, 18, 17]. For general smooth nonlinear systems with limit cycle dynamics, one may obtain the iprc numerically using an adjoint method [19]. If oscillations arise from a dynamical system (1) where F : R n R n is a C 1 differentiable map, then the iprc is a T -periodic vector function of time that obeys an adjoint equation dz/dt = A(t)z(t), A(t) = (DF (γ(t))) (4) together with the boundary condition z(t) = z(t + T ). The adjoint operator A(t) is the transpose of the Jacobian matrix DF evaluated at the limit cycle γ(t) [5, 13]. This adjoint equation method breaks down for nonsmooth systems, i.e. for systems such that the Jacobian DF is not defined at all points around the limit cycle. In particular, if F is a piecewise smooth vector field, the Jacobian linearization may break down at the boundaries separating the regions within which the vector field is smooth. The theory of isochrons, and the subsequent iprc, is well developed for smooth systems [20], with considerably less literature for iprcs in nonsmooth differential equations, such as piecewise smooth systems (differential inclusions [21]). Infinitesimal PRCs have been computed explicitly in some planar systems [15, 17], but no formal development of the iprc exists for differential equations with discontinuous right hand sides. Nonsmooth oscillator models arise in both biological and engineered systems. Examples include planar nonlinear integrate-and-fire neural models 3

4 6 θ 4 2 x1(t) t Figure 1: Direct perturbation (red arrow) and phase response ( θ) for a limit cycle solution of a 2-D Glass network model. For a perturbation of size ε, we recover the iprc value as lim ε 0 θ/ε. See Appendix C. 4

5 [16], piecewise linear approximations to the Hindmarsch-Rose neural model [22], and models of anti-lock braking systems [23, 24]. Section 4 (Discussion) mentions additional examples. Existence of oscillatory solutions in piecewise smooth systems is a question of interest in its own right [25, 26, 21, 24, 27]. In this paper we provide a formula for the infinitesimal phase response curve of stable limit cycles that allows for discontinuities at the domain boundaries of piecewise smooth dynamical systems. In the case of piecewise linear systems, we obtain an explicit expression in terms of the system coefficients for each subdomain through which the limit cycle travels, and the tangent vectors of the surfaces separating the regions where the vector field definition changes (equation (26) in 2.2). To obtain these results we derive a jump condition satisfied by the iprc at the boundaries between subdomains. The main results reported here appeared previously in the Master s thesis of the first author [28]. 1.2 A 1D example To illustrate the necessity of including a jump condition in the phase response for piecewise smooth systems, consider the following one-dimensional example. Let f 1 and f 2 be smooth, strictly positive functions defined on the unit interval. Identify the interval with the circle and let x [0, 1) evolve according to dx dt = { f1 (x), 0 x < a f 2 (x), a x < 1, where 0 < a < 1 marks the location at which the rate law for x changes from f 1 to f 2. The rate law changes back to f 1 when x wraps around from one to zero again. The period of this oscillator is T = a 0 dx 1 f 1 (x) + dx a f 2 (x) We can define a phase variable φ(x) by the condition dφ/dt = 1/T, which gives the form { 1 x dx T 0 f φ(x) φ(0) =, 0 x a 1 (x) φ(a) + 1 x dx, a < x 1 T f 2 (x) a (5) 5

6 Here φ(0) is an arbitrary constant which we may set to zero, without loss of generality. The infinitesimal phase response curve Z x for this system describes the shift in timing of the oscillation upon making a small displacement in the x coordinate, as a function of position. The iprc is Z x = dφ dx = { Z1 (T f 1 (x)) 1, 0 x < a Z 2 (T f 2 (x)) 1, a < x 1. (6) The iprc has a finite jump discontinuity at the location a where the rate law for x changes, namely Z 2 (a) Z 1 (a) = 1 ( 1 T f 2 (a) 1 ). (7) f 1 (a) As a specific example, consider the rate laws f 1 (x) = 1 2αx, f 2 (x) = 1 2α(x 1/2), parameterized by α < 1, with switching point a = 1/2. For this example T = 1 ln ( ) 1 α 1 α, φ(x) = 1 ln ( 1 2 ln(1 α) 1 2αx) for 0 x 1/2, and ( ) φ(x) = 1 1 ln 1 for 1/2 x < 1. The phase response 2 2 ln(1 α) 1 2α(x 1/2) [ α curves in the first and second intervals are Z 1 (t) = 1 2αx(t) ln α] and [ α Z 2 (t) = 1 2α(x(t) 1/2) ln 1 1, 1 α] respectively. The phase φ(x) is continuous across the switch points x = 1/2 and x = 0. The jump in the infinitesimal phase response curve is Z 2 (T a ) Z 1 (T a ) = α2 1 α [ ln 1 1 α] 1; here Ta = T/2 is the time at which the trajectory reaches the switching point a = 1/2. Figure 2 illustrates this scenario for α = In the balance of this paper we calculate the form of the discontinuity in the infinitesimal phase response curve for a limit cycle in a piecewise smooth dynamical system in arbitrary dimensions. In the case of limit cycles arising in n-dimensional piecewise linear dynamical systems, we provide an explicit closed form for the iprc. We illustrate the application of this expression with examples arising in mathematical neuroscience and mathematical cell biology. 6

7 1 1 f(x) 0.5 x(t) φ(x) φ(x(t)) Z(x) 5 Z(t) x t Figure 2: 1D oscillator with piecewise smooth velocity. Top left: Velocity f(x) = 1 2αx for 0 x < 1/2 and f(x) = 1 2α(x 1/2) for 1/2 x < 1, with α = The period is T = 1 ln ( 1 α 1 α) Top right: trajectory x(t). Center: oscillator phase φ plotted versus space (x, left) and versus time (t, right). Bottom: infinitesimal phase response curve Z plotted versus space (x, left) and versus time (t, right). The phase φ is a continuous mapping from x to the circle [0, 1). The iprc jumps by [ α2 1 α ln α] at x = 1/2 and again at x = 0. 7

8 2 Methods 2.1 Definitions and Hypotheses Required to Solve the Adjoint Equation Over Differential Inclusions We introduce notation needed to discuss infinitesimal phase response curves for differential inclusion systems. For a general treatment of Filippov systems see [21]. Limit cycles and asymptotic phase. In a smooth system of the form (1), possessing a stable limit cycle γ, we associate a phase θ [0, 1) with points along the cycle such that dθ(γ(t))/dt = 1/T, (8) where T is the period of the limit cycle and θ(γ(t 0 )) = 0 is chosen arbitrarily. To each point x 0 in the basin of attraction (B.A.) we assign a phase θ(x 0 ) [0, 1) such that the trajectory x(t) with initial condition x(0) = x 0 satisfies lim x(t) γ(t + T θ(x 0)) 0. (9) t The isochrons are level curves of the phase function θ(x 0 ), and foliate the basin of attraction. For a stable limit cycle in a smooth dynamical system, the existence of the phase function is a well known consequence of results from invariant manifold theory [20]. Intuitively, isochrons indicate which points in the basin of attraction eventually converge to the limit cycle solution having a particular phase. Filippov systems. Let D be a path connected subset of R n. We say that an autonomous vector field F : D R n is piecewise smooth on D if there exist a finite number, R, of open sets D r such that the following hypotheses hold: H1. D r is nonempty, simply connected, and open for each r. H2. D i D j =, i j. H3. D R r=1 D r. H4. There exist analytic, bounded vector fields F r : D r R n such that for all x in D r, F r (x) = F (x). 8

9 Note that we require F r = F only on the interior of each open domain D r, while we require that each F r be smooth on the closure D r. The corresponding dynamical system dx dt = F (x) (10) is called a piecewise smooth dynamical system or a Filippov system [21]. Figure 3: Sections of the piecewise smooth dynamical system. Full domain boundaries are omitted for clarity. The first segment of the limit cycle begins at point γ 1 (0) p K 1 (marked ) on the surface Σ K 1. At time T 1 = t 1 the trajectory crosses surface Σ 1 2 at point γ 1 (t 1 ) p 1 2 γ 2 (0) (marked ). The jth segment of the limit cycle travels from γ j (0) Σ j 1 j to γ j (t j ) Σ j j+1 in time t j ; the crossing from region j to region j + 1 occurs at global time T j = t 1 +t t j and at location γ j (t j ) p j j+1 γ j+1 (0). The cycle returns to its starting point, p K 1 (marked ), at time T K = t t K. Thus T = T K is the period of the limit cycle. We further restrict our attention to Filippov systems satisfying the following assumptions: 9

10 1. The system (10) has a stable T -periodic limit cycle that crosses each boundary transversely with nonzero speed. Because the limit cycle could cross the same boundary multiple times, we introduce a separate label for each segment of the limit cycle lying between two boundary crossings (Figure 3). Thus, we label each piecewise smooth portion of the limit cycle by a number k = 1,..., K (see Eq. (16)). The boundary between the kth and (k + 1)st portions of the limit cycle is a surface (with boundary), denoted Σ k k+1. We denote the point at which the limit cycle crosses this boundary by p k k+1 Σ k k Each boundary is at least C 1 in an open ball B(p k k+1, c), centered at p k k+1 with radius c. From this assumption it follows that at each crossing point p k k+1 there exists a tangent hyperplane spanned by n 1 orthonormal basis vectors, denoted ŵ k k+1 i for i = 1,..., n 1, and a unique normal vector ˆn k k+1 directed from region k towards region k + 1. Moreover, the vector fields F k and F k+1 are assumed to satisfy a transverse crossing condition: F k (p k k+1 ) ˆn k k+1 > 0 (11) F k+1 (p k k+1 ) ˆn k k+1 > 0. (12) 3. The system (10) admits a finite period, stable limit cycle, with a welldefined phase function that can be extended to a continuous function θ :B.A. S 1 from the basin of attraction to the circle S 1 [0, 1], satisfying d dt θ(x(t)) = 1 T along trajectories within the basin of attraction. 4. The level sets of the phase function θ (the isochronal surfaces) form a continuous foliation of the basin of attraction. 5. The phase function, θ, is differentiable within the interior of each region for which it is defined. 6. At each boundary crossing point p k k+1 Σ k k+1, the directional derivative of the phase function is defined in the directions of each of the n 1 tangent vectors ŵ k k+1 i. 10

11 For smooth systems the iprc z(t) = θ(γ(t)) may be found using an adjoint equation, dz(t) = A(t)z(t), (13) dt where A(t) = DF T (γ(t)), the negative transpose of the linearization of the vector field F evaluated along the limit cycle γ. To derive (13) one considers an infinitesimal perturbation x(t) = γ(t) + y(t) where y(t) 1 and γ(t) is the limit cycle. As observed in [14], y(t) z(t) is independent of time; setting d (y(t) z(t)) = 0 (14) dt gives an operator equation leading to the adjoint. We note that (14) holds for piecewise smooth systems within the interior of each subdomain. We will develop a parallel method for piecewise smooth vector fields and solve for each limit cycle section γ k, that is, dz k (t) dt = A k (t)z k (t), (15) where A k (t) = DF k (γ k (t)) T, the negative transpose of the linearization of the vector field F k evaluated along the limit cycle portion γ k. The remaining challenge, and the contribution of the paper, is to establish the conditions relating the iprc on either side of each boundary crossing. 2.2 Solving the Boundary Problem of the Adjoint Equation We fix notation and define additional terms. Let F k denote the vector field in which the kth portion of the limit cycle resides, where each F k is numbered sequentially. The limit cycle, γ, is piecewise smooth, consisting of several curves γ 1, γ 2,..., γ K. As illustrated in Figure 3, each γ k spends a time t k in some domain D r. We write the limit cycle γ as a collection of curves, γ 1 (t), 0 = T 0 t < T 1, γ 2 (t T 1 ), T 1 t < T 2, γ(t) = (16). γ K (t T K 1 ), T K 1 t < T K, 11

12 where T i = i j=1 t j is the global time at which the trajectory crosses boundary surface Σ i i+1, and γ k (t k ) = γ k+1 (0) enforces continuity of the limit cycle. At a limit cycle boundary crossing between the kth and (k + 1)st portions of the limit cycle, there exist two adjacent vector fields F k and F k+1. These vector fields evaluated at the limit cycle boundary crossing are denoted F k,tk = lim F k (γ k (t)), t t k F k+1,0 = lim F k+1(γ k+1 (t)). t 0 + (17) In Eq. (17) and for the rest of this section, unless stated otherwise, the value t will refer to the time elapsed within a particular region between boundary crossings, i.e. for the kth limit cycle segment, t [0, t k ). The one-sided limits exist because each vector field is required to be smooth on the closure of its domain. The global iprc, z, will be defined in terms of the phase variable θ [0, 1), but we will view the local iprc z k (t) in terms of local time. The independent variable of these iprcs are related by z(θ) = z k (θt T k 1 ) = z k (t). (18) We will use the local time t in the proofs to follow, so that we only need to consider local dynamics at an arbitrary boundary crossing, without having to refer to the global dynamics. We define additional terms z k,tk and z k+1,0 by z k,tk = lim z k (t), t t k z k+1,0 = lim z k+1(t) t 0 + (19) where z k and z k+1 are the solutions to the adjoint equation (Eq. (15)) over vector fields F k and F k+1, respectively. As a rule, the first entry of the subscript for either z k,t or F k,t denotes the limit cycle section, and the second entry of the subscript (when explicit) denotes the local time. Theorem 2.1 Let γ be a piecewise smooth limit cycle of differential inclusions satisfying hypotheses H1 H4 and assumptions The iprc of γ satisfies the following boundary condition at the boundary p k k+1, z k+1,0 = M k k+1 z k,tk, (20) 12

13 where M k k+1 is an n n matrix, given as follows. Let b j denote the jth coordinate of the vector field F k,tk, let a j be the jth coordinate of the vector field F k+1,0, and let v i,j denote the jth entry of the ith tangent vector ŵ k k+1 i at the limit cycle boundary crossing, p k k+1. The matrix M k k+1 is a product of two matrices, M k k+1 = A 1 k+1 B k. Matrices A k+1 and B k take the form, 1 a 1 a 2... a n v 1,1 v 1,2... v 1,n A 1 k+1 = v 2,1 v 2,2... v 2,n,. v n 1,1 v n 1,2... v n 1,n (21) b 1 b 2... b n v 1,1 v 1,2... v 1,n B k = v 2,1 v 2,2... v 2,n.. v n 1,1 v n 1,2... v n 1,n Existence of the required matrix inverse is guaranteed by the transverse flow condition. See A.1 for the proof of Theorem 2.1. The following two corollaries specialize to the case of piecewise linear vector fields. In this case the vector field is not only piecewise smooth, but the iprc may be obtained in terms of matrix exponentials. Corollary 2.2 With the assumptions of Theorem 2.1 and affine linear vector fields F k, the initial condition of the iprc, z(θ), must satisfy where z 1,0 = Bz 1,0, (22) B = M K 1 e A KtK M 2 3 e A 2t 2 M 1 2 e A 1t 1, (23) t k is the time of flight of the kth portion of the limit cycle, e A kt k is the matrix exponential solution of the adjoint equation with A k = (DF k ) at time t k, and DF k denotes the Jacobian matrix of the vector field F k. Eq. (22) and the normalization condition, F 1,0, z 1,0 = 1 T, (24) yield a unique solution for the initial condition, z 1,0 R n. 13

14 Proof See A.2. Remark 2.3 The Jacobian matrices A k in Corollary 2.2 are distinct from the matrices A k appearing in the jump condition in Theorem 2.1. Corollary 2.4 Let t denote global time. Under the assumptions of Corollary 2.2, the iprc is given by e A1t z 1,0 e A1t Bz 1,0 0 t < T 1 e A 2(t T 1 ) B 1 z 1,0 T 1 t < T 2 z(t) = (25). e A K(t T K 1 ) B K 1 z 1,0 T K 1 t < T K where B 1 = M 1 2 e A 1t 1 B 2 = M 2 3 e A 2t 2 M 1 2 e A 1t 1. B K 1 = M K 1 K e A K 1t K 1 M 2 3 e A 2t 2 M 1 2 e A 1t 1, (26) and k T k = t i, k = 1,..., K, (27) i=1 where t i denotes the time of flight of the k th portion of the limit cycle. Proof See A.3. Remark 2.5 For examples of the matrices M k k+1 see Eqs. (38) (Glass network), (59), (63), (65) (Iris system), and (78), (80), (82) (3D piecewise linear central pattern generator). Shaw et al considered a piecewise linear system satisfying the assumptions of Theorem 2.1 and Corollary 2.2, where the vector field is piecewise differentiable but discontinous at subdomain boundaries. Our theory correctly captures all discontinuities of the iprc at the boundaries (Figure 14

15 8). Coombes (2008) considered systems with continuous vector fields not necessarily differentiable at domain boundaries, and analytically computed continuous iprcs for each system. In the following corollary, we show that a continuous iprc is a general property of limit cycles over continuous, piecewise smooth vector fields. Corollary 2.6 Under the assumptions of Corollary 2.2, if adjacent vector fields evaluated along the limit cycle, F k+1,0 and F k,tk, are continuous at the boundary p k k+1, then the matrix M k k+1 is the identity matrix. Proof See A.4. Remark 2.7 Our analysis therefore includes the iprc calculations of [15] and [17] as special cases. 3 Results We apply our analysis to three examples. The first is a 2D Glass network, a piecewise linear system obtained as the singular limit of a class of models for feedback inhibition and gene regulatory networks [29]. The second example is a planar system introduced in [17], motivated by investigations of heteroclinic channels as a dynamical architecture for motor control. The third example is a 3D piecewise linear system arising as a simplification of a nominal central pattern generator model for regulation of feeding motor activity in the marine mollusc Aplysia californica [30] and related to a Lotka-Volterra system with three populations [31]. 3.1 Two Dimensional Glass Network Glass, Perez, and Pasternack introduced a planar piecewise linear system as a model of feedback inhibition in a genetic regulatory circuit [32, 29]. Figure 4 illustrates several trajectories converging to a stable limit cycle in such a network. The concentration x 1 stimulates the production of x 2, while x 2 inhibits the production of x 1. One may also consider a macroscopic analogue in a predator-prey system, where x 1 is the prey and x 2 is the predator. 15

16 The model equations are: dx 1 dt = Λ 1[ξ(x 1 ), ξ(x 2 )] x 1, dx 2 dt = Λ 2[ξ(x 1 ), ξ(x 2 )] x 2, (28) where ξ : R {0, 1} is a Boolean function, { 1, if x 0 ξ(x) = 0, if x < 0, (29) and Λ i : {0, 1} {0, 1} R is a real-valued function defined by the table, ξ(x 1 ) ξ(x 2 ) Λ 1 Λ a 1 b a 3 b a 2 b a 4 b 4 Within each quadrant of the plane, the trajectories converge towards a target point located in the subsequent quadrant (Figure 4). In local time, the solution to Eq. (28) takes the form, where x i (t) = λ i + (x i (0) λ i )e t, i = 1, 2, (30) λ i = Λ i [ξ(x 1 (0)), ξ(x 2 (0))]. (31) The limit cycle attractor and its points of intersection with each axis is found by choosing an intial condition on the positive x 1 -axis and propagating Eq. (30) forward in time until it returns to the positive x 1 -axis. This method generates a Poincaré map, and the stable limit cycle intersection is a solution to a quadratic equation. We call this point of intersection p 4 1 because the limit cycle intersects the boundary between regions 1 and 4. ( ) a 2 a 4 b 1 b 3 a 1 a 3 b 2 b 4 p 4 1 =, 0. (32) a 2 b 1 b 3 a 2 b 1 b 4 + a 3 b 1 b 4 a 3 b 2 b 4 16

17 (a 1, b 1 ) y (a 4, b 4 ) x θ = 0 (a 2, b 2 ) (a 3, b 3 ) Figure 4: A model of feedback inhibition as discussed in Example 1 of Glass and Pasternack The limit cycle attractor (purple) traverses four quadrants, which serve as the four domains of the model. We call the first quadrant region 1, and because the solutions travel counter-clockwise, the second quadrant is named region 2, the third quadrant region 3, and the fourth quadrant region 4. Within each quadrant, trajectories are attracted to a target point outside the domain, as shown by the black and gray dashed lines. For example, in region 1, the limit cycle trajectory (purple) is attracted to the target point (a 1, b 1 ) until it hits the positive y-axis, at which point the limit cycle trajectory changes direction towards the next target point, (a 2, b 2 ). Two sample trajectories in the same region, one inside (blue) and one outside (red) the limit cycle, demonstrate that the purple loop is indeed a limit cycle attractor. The target points are (a 1, b 1 ) = ( 5, 11), (a 2, b 2 ) = ( 10, 4), (a 3, b 3 ) = (6, 10), and (a 4, b 4 ) = (10, 5). 17

18 Once the point p 4 1 is known, the other intersection points follow immediately. We also solve the limit cycle period in terms of the boundary crossing points, 4 T = t i (33) where t 1 = log t 2 = log t 3 = log t 4 = log i=1 ( a1 [ p 4 1 ] a 1 ( b2 [ p 1 2 ] b 2 ( a3 [ p 2 3 ] a 3 ( b4 [ p 3 4 ] b ) ) ) ), (34) and [ p k k+1 ]i denotes the ith coordinate of the vector p k k+1. Using the limit cycle boundary crossing points, we calculate the jump matrices M k k+1. Starting with the jump matrix between regions 4 and 1, M 4 1, the normal unit vector of this boundary is (0, 1) and we pick the tangent vector to be ( 1, 0). z 1,0 = M 4 1 z 4,t4 ( a1 [ ] ) 1 ( p = 4 1 b 1 1 a4 [ ] p ( ) 1 0 = z 4,t4. a 4 a 1 b 4 b 1 b1 1 b ) z 4,t4 (35) We now express z 4,t4 in terms of its initial condition, z 4,0, to construct the matrix B of Corollary 2.2. Note that the Jacobian matrix of each vector field is identical and takes the form, ( ) 1 0 DF k =, (36)

19 so it is straightforward to solve the adjoint equation (Eq. (15)): ( ) e t k 0 z k (t) = 0 e t z k k (0) = e A kt k z k (0) = e t k z k (0), (37) where t k is the time of flight of the kth portion of the limit cycle. (In this example A k = DFk T is just the identity matrix.) The remaining jump matrices M k k+1 for k = 1, 2, 3 are ( a1 a 2 b 1 b 2 a 2 M 1 2 = 1 0 ( 1 0 M 2 3 = M 3 4 = a 2 a 3 b 2 b 3 b3 ( a3 a 4 b 3 b 4 a ), ), ). (38) With all jump matrices computed, the matrix B of Corollary 2.2 is a product of four matrices and the scalar e T, where T = t 1 + t 2 + t 3 + t 4 is the period of the limit cycle: B = e T M 4 1 M 3 4 M 2 3 M 1 2. (39) The eigenvector associated with the unit eigenvalue is ( ẑ 1,0 = b ) 1, 1. (40) a 1 The unique initial condition vector, z 1,0, by Eq. (24) of Corollary 2.2 is z 1,0 = T ẑ 1,0 [ b 1 a 1 b 1 (a 1 [ p 4 1 ] 1 ) ]. (41) The full iprc is found by combining Eq. (26) with Eqs. (38),(37): e t z 1,0 ( ) 0 t < T 1 1 b 1 b 2 e t a ξ 1 (t) 2 a 0 2 z 1,0 T 1 t < T 2 ( a 1 ) b z(t) = e t 1 1 b 2 a ξ 2 (t) 2 a 2 a 3 z 1,0 T 2 t < T 3 b 3 p ( ) 22 e t 1 q 12 ξ 4 (t) z 1,0 T 3 t < T 4 q 21 q 22 (42) 19

20 z(θ) θ Figure 5: The numerical (dots) and analytical (lines) infinitesimal phase response curves of the planar Glass network model. The analytical solution to the adjoint equation is given by Eq. (42); the numerical iprc is calculated via direct perturbation. Blue curve: iprc for perturbations along the horizontal axis. Gray curve: iprc for perturbations along the vertical axis. where ξ 1 (t) = a 1 a 2 ( ) 2, a 1 + a 1a 3 b 2 b 4 a 2 a 4 b 1 b 3 a 2 b 1 (b 3 b 4 )+a 3 (b 1 b 2 )b 4 ξ 2 (t) = a 1b 2 2(a 2 b 1 (b 4 b 3 ) + a 3 b 4 (b 2 b 1 )) 2 a 2 b 2 1(a 1 b 2 (b 3 b 4 ) + a 4 b 3 (b 1 b 2 )) 2, (43) ξ 3 (t) = κa 1 a 2 3a 4 b 2 2 (a 2 b 2 1(a 2 a 4 b 1 a 3 (a 4 (b 1 b 2 ) + a 1 b 2 )) 2 b 3ˆγ), 20

21 and p 22 = a 2b 1 + a 3 (b 2 b 1 ), a 1 b 3 κ q 12 = a 1ˆγ(a 2 b 1 (b 3 b 4 ) + a 3 (b 1 b 2 )b 4 ), (a 2 a 3 )a 4 q 21 =, a 2 (b 3 b 4 ) + a 3 b 4 q 22 = a 4(a 2 b 1 + a 3 (b 2 b 1 )) a 1 (a 2 (b 3 b 4 ) + a 3 b 4 ), (44) and κ = a 3 (b 2 b 1 )b 4 + a 2 b 1 (b 4 b 3 )) 2, ˆγ = a 2 (b 3 b 4 ) + a 3 b 4, (45) and the global time intervals T k are determined by summing local times of flight, k T k = t i, k = 1,..., 4. (46) i=1 Figure 5 shows iprc obtained analytically together with the iprc obtained by direct numerical perturbation. 3.2 Piecewise Linear Iris System With Non-Uniform Saddle Values In [17], we analyzed the infinitesimal phase response curve (iprc) of a limit cycle arising from a heteroclinic bifurcation. The model, called the sine system in [17], exhibits a stable heteroclinic cycle (SHC) consisting of four planar saddle points related by four-fold rotational symmetry. By introducing a twist in the vector field, the sine system produces a one-parameter family of limit cycles that approach the SHC as the parameter goes to zero. The model arose in an investigation of the role of saddle fixed points in the regulation of timing in central pattern generator circuits. Using numerical simulations, we showed that infinitesimal perturbations to limit cycle trajectories near the heteroclinic bifurcation could cause long delays in the traversal about the limit cycle of the trajectory, leading to large phase offsets. This property 21

22 provides a mechanism by which animals could activate muscles for long times during control of repetitive underlying movements. As part of this study, [17] introduced a piecewise linear analogue (the iris system ) that has a structurally similar bifurcation in which an SHC gives rise to a one-parameter family of limit cycles similar to those in the sine system. The piecewise linear analogue consists of four regions with velocity fields equivalent, under successive 90-degree rotations, to the velocity field in the unit square, dŝ dt = λŝ, dû dt = û, (47) where ŝ represents the local coordinate along the stable manifold of the saddle at (0, 0), and û represents the local coordinate along the unstable manifold of the same saddle. As written, these dynamics represent the flow of region 1 in Figure 6. Existence of a limit cycle is guaranteed as long as the function, û λ û + a = 0, (48) has two roots [17]. The stable root, which we call u, represents the nontrivial entry coordinate of each region, and must be derived numerically for most choices of λ > 1. The piecewise linear iris system allows for the exact derivation of the infinitesimal phase response curve arbitrarily close to the heteroclinic bifurcation. In [17] we derived the iprc by summing an infinite series, a technique that does not readily generalize to systems with fewer symmetries. To illustrate the utility of Theorem 2.1, we alter the iris system so that different squares have different saddle values, λ k for k {1, 2, 3, 4}. The local coordinates must be defined separately for each region, dŝ k dt = λ kŝ k, dû k dt = û k, (49) where ŝ k and û k are the local coordinates of the kth modified iris square (see Figure 7). As in the iris system, the new system also satisfies hypotheses The first hypothesis calls for the existence of a limit cycle, which is 22

23 2 3 =0 θ = (a) (b) θ = 0 (c) (d) Figure 6: Iris system from [17] with uniform saddle values. The bifurcation parameter a determines the transition from stable limit cycles (blue closed curves in panels b-c) to a heteroclinic cycle (panel a). (a) a = 0, (b) a = 0.05, (c) a = 0.2. Red curve and arrow indicate an unstable limit cycle. (d) a = 0.255, limit cycles collide and disappear. Reproduced from [17]; Copyright 2012 Society for Industrial and Applied Mathematics. Reprinted with permission. All rights reserved. shown to exist numerically by constructing a Poincaré map. If the Poincaré map has a stable fixed point, then there exists a stable limit cycle. Without 23

24 loss of generality, we consider the return map function between regions 4 and 1, p(û 1 ) = ( [(ûλ a ) λ 2 + a ] λ3 + a ) λ4 + a. (50) The limit cycle entry coordinate of the southwest square is a value u 1 such that the equation, q(û) = p(û 1 ) û 1, (51) is zero at u 1, and such that u 1 is a stable fixed point of (50). The zero is readily found using Newton s method. We are now ready to apply Theorem 2.1 and Corollary 2.2 to derive the iprc of the modified iris system analytically. We begin our calculations with the southwest square of Figure 7; the NW, NE, and SE squares are labeled region 2, 3, and 4, respectively. In region 1, the global coordinates with their local coordinate counterparts are, ( ) ( ) λ1 (x(t) + α F 1 (x(t), y(t)) = 1 ) λ1 ŝ = 1, (52) y(t) + β 1 û 1 where α 1 and β 1 are constants that offset the vector field F 1. The Jacobian matrix J 1 of vector field F 1 is ( ) λ1 0 J 1 =, (53) 0 1 and in local time the solution to the adjoint equation (Eq. (15)) for region 1 is ( ) e λ 1 t 0 z 1 (t) = 0 e t z 1 (0). (54) The vector field F 2 of region 2 is ( ) x(t) + α2 F 2 (x(t), y(t)) = = λ 2 (y(t) + β 2 ) and the adjoint solution is z 2 (t) = ( û2 λ 2 ŝ 2 ), (55) ( ) e t 0 0 e λ 2t z 2 (0). (56) 24

25 2 3 θ = (a) (b) θ = 0 (c) (d) Figure 7: The modified iris system with non-uniform saddle values at various bifurcation parameter values. Unlike the original iris system [17], limit cycles do not have four-fold symmetry. Times of flight and ingress/egress coordinates differ from region to region. Limit cycle segments are numbered (1-4) moving clockwise from the southwest domain. Parameter values are (a) a = 0, λ 1 = 2.5, (b) a = 0.05, λ 2 = 1.5, (c) a = 0.2, λ 3 = 3, and (d) a = 0.33, λ 4 = 4. The unstable limit cycle in panel (c) (not shown), collides and destroys the stable limit cycle, resulting in a stable spiral (panel d). 25

26 For simplicity, we let u k and s k represent the nontrivial limit cycle entry and exit coordinates for each region, respectively. The final vector field value of F 1 evaluated along the limit cycle is F 1,t1 = ( λ 1 s 1, 1), (57) and the initial vector field value of F 2 evaluated along the limit cycle is F 2,0 = (u 2, λ 2 ), (58) where u 2 and s 1 are related by u 2 a + s 1. By Figure 7, the unit normal vector at the boundary crossing between vector fields F 1 and F 2 is ˆn 1 2 = (0, 1), and it follows that the unit tangent vector at the same boundary is ŵ 1 2 = ( 1, 0). Therefore, by Theorem 2.1, the jump matrix M 1 2 is ( ) 1 ( ) u2 λ M 1 2 = 2 λ1 s = 1 ( ) ( ) 0 λ2 λ1 s 1 1 λ 2 1 u ( ) 1 0 =. λ 1s 1 +u 2 λ 2 1 λ 2 Combiming Eqs. (54), (56), and (59) gives ( ) ( ) ( e t e λ 1 t 1 0 z 2 (t) = 0 e λ 2t 0 e t 1 λ 1s 1 +u 2 λ 2 1 λ 2 (59) ) z 1,0. (60) At the next boundary, the jump condition involves F 2,t2 and F 3,0 (Eq. (17)). The vector field F 2 is given above; the vector field F 3, written in local coordinates, takes the form, F 3 = ( λ 3 ŝ 3, û 3 ). (61) Therefore, the values F 2,t2 and F 3,0 are F 2,tf = (1, λ 2 s 2 ), F 3,0 = (λ 3, u 3 ), (62) where s 2 represents the exit position of the limit cycle in local coordinates along the right edge of region 2, and u 3 represents the entry position of 26

27 the limit cycle in local coordinates along the left edge of region 3 (note u 3 a + s 2 ). Therefore ( 1 λ 2 s 2 +u 3 ) M 2 3 = λ 3 λ 3. (63) 0 1 The iprc through region 3, given in terms of z 1,0, is ( ) ( e λ 3 t 0 1 λ 2 s 2 +u 3 ) ( ) z 3 (t) = 0 e t λ 3 λ 3 e t e λ 2t 2 ( ) ( ) 1 0 e λ 1 t e t z 1 1,0. λ 1s 1 +u 2 λ 2 1 λ 2 For the remaining boundaries, p 3 4 and p 4 1, the jump matrices are ( ) 1 0 M 3 4 =, M 4 1 = λ 3s 3 +u 4 1 λ 4 λ 4 ( 1 λ 1 λ 4 s 4 +u 1 λ ), (64) (65) and the solution to the adjoint equation for region 4, ( ) e t 0 z 4 (t) = 0 e λ 4t z 4,0. (66) To obtain the matrix B of Corollary 2.2, we combine Eqs. (64)-(66): B = 1 ( u1 (υ + ζ) + ξ u 1(u 1 ) ζ+ξ) λ 1, (67) ξ λ 1 (υ + ζ) u 1 ζ where υ = u 2 u 3 u 4, ξ = s 1 s 2 s 3 s 4 λ 1 λ 2 λ 3 λ 4, and ζ = s 1 λ 1 (u 3 u 4 + s 2 λ 2 (u 4 + s 3 λ 3 )). Note that for all k, we have made the substitutions e t k = uk and e λ kt k = 1/sk. The matrix B has unit eigenvalue with associated eigenvector, ( ) u1 ζ + ξ ẑ 1,0 = λ 1 (υ + ζ), 1, (68) where z 1,0 ẑ 1,0. We scale Eq. (68) using Corollary 2.2, leading to the unique initial condition of the iprc, z 1,0 = ẑ 1,0 [ ], (69) T u 1 u 1ζ+ξ (υ+ζ) 27

28 where the period T is the sum of local traversal times, T = 4 t k = k=1 4 log(1/u k ), (70) k=1 and u 1 is the numerically derived limit cycle initial condition. Each u k for k > 1 may be calculated iteratively from u 1. The exact iprc is found by combining Eq. (26) with the jump matrices Eqs. (59), (63), (65) and the matrix exponentials Eqs. (54), (60), (64), (66). Figure 8 shows the analytically obtained iprc plotted against a numerical iprc derived via direct perturbations. 3.3 Nominal Biting Model of Aplysia Californica Shaw et al. developed a model for a feeding pattern generator comprising three pools of motor neurons interacting with a nominal biomechanical model of the feeding apparatus of the marine mollusc Aplysia californica [33, 30]. The motor pools interact through quadratic (Lotka-Volterra) asymmetric lateral inhibitory coupling [34, 35], given by the equations db i dt = f i = b i b 2 i + µ ρb i b i+1, i = 1, 2, 3. (71) The state variable b i for i = 1, 2, 3, represents activation of the ith motor pool, the parameter µ 0 acts as a bifurcation parameter, and ρ represents coupling strength between adjacent nodes b i. The dynamics of the model depend on the parameters µ and ρ. When µ and ρ are zero, each equation reduces to three independent logistic curve equations. For some nonzero values of ρ, and µ = 0, the system forms an attracting stable heteroclinic channel (SHC) consisting of three saddle points with the unstable manifold of each saddle intersecting the stable manifold of the next saddle. When both µ and ρ are chosen appropriately, the heteroclinic connections break and the unstable manifolds spiral into a stable limit cycle [33]. For small positive values of µ, trajectories slow significantly while passing near the succession of saddle equilibria. This behavior allows the model to reproduce the extended dwell times in localized areas of phase space observed in in vivo recordings of Aplysia motor activity. Linearization of Eq. (71) about the three saddle equilibriua leads to a piecewise linear differential inclusion (Appendix B). Written in the order of 28

29 Figure 8: Modified iris system iprc for bifurcation parameters (from top to bottom) a =.01,.1,.2,.24. Of the two columns, the left column shows the numerical iprc (dots) superimposed on the analytical iprc (lines), with each curve plotted against phase. The blue dots and black line in the foreground represent the iprc for perturbations in the positive x direction and the gray dots and gray line represent the iprc for perturbations in the positive y direction. Right column: limit cycle for which the iprc is plotted on the left. As in [17], increasing absolute iprc value indicates enhanced sensitivity to small perturbations near the heteroclinic bifurcation. 29

30 regions 1, 2, and 3, respectively, we consider the following system: 1 x (y + a)ρ y + a, x y + a, x z a, (z a)(1 ρ) dr dt = (x a)(1 ρ) 1 y (z + a)ρ, y > x a, y z + a, z + a (72) x + a (y a)(1 ρ), z > x + a, z > y a, 1 z (x + a)ρ where r = (x, y, z), a 0 is the bifurcation parameter (analogous to µ in the smooth system (71)), and ρ is the coupling strength. The domains of Eq. (72) lie in equal thirds of the unit cube, which, when a = 0, all share an edge along the vector (1, 1, 1). For a = 0, the domain of region 1 is the convex hull of the vertices (1, 0, 0), (1, 0, 1), (1, 1, 0), and (1, 1, 1). Similarly, the domain of region 2 is the convex hull of the vertices (0, 1, 0), (1, 1, 0), (0, 1, 1), and (1, 1, 1), and the domain of region 3 is defined by the vertices (0, 0, 1), (1, 0, 1), (0, 1, 1), and (1, 1, 1). The saddle points of the system lie on a vertex of each domain, namely at (1, 0, 0), (0, 1, 0), and (0, 0, 1), for regions 1, 2, and 3, respectively. Each saddle point has a two dimensional stable manifold and one dimensional unstable manifold (Figure 9). Because the vector field is linear within each region, the stable manifold is a plane spanned by the two stable eigenvectors of the Jacobian for each region, and the unstable manifold is the (half) line in the direction of the unstable eigenvector. The two stable eigenvectors and the unstable eigenvector for region 1 are, respectively, (1, 0, 0), (0, 0, 1), ( ρ ) 2, 1, 0. (73) The vectors for the saddle in region 2 are, respectively, (0, 1, 0), (1, 0, 0), (0, ρ ) 2, 1, (74) and for region 3, ( (0, 0, 1), (0, 1, 0), 1, 0, ρ ). (75) 2 30

31 z z y x y x Figure 9: Piecewise linear model of Aplysia motor system for two bifurcation parameter values, ρ = 3, a = 0.02 (left) and a = (right) The three domain boundaries are defined according to Eq. (72), with (x, y, z) 1 1 when a = 0. Regions 1, 2, and 3 are colored magenta, green, and orange, respectively (color online). Each region includes one saddle point, denoted by a circle. The two arrows pointing into the saddle point indicate the two eigenvectors generating the stable manifold, and the arrow pointing away from the saddle indicates the eigenvector generating the unstable manifold (Eqs. (73)-(75)). The black loop in both figures represents the stable limit cycle, with the black arrow denoting the direction of flow. The tip of the black arrow marks the point p 3 1 at the boundary between regions 3 and 1. We define the phase at this point to be zero. 31

32 When a > 0 the heteroclinic cycle is broken, and the unstable manifold of each vector field F k flows into the boundary surface between vector fields F k and F k+1 (as opposed to flowing along the boundary edge when a = 0, where there is a nonempty intersection of the unstable manifold of F k and the stable manifold of F k+1 ). The vector fields F 1, F 2, and F 3, are shifted by vectors s 1, s 2, and s 3, respectively, where s 1 = (0, a, a), s 2 = (a, 0, a), s 3 = ( a, a, 0). (76) As in the example of the modified iris system, the limit cycle of this nominal piecewise linear SHC model is not analytically computable. The limit cycle coordinates are obtained numerically; we denote them p k k+1 = (η k+1, κ k+1, ν k+1 ), i.e., for the kth portion of the limit cycle, its initial value is the vector γ k,0 = (η k, κ k, ν k ). We now calculate the jump matrices M k k+1 for each boundary p k k+1, beginning with region 1. The normal vector at p 1 2 and its two tangent vectors, û 1 2 and ŵ 1 2, are ( ˆn 1 2 = 1 ) 1,, 0, 2 2 û 1 2 = (0, 0, 1), ( ŵ 1 2 = 1, 1 ), (77) Then the matrix M 1 2, written as a product of A 1 2 and B 1 of Theorem 2.1 is a 11 a 12 a 13 M 1 2 = a 21 a 22 a 13, (78) 0 0 a 33 where a 11 = η 2 + (ρ 1)κ 2 ρν 2, a 12 = 1 a(1 + ρ) 2κ 2 ρν 2, a 13 = a(ρ 2) + ρν 2, a 21 = 1 + a(1 2ρ) + (ρ 2)η 2 ρκ 2, a 22 = κ 2 a(ρ 2) + (ρ 1)η 2, a 33 = 1 + a(1 2ρ) + (ρ 1)η 2 κ 2 ρ 32 (79)

33 The remaining jump matrices are b M 2 3 = b 21 b 22 b 23, (80) b 21 b 32 b 33 where and where b 11 = a(2ρ 1) + ρη 3 (ρ 1)κ 3 + ν 3 1, b 21 = a(ρ 2) ρη 3, b 22 = ρη 3 κ 3 (ρ 1)ν 3, b 23 = a(ρ + 1) + ρη 3 + 2ν 3 1, b 32 = a(2ρ 1) (ρ 2)κ 3 + ρν 3 1, b 33 = a(ρ 2) (ρ 1)κ 3 ν 3, M 3 1 = c 11 c 12 c 13 0 c 22 0 c 31 c 12 c 33 c 11 = a(ρ 2) η 1 (ρ 1)ν 1, c 12 = ρκ 1 a(ρ 2), c 13 = 2ρa a + ρη 1 (ρ 2)ν 1 1, c 22 = 2ρa a + η 1 + ρκ 1 ρν 1 + ν 1 1, c 31 = ρa + a + 2η 1 + ρκ 1 1, c 33 = (ρ 1)η 1 + ρκ 1 ν 1. The solution to the adjoint equation for each region is e t 0 0 z 1 (t) = ρ sinh(t) e t 0 z 1,0, 0 0 e t(ρ 1) z 2 (t) = et(1 ρ) e t 0 z 2,0, 0 ρ sinh(t) e t e t 0 ρ sinh(t) z 3 (t) = 0 e t(ρ 1) 0 z 3, e t 33 (81), (82) (83) (84)

34 The time of flight for each portion of the limit cycle, t k, must be derived numerically for each k and for fixed parameter values a and ρ. We continue with ρ = 3 and a = 0.01 which are in the range giving a stable limit cycle (Figure 9). The matrix B of Corollary 2.2 is obtained from Eqs. (78), (80), (82), and (84). When a = 0.01, B takes the form B , (85) with a near-unit eigenvalue of approximately The associated eigenvector, ẑ 1,0 is, ẑ 1,0 ( , 1, ). (86) As in the previous two examples, the initial condition of the iprc, z 1,0, comes from scaling this eigenvector of matrix B, ẑ 1,0, by Eq. (24) of Corollary 2.2: ẑ 1,0 z 1,0 = T. (87) ẑ 1,0, p 3 1 The values p 3 1 and T represent the initial condition of the limit cycle and the total period of the limit cycle, respectively, and are found numerically. The iprc is found by combining (26) with the jump matrices (78), (80), (82) and matrix exponentials Eqs. (84). Figure 10 plots the resulting iprc together with the iprc obtained numerically by the direct perturbation method, showing good agreement. 4 Discussion In this paper we have derived the form of the discontinuity in the infinitesimal phase response curve at domain boundaries for a generic limit cycle arising in a piecewise smooth dynamical system. Our results apply broadly, because many systems have the structure we discuss. Nevertheless some caveats are in order. The iprc does not always capture the response to a stimulus. In situations in which the linear approximation to the asymptotic phase function breaks down, for instance when the stimulus drives the oscillator s trajectory close to the stable manifold of a saddle point on the boundary of the basin of 34

35 z(θ) θ Figure 10: Aplysia motor control model iprc, for parameters ρ = 3 and a = The blue dots, gray squares, and light green diamonds represent the numerical iprc, found by the direct method of perturbations, of the first, second, and third components of the iprc, respectively. The solid black, solid gray, and dashed light gray lines represent the analytical iprc derived using Theorem 2.1 and Corollary 2.2 of the first, second, and third components of the iprc, respectively. attraction, mechanisms such as shear-induced chaos can lead to complicated responses to periodic forcing that cannot be predicted via iprc analysis. This scenario can arise near the homoclinic bifurcation in the Morris-Lecar model, for example [18]. Nevertheless, in many systems the iprc plays an important role in understanding oscillator entrainment and synchronization. Limit cycles with a sliding component, see [36], do not satisfy the trans- 35

36 verse boundary crossing assumption, and our theory does not apply. Moreover, our results presume the existence and uniqueness of an asymptotic phase function on the basin of attraction of an oscillator; and that the phase function is continuous throughout the basin of attraction, C 1 within the domains, and differentiable at the domain boundaries in all directions tangent to the boundary (Assumptions 3. 6.). For smooth systems, existence, uniqueness and differentiability of the phase function follows from classical results of invariant manifold theory [20]. To our knowledge, parallel results have not been established in full generality for piecewise smooth systems. However, for the examples considered here, the theoretically obtained iprcs coincide with the iprcs obtained through direct numerical simulation. Specifically, we validated our calculations for three piecewise linear examples: a 2-dimensional Glass network, a family of 2-dimensional piecewise linear models verging on a heteroclinic cycle, and a 3-dimensional piecewise linear model motivated by motor control in the Aplysia californica feeding system. In each example we computed the iprcs analytically and found strong agreement with numerically derived iprcs (Figures 5, 8, and 10). In the examples we consider, the decomposition of the vector field into piecewise linear domains is specified in the statement of the original system. The approximation of limit cycles in a smooth system, with limit cycles in a piecewise linear system, has been investigated in a general setting [37]. However, there is no a priori heuristic for how to approximate an arbitrary nonlinear system with a piecewise linear approximation. The extension of results from classical dynamical systems to nonsmooth systems is an active area of research with applications in a wide variety of contexts. Carmona et al. studied a canonical form for limit cycles in planar PWL dynamical systems with two regions [38], using Melnikov methods to study existence and bifurcations of limit cycles. Ponce et al. studied bifurcations leading to limit cycles in PWL planar systems in [39]. Existence of limit cycles has been shown for planar PWL systems with two regions in [26] and [27], for planar PWL systems with an arbitrary but finite number of separate regions in [25], and for a PWL system in R 4 with three regions in [40]. Stability of piecewise linear limit cycles in R n with m + 1 regions is analyzed in [41] using Poincaré map techniques. The review [42] discusses necessary and sufficient conditions for asymptotic stability of piecewise linear systems in R n ; [43] adapts Lyapunov functions for piecewise linear systems. Limit cycles in piecewise linear systems occur not only in biology but also in control engineering [44]. Piecewise linear systems arise naturally in anti- 36

37 lock braking systems [24], which are themselves engineered to produce limit cycle oscillations [23]. Piecewise smooth relay feedback systems, first used in heating [45] and more recently in (PID) control [46], can give rise to limit cycles. The exact conditions for limit cycle existence in relay feedback systems is given in [47]. Many piecweise linear biological models exist as well. A piecewise linear version of the Fitzhugh-Nagumo model (also called the McKean model) and a piecewise linear version of the Morris-Lecar model are studied in [15]. The authors in [48] convert the Hindmarsch-Rose model into a piecewise linear version and analyze its stability. Gene regulatory networks are a classic example of piecewise linear models exhibiting limit cycle oscillations [29], and a subject of ongoing research. For instance, [49] analyzes the stability of synchronous periodic solutions, assuming weak symmetric coupling of two Glass networks. Rigorous investigations of Glass networks have considered them within the framework of differential inclusions [50, 51]. To facilitate construction of networks with customized dynamics [52] systematically classified cyclic attractors on Glass networks with up to six switching units. In summary, there is a rich collection of contexts in which piecewise linear and piecewise smooth systems arise. 5 Conclusion The infinitesimal phase response curve provides a linear approximation to the geometry of the asymptotic phase function in the vicinity of a stable limit cycle. The classical method for obtaining iprcs from the adjoint [53] breaks down with nonsmooth dynamics because the Jacobian may not be well defined at the domain boundaries. In this paper we have introduced a general theory for the iprcs for limit cycles arising in piecewise smooth systems, provided the limit cycle intersects the domain boundaries transversely and the boundaries are smooth at the points of intersection. In the case of piecewise smooth system which are also continuous across the domain boundaries, we obtain continuous iprcs. Discontinuities in the iprcs may arise when the vector field is discontinuous across domain boundaries, and our analysis provides the explicit form of the discontinuity. Our results are consistent with, and extend, existing findings, such as [15]. Because piecewise smooth and piecewise linear systems arise in a wide variety of fields, from biology to engineering, our analysis has the potential for broad application. 37