A Poincaré–Bendixson theorem for hybrid dynamical systems on directed graphs


The purpose of this work is to obtain restrictions on the asymptotic structure of two-dimensional hybrid dynamical systems. Previous results have been achieved by the authors concerning hybrid dynamical systems with a single impact surface and a single state space. Here, this work is extended to hybrid dynamical systems defined on a directed graph; each vertex corresponds to a state space and each directed edge corresponds to an impact.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6


  1. 1.

    Barreiro A, Baños A, Dormido S, González-Prieto JA (2014) Reset control systems with reset band: well-posedness, limit cycles and stability analysis. Syst Control Lett 63:1–11. https://doi.org/10.1016/j.sysconle.2013.10.002, http://www.sciencedirect.com/science/article/pii/S0167691113002107

  2. 2.

    Bendixson I (1901) Sur les courbes definies par des equations differentielles. Acta Mathematica 21:1–88

    MathSciNet  Article  Google Scholar 

  3. 3.

    Chartrand G, Lesniak L, Zhang P (2010) Graphs & digraphs, 5th edn. A Chapman & Hall book, London

    Google Scholar 

  4. 4.

    Clark W, Bloch A, Colombo L (2019) A Poincaré–Bendixson theorem for hybrid systems. Math Control Relat Fields. https://doi.org/10.3934/mcrf.2019028

  5. 5.

    Crampin M, Heal B (1994) On the chaotic behaviour of the tent map. Teach Math Appl Int J IMA 13(2):83–89

    Google Scholar 

  6. 6.

    Dulac H (1923) Sur les cycles limites. Bulletin de la Société Mathématique de France 51:45–188

    MathSciNet  Article  Google Scholar 

  7. 7.

    Efimov D, Perruquetti W, Shiriaev A (2014) On existence of oscillations in hybrid systems. Nonlinear Anal Hybrid Syst 12:104–116. https://doi.org/10.1016/j.nahs.2013.11.005, http://www.sciencedirect.com/science/article/pii/S1751570X13000629

  8. 8.

    Goebel R, Sanfelice RG, Teel AR (2012) Hybrid dynamical systems. Princeton University Press, Princeton

    Google Scholar 

  9. 9.

    Lamperski A, Ames AD (2007) Lyapunov-like conditions for the existence of Zeno behavior in hybrid and Lagrangian hybrid systems. In: 2007 46th IEEE conference on decision and control, IEEE, pp 115–120

  10. 10.

    Lou X, Li Y, Sanfelice RG (2017) Existence of hybrid limit cycles and Zhukovskii stability in hybrid systems. In: 2017 American control conference (ACC), pp 1187–1192, https://doi.org/10.23919/ACC.2017.7963114

  11. 11.

    Morris B, Grizzle JW (2009) Hybrid invariant manifolds in systems with impulse effects with application to periodic locomotion in bipedal robots. IEEE Trans Autom Control 54(8):1751–1764. https://doi.org/10.1109/TAC.2009.2024563

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Perko L (1991) Differential equations and dynamical systems. Texts in applied mathematics, Springer. https://books.google.com/books?id=xftQAAAAMAAJ

  13. 13.

    Poincaré H (1886) Sur les courbes definies par les equations differentielles. J Math Pures Appl 2:151–217

    MATH  Google Scholar 

  14. 14.

    Rudin W (1976) Principles of mathematical analysis. International series in pure and applied mathematics, McGraw-Hill, New York. https://books.google.com/books?id=kwqzPAAACAAJ

  15. 15.

    Saglam CO, Teel AR, Byl K (2014) Lyapunov-based versus Poincaré map analysis of the rimless wheel. In: 2014 IEEE 53rd annual conference on decision and control (CDC), IEEE, pp 1514–1520

  16. 16.

    Sanfelice RG, Goebel R, Teel AR (2007) Invariance principles for hybrid systems with connections to detectability and asymptotic stability. IEEE Trans Autom Control 52(12):2282–2297. https://doi.org/10.1109/TAC.2007.910684

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Simic SN, Sastry S, Johansson KH, Lygeros J (2002) Hybrid limit cycles and hybrid Poincaré-Bendixson. IFAC Proc Vol 35(1):197–202. https://doi.org/10.3182/20020721-6-ES-1901.01104, http://www.sciencedirect.com/science/article/pii/S1474667015395252, 15th IFAC World Congress

  18. 18.

    Strogatz S (2014) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering. Studies in Nonlinearity, Avalon Publishing. https://books.google.com/books?id=yRAjnAEACAAJ

  19. 19.

    Tang JZ, Manchester IR (2014) Transverse contraction criteria for stability of nonlinear hybrid limit cycles. In: 53rd IEEE Conference on decision and control, pp 31–36. https://doi.org/10.1109/CDC.2014.7039355

  20. 20.

    Tu L (2010) An introduction to manifolds. Universitext. Springer, New York

    Google Scholar 

  21. 21.

    Zhang Z (1981) On the existence of exactly two limit cycles for the Liénard equation. Acta Math Sinica 24:710–716

    MathSciNet  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to William Clark.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was funded by NSF DMS 1613819 and AFSOR FA9550-18-10028.



A Technical lemmas

Before we state the following lemmas, we need to define the discrete \(\omega \)-limit set. Given a discrete dynamical system, \(T:S\rightarrow S\), the discrete \(\omega _d\)-limit set is defined as

$$\begin{aligned} \omega _d(x) = \left\{ y\in S : \exists N_n\rightarrow \infty ~s.t.~ \lim _{n\rightarrow \infty } \, T^{N_n}(x)=y\right\} . \end{aligned}$$

Lemma 1

(See, Lemma 4.2 in [4]) Let \(P:[a,b]\rightarrow [a,b]\) be \(C^1\) and injective. Then for all \(x\in [a,b]\), \(\omega _d(x)\) is either a single point or two points. In either case, all trajectories approach a periodic orbit.


By being injective, the map P is automatically monotone. Let \(I:=[a,b]\) and define the fixed-point set \(F:=\{x\in I : P(x)=x\}\) and consider \(x\in I\). If \(x\in F\) we are done, so assume \(x\not \in F\).

First, assume that P is (not necessarily strictly) increasing. Then, since \(P(x)>x\) implies \(P^2(x)>P(x)\) and \(P(x)<x\) implies \(P^2(x)<P(x)\), it follows (since \(x\not \in F\)) that \(\{P^n(x)\}\) is a monotone and bounded sequence and therefore converges. Hence, \(\omega _d(x)\) is a singleton.

To complete the proof, consider the case where P is (not necessarily strictly) decreasing. Since \(\omega _d(x;P)=\omega _d(x;P^2)\cup \omega _d(P(x);P^2)\), the previous paragraph shows that \(\omega _d(x)\) is either two points or a singleton. \(\square \)

Lemma 2

(See, Lemma 6.1 in [4]) Let \(S\subset {\mathbb {R}}\) be a finite set and \(P:S\rightarrow S\). Then, for all \(x\in S\), there exists \(N\ne M\) large enough such that \(P^N(x)=P^M(x)\) where

$$\begin{aligned} P^N(x) = \underbrace{P\circ P\circ \ldots \circ P}_{N\mathrm {~times}}(x). \end{aligned}$$

In particular, \(\omega _d(x)\) is a periodic orbit.


This follows immediately from the Pigeonhole principle. We have a function \(f_x:{\mathbb {Z}}^+\rightarrow S\), \(m\mapsto P^m(x)\), which cannot be injective. Therefore, there exists N and M such that \(f_x(N)=f_x(M)\). \(\square \)

Lemma 3

The set \(S_{(1,2)}^\infty \) is either a point or an interval.


We start by defining the sets \(S_{(1,2)}^m\). Let P be the return map, \(P:U\subset S_{(1,2)} \rightarrow S_{(1,2)}\). Then, \(S_{(1,2)}^1 := \{x\in S_{(1,2)} : P(x)\in S_{(1,2)}\}\), that is, points in \(S_{(1,2)}\) that return to \(S_{(1,2)}\) at least once. Similarly, \(S_{(1,2)}^m\) are points of \(S_{(1,2)}\) that return to \(S_{(1,2)}\) at least m times. This allows us to express \(S_{(1,2)}^\infty \) as

$$\begin{aligned} S_{(1,2)}^\infty = \bigcap _{m=1}^\infty \,S_{(1,2)}^m. \end{aligned}$$

If we can show that each \(S_{(1,2)}^m\) is an interval, then the desired result follows due to nesting. The base case is satisfied by (Q.4) via \(S_{(1,2)}^0 := S_{(1,2)}\). We will continue by induction. For each m, we can iterate by

$$\begin{aligned} S_{(1,2)}^{m+1} = S_{(1,2)}^m \cap P^{-1}\left( S_{(1,2)}^m \right) . \end{aligned}$$

Therefore, if \(S_{(1,2)}^m\) and \(P^{-1}\left( S_{(1,2)}^m \right) \) are both intervals, so is \(S_{(1,2)}^{m+1}\). All that is left to prove is \(P^{-1}\left( S_{(1,2)}^m \right) \) is an interval. Before we can study the structure of \(P^{-1}\left( S_{(1,2)}^m \right) \), we first define the family of functions

$$\begin{aligned} \mu _i : {\mathscr {D}}_i\subset {\mathscr {X}}_i\rightarrow S_{(i,i+1)}. \end{aligned}$$

where \({\mathscr {D}}_i = \{ x\in {\mathscr {X}}_i : \exists t>0 \text { with } \varphi _t^i(x)\in S_{(i,i+1)}\}\) and \(\mu _i(x) = \varphi _t^i(x)\in S_{(i,i+1)}\). These functions can be thought of as “projecting” onto the sets \(S_{(i,i+1)}\). What remains to show is that \(\varDelta _n(S_{(n,1)})\cap {\mathscr {D}}_1\) is an interval. If we can show this, we are done because \(\varDelta _{(n,1)}^{-1}\left( \varDelta _{(n,1)}(S_{(n,1)})\cap {\mathscr {D}}_1\right) \) is also an interval, and we can march backward around the cycle and end up on with a subinterval of \(S_{(1,2)}\).

Since \(S_{(n,1)}\) is diffeomorphic to an interval, so is the set \(\varDelta _{(n,1)}(S_{(n,1)})\). Let the map \(h:\varDelta _{(n,1)}(S_{(n,1)})\rightarrow [a,b]\) be a diffeomorphism. Define the points \({\tilde{a}}\) and \({\tilde{b}}\) as

$$\begin{aligned} \begin{aligned} {\tilde{b}}&= \max \{ x\in [a,b] : o^+_c(h^{-1}(x))\cap S_{(1,2)} \ne \emptyset \}, \\ {\tilde{a}}&= \min \{ x\in [a,b] : o^+_c(h^{-1}(x))\cap S_{(1,2)} \ne \emptyset \}. \end{aligned} \end{aligned}$$

We claim that \(\varDelta _{(n,1)}(S_{(n,1)})\cap {\mathscr {D}}_1 = h^{-1}[{\tilde{a}},{\tilde{b}}]\). By the choice of \({\tilde{a}}\) and \({\tilde{b}}\), we know that \(\varDelta _{(n,1)}(S_{(n,1)})\cap {\mathscr {D}}_1\subset h^{-1}[{\tilde{a}},{\tilde{b}}]\). To show the other direction, we consider the region bounded by the four curves: \(h^{-1}[{\tilde{a}},{\tilde{b}}]\), \({\tilde{o}}^+_c(h^{-1}({\tilde{a}}))\), \({\tilde{o}}^+_c(h^{-1}({\tilde{b}}))\), and \(S_{(1,2)}\) (where \({\tilde{o}}^+(x)\) is the forward orbit of x until it impacts \(S_{(1,2)}\)). By assumption (Q.5) and uniqueness of solutions, any curve starting on \(h^{-1}[{\tilde{a}},{\tilde{b}}]\) must intersect \(S_{(1,2)}\) before leaving the region described above (see, Lemma 10 in [4]). Therefore, \(h^{-1}[{\tilde{a}},{\tilde{b}}]\subset \varDelta _n(S_n)\cap {\mathscr {D}}_1\). \(\square \)

Lemma 4

Let \(({\mathscr {N}},{\mathscr {E}},{\mathscr {X}},S,\varDelta ,f)\) be a GHDS where \(({\mathscr {N}},{\mathscr {E}})\) is a cycle. Let \(x_0\in S_{(i-1,i)}{\setminus } \text {fix}(f_{i-1})\) be such that there exists a time, \(T_0>0\), where \(\varphi ^i_{T_0}(\varDelta _{(i-1,i)}(x_0))\in S_{(i,i+1)}\) (\(\varphi _t^i\) is the continuous flow corresponding to \({\dot{x}}=f_i(x)\) in vertex i). Additionally, assume that the flow intersects the surface, \(S_{(i,i+1)}\), transversely. Then there exists an \(\varepsilon > 0\) and a \(C^1\) function \(\tau :{\mathscr {B}}_{\varepsilon }(x_0)\cap S_{(i-1,i)} \rightarrow {\mathbb {R}}^+\) such that for all \(y\in {\mathscr {B}}_{\varepsilon }(x_0)\cap S_{(i-1,i)}\), \(\varphi _{\tau (y)}^i(\varDelta _{(i-1,i)}(y))\in S_{(i,i+1)}\).


Define the function

$$\begin{aligned} F_i:(0,+\infty )\times S_{(i-1,i)} \rightarrow {\mathscr {X}}_i, \end{aligned}$$

by \(F_i(t,x) = H_{(i,i+1)}(\varphi _t^i(\varDelta _{(i-1,i)}(x)))\). It follows from Theorem 1 in Section 2.5 in [12] that \(F_i\in C^1({\mathbb {R}}^+\times S_{(i-1,i)})\). This allows the use of the implicit function theorem. By the assumptions of the lemma, we know that \(F_i(T_0,x_0)=0\). Differentiating \(F_i\) with respect to time yields

$$\begin{aligned} \frac{\partial F_i}{\partial t}(T_0,x_0) = \left. \frac{\partial H_{(i,i+1)}}{\partial y}\right| _{y=\varphi _{T_0}^i(\varDelta _{(i-1,i)}(x_0))\in S_{(i,i+1)}} \cdot f_i\left( \varphi _{T_0}^i(\varDelta _{(i,i+1)}(x_0))\right) \ne 0.\nonumber \\ \end{aligned}$$

The first factor is nonzero because of assumption (G.4) and the second is nonzero because we are away from fixed points of the continuous flow. Their inner product is nonzero because of the transversality condition. This allows the use of the implicit function theorem (see, Theorem 9.28 in [14]) to show that there exists a neighborhood of \(x_0\) and a \(C^1\) function \(\tau \) with the desired properties. \(\square \)

Note that this implies that the maps \(P_i:U_i\subset S_{(i-1,i)}\rightarrow S_{(i,i+1)}\) are all \(C^1\). Because the composition of \(C^1\) maps is still \(C^1\), the map \(P:= P_1\circ \ldots \circ P_n : U\subset S_{(1,2)} \rightarrow S_{(1,2)}\) is \(C^1\).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Clark, W., Bloch, A. A Poincaré–Bendixson theorem for hybrid dynamical systems on directed graphs. Math. Control Signals Syst. 32, 1–18 (2020). https://doi.org/10.1007/s00498-019-00251-w

Download citation


  • Poincaré–Bendixson theorem
  • Hybrid systems
  • Periodic orbits
  • Directed graphs

Mathematics Subject Classification

  • 37E25
  • 37C27
  • 34A38
  • 37C10
  • 34A34