Markus–Yamabe Conjecture for Asymptotic Stability of Planar Piecewise Linear Systems

We present in this paper a detailed study on the Markus–Yamabe conjecture in planar piecewise linear systems. We consider discontinuous piecewise linear systems with two zones separated by a straight line, in which every subsystem is asymptotically stable. We prove the existence of limit cycles under explicit parameter conditions and give more different counterexamples to the Markus-Yamabe conjecture in addition to the counterexamples given by Llibre and Menezes. In particular, we consider continuous planar piecewise linear systems. For such a system with n + 1 zones separated by n parallel straight lines in phase space, we prove that if each of subsystems is asymptotically stable, then this system has a globally asymptotically stable equilibrium point, therefore the Markus–Yamabe conjecture still holds. Some examples are given to illustrate the main results.


Introduction and main results
The Markus-Yamabe conjecture [2,8,9,13,19,20,34,36,37] is that if all eigenvalues of the Jacobian matrix Df (x) have negative real part for any x ∈ R n where f : R n → R n is a C 1 vector field with f (0) = 0, then the origin is globally asymptotically stable. The conjecture is true when n ≤ 2, which was completely solved by Fessler [13], Gutierrez [21] and Glutsyuk [20], and false [1,11] when n ≥ 3. For two-dimensional cases, it is of interest to study whether the Markus-Yamabe conjecture still holds if the differentiability condition of the vector field is weakened. Following Llibre and Menezes [28], we call a planar piecewise linear (PWL) Markus-Yamabe system if every subsystem is asymptotically stable.
For continuous planar PWL Markus-Yamabe systems with two zones separated by a straight line, the authors in [5] proved that the unique equilibrium point in the separation straight line is globally asymptotically stable and the authors in [40] proved that the Markus-Yamabe conjecture still holds no matter where the unique equilibrium point is. Recently, Llibre and Menezes [28] proved that discontinuous planar PWL Markus-Yamabe systems with two zones separated by a straight line have at most one limit cycle and gave some examples with one limit cycle, thus presented a counterexample to the Markus-Yamabe conjecture. As shown in [28], the theory of A matrix is said to be Hurwitz if all of its eigenvalues have negative real part. If the left or right subsystem has a boundary node, then system (1.1) has no periodic orbits. The case that one of subsystems has a boundary focus was discussed in [28]. Note that if a 12 = 0 or b 12 = 0, then the component of the corresponding vector field normal to Σ has the same sign. Thus system (1.1) can not have any periodic orbits. In this paper we consider the case that each subsystem has no boundary singularities and the main results are shown in the following: Theorem 1 For system (1.1), suppose that all of the following conditions hold: (i) A and B are Hurwitz and a 12 b 12 > 0; (ii) p 1 < 0, q 1 > 0,x = (0,x 2 ) is an invisible fold point of the right subsystem andȳ = (0,ȳ 2 ) is an invisible fold point of the left subsystem.
Then the following holds, (a) if a 12 (x 2 −ȳ 2 ) < 0, then this system has only one periodic orbit which is a stable limit cycle and a repulsive sliding segment; (b) if a 12 (x 2 −ȳ 2 ) = 0, then each trajectory tends to a pseudo-equilibrium point as t → +∞; (c) if a 12 (x 2 −ȳ 2 ) > 0, then each trajectory enters an attractive sliding segment for some positive time and tends to a stable pseudo-equilibrium point as t → +∞.
Theorem 1 will be proved in Section 2. This theorem shows that system (1.1) undergoes a pseudo-Hopf bifurcation [7] atx 2 =ȳ 2 . Secondly, we consider continuous planar PWL systems with each subsystem asymptotically stable. Recall that if the planar PWL system is continuous on the dividing line and each of subsystems is asymptotically stable, then this system has a globally asymptotically stable equilibrium point [40]. This shows that the Markus-Yamabe conjecture is true. Now we consider more general case: is the Markus-Yamabe conjecture is true for a continuous planar PWL system with n + 1 zones separated by n parallel straight lines?
Theorem 2 System (1.2) has a globally asymptotically stable equilibrium point if for each i ∈ {1, 2, · · · , n}, A i is Hurwitz and The proof of this theorem is given in Section 3. Some examples and simulations are provided in Section 4.
2 Proofs of Theorem 1

Preliminaries
To pave the way for the proof of Theorem 1, we need some preliminaries. System (1.1) can be classified according to the type of the singular point [29]: a focus (F ); a diagonalizable node (N ) with distinct eigenvalues; an improper node (iN ), i.e. a non-diagonalizable node. The right subsystem has a boundary singularity if p 1 = 0, a virtual singularity if p 1 < 0 and a real singularity if p 1 > 0. The subscript v for the singularity type represents a virtual singularity, e.g., F v a virtual focus. Denote by ϕ A (t, x 0 ) and ϕ B (t, x 0 ) the solutions generated by the right and left subsystems given bẏ with initial condition ϕ A (0, x 0 ) = x 0 and ϕ B (0, x 0 ) = x 0 , respectively.
Without loss of generality, set a 12 > 0. Let It is easy to see that the positive trajectories of the points in I 2 under the flow of the left subsystem intersect Σ first in I 1 and the positive trajectories of the points in I 4 under the flow of the right subsystem intersect Σ first in I 3 .
Let P B denote the map that associates the points in I 2 with their points of first return to Σ under the flow of the left subsystem, which is given by where τ (x 2 ) > 0 is the time of first return of the point (0, x 2 ) to Σ. Similarly, the map P A that associates the points in I 4 with their points of first return to Σ under the flow of the right subsystem can be given by Ifȳ 2 ≥x 2 , then I 1 ⊂ I 4 , I 3 ⊂ I 2 and a Poincaré map can be defined by where P (x 2 ) = h 2 (h 1 (x 2 )). Based on the chain rule, we have In order to prove the statement (a) of Theorem 1, we first prove the following proposition.
Proposition 1 Suppose that a 12 > 0. If the left subsystem has a stable singularity of type L and the right subsystem of system (1.1) has a stable singularity of type R where L, The proof of this proposition is completed if we show the following several results. The proof of this lemma is straightforward and so is omitted.
Lemma 2 Suppose that a 12 > 0. If the right subsystem of system (1.1) has a stable singularity Proof If the right subsystem of system (1.1) has a stable singularity, then A is Hurwitz. According to Proposition 5 of [29], the right subsystem of (1.1) can be transformed into the following simple normal formẋ by a vertical lines-preserving linear transformation which means a linear transformation of variables in the plane preserving the vertical lines. If M = M 1 , then it is easy to get that According to the proof of Proposition 3 in [40], we have From the proof of Proposition 4 in [40], we have λt z and t z satisfies Based on the proof of Proposition 5 in [40], we have According to Lemma 1, µ A (z 2 ) is invariant under any vertical lines-preserving linear change of variables, time-rescaling or translation along the y axis. Therefore, this lemma is proved.
Based on Lemma 1 and Lemma 2, we have the following result.
Using Lemma 2 and Lemma 3, Proposition 1 can be proved easily.

Proof of Theorem 1
We now give the proof of Theorem 1.
Proof Without loss of generality, assume that a 12 > 0. Ifȳ 2 >x 2 , then it is easy to find that the set {(0, x 2 ) |x 2 < x 2 <ȳ 2 } is a repulsive sliding segment. Moreover, we have from Proposition 1 that for each point n=0 has a limit which corresponds to the fixed point of P A • P B where P A • P B is given by (2.2). Based on Banach Contraction Theorem [10], the continuous map P A • P B has only one fixed point in I 3 . This means that system (1.1) has only a periodic orbit. It follows from Proposition 1 that this periodic orbit is a stable limit cycle. Thus the statement (a) is proved.
Ifȳ 2 =x 2 , then the invisible fold point of the left subsystem is also the invisible fold point of the right subsystem. According to the definition of Filippov vector field [15], we define the Filippov vector field atx as f 0 (x) = 0 which meansx is a pseudo-equilibrium point of system (1.1) and P A • P B (x) =x. Similar to the proof of statement (a), we have that P A • P B has only one fixed point and this point is asymptotically stable. Therefore, each trajectory tends to the equilibrium pointx as t → +∞.
To prove the statement (c), let (2.4) Similar to the proof of Theorem 1 in [40], we get that each trajectory intersects I A,i or I B,i , i ∈ N for some positive time and will eventually fall into the segment {(0, x 2 ) ∈ R 2 |ȳ 2 ≤ x 2 ≤x 2 } at a large time and remain for all later times. According to Filippov convex method [14], [15], [26], along the attractive sliding set, the sliding motion can be defined bẏ It is easy to find that According to (2.6), we have that f 0 2 (x) has two zeros and only one zero x e = (0, x e,2 ) T lies in Σ as which is a pseudo-equilibrium point of system (1.1). Since we obtain that x e is asymptotically stable and each trajectory in the sliding segment tends to x e as t → +∞. Therefore, Theorem 1 is proved.
3 Proof of Theorem 2

preliminaries
It is easy to find that the linear transformation of variables can be done to change the angle of n parallel lines such that these n lines are parallel to the y axis.
n be the n switched lines where a n < a n−1 < · · · < a 2 < a 1 . Therefore, system (1.2) can be transformed into the following systeṁ Denote by Φ(t, x 0 ) the solutions of system (3.1) with initial condition Φ(0, x 0 ) = x 0 . The singular point p i is called a real singular point when p ∈ Γ i , a virtual singular point when p i / ∈ Γ i . To complete the proof of Theorem 2, we need to prove the following propositions. (3.2) If e T 1 A i e 2 = 0, then we find from (3.2) that Combining (3.3) and (3.4), we obtain that If p i , i = 1, 2, · · · , n are all virtual singular points, then according to (3.5) we get that p n−1,1 − a n−1 < 0, p n,1 − a n−1 < 0, p n,1 − a n < 0, p n+1,1 − a n < 0, which implies that system (3.1) has a real singular point p n+1 . Thus the case that all subsystems have virtual singular points can not happen, that is system (3.1) has at least one real singular point.
It follows that p i , 1 ≤ i ≤ n + 1, i = k 1 are virtual singular points.
Therefore, system (3.1) has a unique real singular point.
since γ is a periodic orbit of system (3.1).

Proposition 4 Consider a general linear systeṁ
be two lines parallel to the y-axes and Γ 0 be the region between Σ 0 1 and Σ 0 2 . If A is Hurwitz, then for each point Proof If a 12 = 0, then the proof is trivial and so is omitted. If a 12 = 0, according to For convenience, let If A = M 1 , it is easy to obtain that the x 1 -component of solution ϕ A (t, x 0 ) which is given by Considering the derivative of g(t) with respect to t, we get If δ 2 > −aδ 1 , then g ′ 1 (t) has a unique zero with g 1 (0) > 0. It follows that g 1 (t) increases first and then decreases with respect to t. In consideration of g 1 (t) → p 1 as t → +∞, there exists a t 1 > 0 such that g 1 (t 1 ) = m 1 .
If A = M 2 , it is easy to find that the x 1 -component of the solution ϕ A (t, x 0 ) is g 2 (t) = p 1 + e at (δ 1 cos t + δ 2 sin t).
Owing to the property of trigonometric function δ 1 cos t + δ 2 sin t, we obtain that there exists a t 3 > 0 such that g 2 (t 3 ) = m 1 .
Taking the derivative of g 2 (t) gives .
Therefore, the proof is completed.

Proof of Theorem 2
Proof If e T 1 A 1 e 2 = 0, then the conclusions are obviously true. Thus we only need to prove the case of e T 1 A 1 e 2 = 0. Without loss of generality, set e T 1 A 1 e 2 > 0, e T 2 A 1 e 2 < 0. According to Proposition 2, system (3.1) has a unique equilibrium point. We shall prove the case k 1 = 2. That is system (3.1) has a unique equilibrium point p 2 in Γ 2 . The other cases can be proved similarly, so we omit it.
For convenience, we define the following regions: R s 2 ={x 0 ∈ Γ 2 | the trajectory starting at x 0 remains in Γ 2 for all positive time}, R r i ={x 0 ∈ Γ i | the positive trajectory starting at x 0 leaves Γ i through the line Σ i−1 }, R l i ={x 0 ∈ Γ i | the positive trajectory starting at x 0 leaves Γ i through the line Σ i }, where i = 2, 3, 4, · · · , n.
Based on Proposition 4 and asymptotic stability of subsystems, we get the following results, as illustrated in Figure 1: 1) each trajectory starting in Γ 1 will enter one of the regions R s 2 , R r 2 , R l 2 for some positive time; 2) each positive trajectory starting in R l i will enter either R r i+1 or R l i+1 for every i ∈ {2, 3, · · · , n}; 3) each positive trajectory starting in Γ n+1 can only enter R r n and each positive trajectory starting in the region R r i must enter R r i−1 where i = 4, 5, · · · , n. Thus each trajectory starting in Γ n+1 or n i=3 R r i will enter R s 2 or R r 2 for some positive time; 4) each trajectory starting in R s 2 converges to p 2 as t → +∞. Fig. 1: Each trajectory of system (3.1) staring in one region may enter the region that the arrow points to for some positive time.
Note that if there exists an arbitrarily large positively invariant set, then according to Poincaré-Bendixson Theorem [39] each trajectory starting in this set approaches either the equilibrium point p 2 or a periodic orbit as t → +∞. According to Proposition 3, system (3.1) has no periodic orbits. Therefore, each trajectory starting in R 2 \{p 2 } approaches the equilibrium point p 2 as t → +∞.
To complete the proof, it suffices to prove that for any compact set K containing the equilibrium point p 2 , there exists a positively invariant set M K ⊃ K. Now we give the construction process of the positively invariant sets.
We need to define several maps. Recall that for any given positive definite matrices W i , i = 1, 2, 3, · · · , n + 1, there exist positive definite matrices Q i , such that is a quadratic Lyapunov function of i-th subsystem. Let = c}, c > 0 be the level set. It is easy to see that V −1 i (c) is convex. See [38] for more details.
It can be easily seen that for any given point Let γ 4 (x 1 , x 2 ) be the line between G 3 (x 1 ) and G 4 (x 2 ), and γ 8 (x 1 , x 2 ) be the line between x 1 and G 6 (x 2 ). It is easy to verify that the region R(x 1 , x 2 ) bounded by is a positively invariant set, as shown in Figure 2. According to the above construction process of positively invariant sets, we obtain that for any compact set K containing the equilibrium point p 2 , there exist x 1 = (x 1,1 , x 1,2 ) ∈ Σ n and x 2 = (x 2,1 , x 2,2 ) ∈ Σ 2 , such that the region R(x 1 , x 2 ) ⊃ K is positively invariant. We conclude that system (3.1) has a globally asymptotically stable equilibrium point. Therefore, the proof of Theorem 2 is completed.
We remark that if system (3.1) has a unique equilibrium point p k1 in Γ k1 where 1 ≤ k 1 ≤ n+1, then each trajectory starting in zone Γ i , i = k 1 leaving Γ i at some point must enter Γ k1 at a large time and remain for all positive time. It is easy to notice that solutions remaining in Γ k1 for all positive times will approach the equilibrium point p k1 asymptotically as t → ∞. If the subsystem of system (3.1) has a boundary equilibrium point p k2 in Σ k2 where 1 ≤ k 2 ≤ n, then each positive trajectory shall enter a positively invariant set in Γ k2 ∪ Σ k2 ∪ Γ k2+1 at a large time and tends to the equilibrium point p k2 asymptotically as t → ∞, as shown in the Figure 3.

Examples and Simulations
To illustrate the results, we give some examples.

Example 1
Consider the following discontinuous PWL systeṁ It is easy to see that (−4, 2) is the virtual stable node of the right subsystem, (1, q 2 ) is the virtual node of the left subsystem, (0, 6) is the invisible fold point of the right subsystem and (0, q 2 − 3 2 ) is the invisible fold point of the left subsystem.
If q 2 = 8, then it is easy to obtain that system (4.1) has only one limit cycle. The trajectory passing through (−0.1, 6) and the trajectory passing through (−0.1, 5) both approach the limit cycle as t → +∞, as shown in Figure 4a. By numerical calculation, we obtain that the limit cycle intersects Σ at (0, 6.8901) and (0, 5.3141) approximately.
If q 2 = 6, then the attractive sliding set is Σ as = {(0, x 2 ) ∈ R 2 | 9 2 < x 2 < 6} and each trajectory will enter the sliding segment for some positive time, as shown in Figure 4b. Moreover, the trajectory on the sliding segment will approach to a new asymptotically stable equilibrium point (0, 3 + √ 3) as t → +∞. If q 2 = 15 2 , then the invisible fold points of the left and right subsystem are the same point and each trajectory converges to this pseudo-equilibrium point as t → +∞, as shown in Figure  4c.

Example 2
Consider the following discontinuous PWL systeṁ Note that (−1, 2) is the virtual stable node of the right subsystem, (2, q 2 ) is the virtual improper node of the left subsystem, (0, 10 3 ) is the invisible fold point of the right subsystem and (0, q 2 − 4) is the invisible fold point of the left subsystem.
If q 2 = 9, then it is easy to get that system (4.2) has only one limit cycle. The trajectory passing through (−0.1, 5) and the trajectory passing through (0.1, 5) will both approach the limit cycle as t → +∞, as shown in Figure 5a. By means of numerical calculation, we find that the limit cycle intersects Σ at (0, 6.3694) and (0, 3.0330) approximately.
If q 2 = 7, then the attractive sliding set is Σ as = {(0, x 2 ) ∈ R 2 | 3 < x 2 < 10/3} and each trajectory enters the sliding segment for some positive time and eventually approaches to a new asymptotically stable point (0, 137−  It is easy to see that (1, 0) is the virtual stable node of the right subsystem, (−3, 2) is the real stable focus of the middle system and (− 7