The qualitative analysis of a class of planar Filippov systems

Nonlinear Analysis 73 (2010) 1277–1288

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The qualitative analysis of a class of planar Filippov systemsI Shuliang Shui ∗ , Xuyang Zhang, Jingjing Li College of Mathematics, Physics and Information Engineering, Zhejiang Normal University, Jinhua 321004, Zhejiang, China

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Article history: Received 10 August 2009 Accepted 24 April 2010

We use the theory of differential inclusions, Filippov transformations and some appropriate Poincaré maps to discuss the special case of two-dimensional discontinuous piecewise linear differential systems with two zones. This analysis applies to uniqueness and nonuniqueness for the initial value problem, stability of stationary points, sliding motion solutions, number of closed trajectories, existence of heteroclinic trajectories connecting two saddle points forming a heteroclinic cycle and existence of the homoclinic trajectory © 2010 Elsevier Ltd. All rights reserved.

MSC: 34A25 34A34 34A60 34C37 Keywords: Filippov system Homoclinic trajectory Heteroclinic cycle Closed trajectory Sliding motion solution

1. Introduction There are a lot of applications in different parts of science, such as in mechanics, in engineering, in the social and financial sciences, etc., which are modelled by piecewise smooth systems of ordinary differential equations. These systems are given by x˙ = f (x, µ), with f = fi : Rm+n → Rn being smooth on a finite number of domains Gi , i = 1, . . . , n, and losing smoothness on the boundaries Mij , between adjacent domains Gi and Gj . Thereby, µ ∈ Rm is a parameter vector and x ∈ Rn , the state vector. Many examples come from mechanical systems with dry friction between two surfaces, noise generation in relay wheels, chattering of machine tools, squealing car brakes, electric circuits and so on; e.g., see [1–10]. For convenience, we study a planar piecewise linear system with a discontinuous line and for simplicity, we set piecewise linear differential system (1) with X˙ = A± X + sgn(w T X )C

(1)

where

 A± =

a± 11 a± 21

a± 12 a± 22



,

C = (c1 , −c2 )T ,

w = (w1 , w2 )T ,

  X =

x . y

I Project supported by the National Natural Science Foundation of China (No 10771196) and the Zhejiang Provincial Natural Science Foundation of China (No Y7080198). ∗ Corresponding author. Tel.: +86 579 82282629; fax: +86 579 82282629. E-mail addresses: [email protected], [email protected] (S. Shui).

0362-546X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2010.04.053

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Many papers study this system with A+ = A− (see [1,11–14,7,15,16,9,17]), and the first analytical results on this systems go back to Andronov et al. in 1930 who gave the conditions for sliding motion solutions. Most papers deal with piecewise linear differential systems with right-hand sides discontinuous in R2 ; either the systems considered are continuous (see [18–20]) or only particular cases are investigated (see [1,21,22]). In [1], the behavior of Eq. (1) with





0 0

A+ =

1 , −1



A− = A+ ,

C =



0 , −1

w=

and

  1

ρ

,

with respect to the parameter ρ was studied. In [21], the author studied Eq. (1) with

 A+ =



0

1 , 2k

−µ



A− = A+ ,

C =

0

−v



,

and

w=

  1

ρ

,

where µ, k, v > 0 and ρ ∈ R,

considered the existence and non-existence of periodic solutions of this system, and obtained neither unstable closed trajectories nor periodic trajectories with sliding motion. In [21], the authors studied Eq. (1) with

 

A− = A+ ,

C =

b1 b2

,

and w =

 

1 , 0

applied the theory of differential inclusions and the method of point transformation to study the stability of equilibrium points, existence of sliding motion, number and stability of periodic trajectories and existence of homoclinic solutions in systems of focus type. In [19], the authors applied the theory of differential inclusions and an appropriate Poincaré map to study the stability of equilibrium points, existence of sliding motion, number and stability of periodic trajectories and existence of homoclinic solutions in the systems of saddle type, and obtained the following results. (1) If δ = 0, b1 = 0 and b2 < 0, system (1) has a heteroclinic cycle and for any p0 (0, y0 ) (where − system (1) has a closed trajectory. (2) If δ = 1, b1 > 0 and b2 < 2pb1 , system (1) has exactly one stable closed trajectory. (3) If δ = 1, b1 > 0 and b2 = 2pb1 , system (1) has a heteroclinic cycle. (4) In all the other cases there are no closed, heteroclinic or homoclinic trajectories.

b2

√ p

−p

< y0 <

b2

√ −p p

),

In [20], the author studied Liénard equations for the existence conditions for type I and type II periodic orbits. Following [23] we know the definitions of type I periodic orbits and type II periodic orbits. (1) The closed trajectoryTγ lies in G+ and G− and crosses M twice, i.e. γ ⊂ G+ M G− , γ G± 6= Ø, and there exist y1 < y2 such that γ M = {(0, y1 ), (0, y2 )}; this closed trajectory γ is also called a type I closed trajectory. ¯ + and/or G¯ + and the corresponding solution stays for a whole time interval in M (‘sliding’ (2) The closed trajectory γ lies in G S ¯ ¯ ¯ along the y-axis). This T means that γ ⊂ G± or γ ⊂ G+ G− and there exist at least y1 < y2 such that {(0, y)2 : y1 ≤ y ≤ y2 } ⊂ γ M; this closed trajectory γ is also called a type II closed trajectory, where G+ = {(x, y) ∈ R : ¯ ± is the closure set of G± . Furthermore, we define the line of discontinuity: x > 0}, G− = {(x, y) ∈ R2 : x < 0}, and G M = {(x, y) ∈ R2 : x = 0}.

S

S

T

± + + − − T Now, we consider the case a± 11 = 0, a12 = 1, a21 = a, a22 = b, a21 = A, a22 = B, w = (1, 0) in the system (1). And system (1) changes to

X˙ =



A+ X + C A− X − C

x>0 x<0

(2)

where

 A+ =

0 a



1 , b

 A− =



0 A

1 , B

C = (c1 , −c2 )T ,

  X =

x . y

We set f + (x, y) = f (x, y) = −

 

y + c1 ax + by − c2



y − c1 Ax + By + c2

 =



f1+ f2+

 =



f1− f2−



,

(x, y) ∈ G¯ + ,

,

(x, y) ∈ G¯ − ,

where a, A > 0, b, B, c1 , c2 are parameters. G± and M are as mentioned above. Replacing the right-hand side of Eq. (2) with the set-valued function

 + {f (x, y)}, (x, y) ∈ G+ F (x, y) = {α f + (x, y) + (1 − α)f − (x, y) : α ∈ [0, 1]}, {f − (x, y)}, (x, y) ∈ G −

(x, y) ∈ M

(20 )

S. Shui et al. / Nonlinear Analysis 73 (2010) 1277–1288

1279

with f ± (x, y) as above, we obtain the differential inclusion

  x˙ ∈ F (x, y). y˙ 0

From the definitions (2) and (2 ) it follows that F (x, y) =

 A− +

Sgn(x) + 1

  x (A+ − A− ) + Sgn(x)C , y

2

where if x > 0 if x = 0 if x < 0.

( {1}, Sgn(x) = [−1, 1], {−1},

 x˙ y˙

∈ F (x(t ), y(t )). If

 0 0

x(t ) y(t )





, defined on an I ⊂ R which satisfies   ∈ F (x, y) = A− + Sgn(2x)+1 (A+ − A− ) yx + Sgn(x)C , then (x, y) ∈ R2 is an equilibrium point of

Following [23], we define a solution of Eq. (2) as an absolutely continuous function



system (2). Equilibrium which belong to M are sometimes called quasi-equilibrium. We define the sliding motion interval Is as the subset of M in which the vector field is not transversal, i.e. Is = {0} × {y ∈ R2 : f1+ (0, y)f1− (0, y) < 0}. We obtain that (0, 0) is a quasi-equilibrium of system (2), with the sliding motion intervals in M : Is = {(0, 0)} for c1 = 0 and the sliding motion intervals in M : Is = {(0, y) : −|c1 | ≤ y ≤ |c1 |} for c1 6= 0, and if y ∈ [−|c1 |, |c1 |], we define f (y) = − 0

y(b − B)

 y−

2c1

2c2 + (b + B)c1

2c2 +(b+B)c1 b −B

We only consider the case |



b−B

.

| > |c1 | in this paper.

In this paper, we provide a general class of systems satisfying | following one.

2c2 +(b+B)c1 b −B

| > |c1 |. The main result of this paper is the

Theorem 1. If λ1 , λ2 , ξ1 , ξ2 > 0, c1 ≥ 0, then system (2) has the times s1 , s2 > 0 such that (i) x+ p (s1 ) = 0, y+ p (s1 ) =

−c1 ξ1 +ξ1 e−ξ2 s1 c1 +e−ξ2 s1 ξ2 y−c2 +e−ξ2 s1 c2 , ξ2

(ii) x− p (−s2 ) = 0,

e−λ1 s2 λ y+e−λ1 s2 c −c −e−λ1 s2 λ c +λ c

2 1 2 1 2 2 1 y− , p (−s2 ) = λ1 and it has the following results: c (λ2 −ξ1 ) (a) if ξ1 + ξ2 = λ2 + λ1 , c1 = ξ 2λ +λ , then system (2) has a heteroclinic cycle and 2 2 1 ξ1 ¬ if c1 > 0, then (2) doesn’t have any closed trajectory, ­ if c1 = 0, i.e. ξ2 = λ1 , λ2 = ξ1 , then (2) has a closed trajectory through the point p(0, y) (where y ∈ (yΠ ((0,−c −

(ξ −λ )c

(λ −ξ )c

(b) ¬ if c1 ∈ ( ξ λ2 +λ1 ξ2 , ξ λ2 +λ1 ξ2 ) and s1 = 1 1 2 2 1 1 2 2 trajectory, (λ −ξ )c

(ξ −λ )c

­ if c1 ∈ ( ξ λ2 +λ1 ξ2 , ξ λ2 +λ1 ξ2 ) and s2 = 1 1 2 2 1 1 2 2 (c) if one of the conditions (ξ −λ )c

ξ (λ c ξ2 −c2 ξ1 +λ2 c1 ξ1 −λ1 c2 ) ln ξ 2(−ξ1 c1 +λ 1 2 2 1 c1 ξ1 +c1 ξ2 λ2 +λ1 c2 )

λ1

(ξ −λ )c

(λ −ξ )c (ξ −λ )c min{ ξ λ2 +λ1 ξ2 , ξ λ2 +λ1 ξ2 } 1 1 2 2 1 1 2 2

λ1

, λ1 cλ12+c2 )),

, where x+ (s1 ) = 0, then (2) has a homoclinic mu

, where x− (s2 ) = 0, then (2) has a homoclinic trajectory, mu +

ξ (λ c ξ2 −c2 ξ1 +λ2 c1 ξ1 −λ1 c2 ) ln ξ 2(−ξ1 c1 +λ 1 2 2 1 c1 ξ1 +c1 ξ2 λ2 +λ1 c2 )

ξ2

1 ))

−

λ (ξ λ c +ξ λ c +ξ c +λ c ) ln λ1 (ξ1 c 2λ1 +c 2 λ 1 ξ1 +λ2 c2 −ξ2 c2 ) 2 1 1 1 1 2 2 1 2 2 2

­ c1 ∈ ( ξ λ2 +λ1 ξ2 , ξ λ2 +λ1 ξ2 ) and s02 < 1 1 2 2 1 1 2 2 ® c1 <

ξ2

(λ −ξ )c

¬ c1 ∈ ( ξ λ2 +λ1 ξ2 , ξ λ2 +λ1 ξ2 ) and s01 < 1 1 2 2 1 1 2 2 (λ −ξ )c

λ (ξ λ c +ξ λ c +ξ c +λ c ) ln λ1 (ξ1 c 2λ1 +c 2 λ 1 ξ1 +λ2 c2 −ξ2 c2 ) 2 1 1 1 1 2 2 1 2 2 2

, where s01 satisfy x+u (s01 ) = 0, m−

, where s02 satisfy x−u (s02 ) = 0, m+

holds, then system (2) has at least one closed trajectory, (d) if one of the following conditions holds, then system (2) has at most one closed trajectory: ¬ bB ≥ 0 and b + B 6= 0, q q ­ Bb < 0, (2c1 + bx1 − Bx0 )(2c1 − ba a2 x21 − 2Aax20 ) < 0 (or (2c1 + bx1 − Bx0 )(2c1 − AB A2 x20 − 2Aax21 ) < 0) and one of the following conditions holds:

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(I) (λ1 ξ1 − ξ2 λ2 )(ax21 − Ax20 ) < 0 and (λ1 ξ1 − ξ2 λ2 )(b − B) < 0, (II) bAx0 < Bax1 and (λ1 ξ1 − ξ2 λ2 )(b − B) > 0. The paper is organized as follows. In Sections 2–5, we introduce the stationary point, the sliding motion, the flow in

{x > 0} and {x < 0}, the stable and unstable manifold of the equilibrium, respectively. In Section 6, we give the proof of Theorem 1. Now, we note that





0 f 0 (y)

= F (0, y)

\

M.



0



˙ t ) = f 0 (ψ(t )), and ψ(t ) ∈ [−|c1 |, |c1 |] for all t ∈ [0, T ](T > 0), then If ψ(t ) satisfies ψ( ψ(t ) is a sliding motion solution of (2). Theorem 2 (Existence). (1) (a) (b) (c) (i)

In case c1 > 0, if (x0 , y0 ) ∈ R2 \ {(x, y) ∈ R2 : x = 0, −|c1 | ≤ y ≤ |c1 |}, then system (2) with initial value (x0 , y0 ) has a unique solution, if (x0 , y0 ) ∈ {(x, y) ∈ R2 : x = 0, −|c1 | < y < |c1 |}, then system (2) with initial value (x0 , y0 ) has at least three solutions, if (x0 , y0 ) ∈ {(0, −c1 ), (0, c1 )}, then system (2) with initial value (x0 , y0 ) −c has at least three solutions for max{B, b} < c 2 , 1

(ii) has at least one solution for min{b, B} ≥ − c2 ; c

1

(2) in case c1 < 0, system (2) with initial value (x0 , y0 ) ∈ R2 has a unique solution; (3) in case c1 = 0, −c (a) system (2) with initial value (x0 , y0 ) ∈ R2 has a unique solution for max{B, b} ≤ c 2 , 1

(b) system (2) with initial value (x0 , y0 ) ∈ R2 has a unique solution if and only if (x0 , y0 ) 6= (0, 0) for min{b, B} > − c2 . c

1

Proof. By [23] and Section 3 in this paper, we can easily establish that the above assertions are true.



2. The stationary point By the definition of an equilibrium point, we can easily obtain that if bc1 + c2 > 0, Bc1 + c2 > 0, then system (2) bc +c Bc +c has two equilibrium points, e+ = ( 1a 2 , −c1 ), e− = (− 1A 2 , c1 ), and both of them are saddle points. Furthermore, if bc1 + c2 < 0, then system (2) doesn’t have any equilibrium points in G+ . If Bc1 + c2 < 0, then system (2) doesn’t have any equilibrium points in G− . Next, we have: Lemma 3. (1) If c2 6= 0, (0, 0) is a unique equilibrium point of system (2) in M and (a) for c1 6= 0, (i) if Bc1 + c2 , bc1 + c2 > 0, then (0, 0) is an unstable (a stable) node point for c1 > 0 (c1 < 0), (ii) if Bc1 + c2 , bc1 + c2 < 0, then (0, 0) is a saddle point, (b) for c1 = 0, (0, 0) is a focus point or a center point; (2) if c1 = c2 = 0, then (0, 0) is an equilibrium point of system (2) and this is a saddle point. For the proof of Lemma 3, see [18]. 3. The sliding motion Lemma 4. (1) System (2) has a sliding motion solution if and only if bc1 6= Bc1 . (2) If bc1 + c2 > 0 and Bc1 + c2 > 0, ◦

if c1 > 0, there are no solutions of Eq. (2) which enter the interior I s of the sliding motion interval Is (see Fig. 1(a)), if c1 < 0, all sliding motion solutions of Eq. (2) converge to (0,0) as t → ∞ (see Fig. 1(b)). If bc1 + c2 and Bc1 + c2 < 0, if c1 < 0, all sliding motion solutions of Eqs. (2) leave the sliding motion intervals Is at (0, ±c1 ) after a finite time and become unbounded as t → ∞ (see Fig. 1 (c)), (b) if c1 > 0, any sliding motion solution (x(t ), y(t )) of Eq. (2) satisfies either (x(t ), y(t )) → (0, 0) or |x(t )| + |y(t )| → ∞ as t → ∞ (see Fig. 1 (d)). (a) (b) (3) (a)

Corollary 5. There are no type II periodic solutions of system (2). Now we consider the case a = ξ1 ξ2 , b = ξ1 − ξ2 , A = λ1 λ2 , B = λ1 − λ2 and Bc1 + c2 > 0, bc1 + c2 > 0 (where λ1 , λ2 , ξ1 , ξ2 > 0). Without loss of generality, we assume that c1 ≥ 0. Because a = ξ1 ξ2 , b = ξ1 − ξ2 , A = λ1 λ2 , B = λ1 − λ2 , the eigenvalues of A± are ξ1 , −ξ2 and λ1 , −λ2 , respectively.

S. Shui et al. / Nonlinear Analysis 73 (2010) 1277–1288

(a) Bc1 + c2 > 0, bc1 + c2 > 0, c1 > 0.

(b) Bc1 + c2 > 0, bc1 + c2 > 0, c1 < 0.

(c) Bc1 + c2 < 0, bc1 + c2 < 0, c1 < 0.

(d) Bc1 + c2 < 0, bc1 + c2 < 0, c1 > 0. Fig. 1. Phase portraits in a neighborhood of Is .

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S. Shui et al. / Nonlinear Analysis 73 (2010) 1277–1288

4. The flow in {x ≥ 0} and {x ≤ 0} For any point p = (xp , yp )T we denote by γp the orbit through p. If p is in the half-plane {x ≥ 0}, let Xp+ (s) = (x+ p (s),

yp (s))T be the solution of system (2) with initial condition X (0) = p. When x+ p (s) ≥ 0 we have +

x+ p (s) =

 ξ2 x − 

y+ p ( s) =

ξ2 x −

bc1 +c2 a



ξ1 + ξ2 

bc1 +c2 a

+ y + c1

+ y + c1

ξ1 + ξ2

e ξ1 s +

 ξ1 x −

bc1 +c2 a



− y − c1

ξ1 + ξ2 

ξ1 eξ1 s −

ξ1 x −

bc1 +c2 a



e−ξ2 s +

− y − c1

ξ1 + ξ2

bc1 + c2 a

ξ2 e−ξ2 s − c1 .

(3)

If xp = 0, yp ∈ M \ Is , and yp > 0, eT1 p˙ > 0 (where eT1 = (1, 0)T and p˙ is the value of the vector field associated with system (2) at the point p), then γp crosses through the line {x = 0} from the half-plane {x < 0} to the half-plane {x > 0}. Differential system (2) is linear in the half-plane {x ≥ 0} and stable and unstable manifolds of the saddle point e+ intersect the line {x = 0} (see Section 5). − T If p is in the half-plane {x ≤ 0}, let Xp− (s) = (x− p (s), yp (s)) be the solution of system (2) with initial value X (0) = p. − When xp (s) ≤ 0 we have x− p (s) =

 λ2 x + 

y− p ( s) =

λ2 x +

Bc1 +c2 A



+ y − c1

λ1 + λ2  Bc1 +c2 A

+ y − c1

λ1 + λ2

eλ1 s +

 λ1 x +



− y + c1

λ1 + λ2 

λ1 eλ1 s −

Bc1 +c2 A

λ1 x +

Bc1 +c2 A



e−λ2 s −

− y + c1

λ1 + λ2

Bc1 + c2 A

λ2 e−λ2 s + c1 .

(4)

If xp = 0, yp < 0 and yp ∈ M \ Is , eT1 p˙ < 0, then γp crosses through the line {x = 0} from the half-plane {x > 0} to the half-plane {x < 0}. Differential system (2) is linear in the half-plane {x ≤ 0} and stable and unstable manifolds of the saddle point e− intersect the line {x = 0} (see Section 5). 5. The stable and unstable manifolds of equilibrium We note that invariant manifolds of singular points e+ and e− are linear manifolds in the neighborhood of e+ (e− ). Thus, the unstable manifold W u (e+ ) (W u (e− )) of e+ (e− ) contains the line

 − c1 , x ≥ 0 . a      Bc 1 + c2 u L− = (x, y)|y = λ1 x + + c1 , x ≤ 0 



Lu+ = (x, y)|y = ξ1 x −

bc1 + c2



a

generated by the eigenvalue ξ1 (λ1 ) associated with the eigenvectors (1, ξ1 )((1, λ1 )). This half-line intersects the plane {x = 0} at the point

 − c1 . ξ2    Bc + c2 1 u m− = 0, + c1 . λ2 mu+ =



0, −

bc1 + c2

The stable manifold W s (e+ ) (W s (e− )) of e+ (e− ) contains a half-line





 − c1 , x ≥ 0 . a      Bc 1 + c2 Ls− = (x, y)|y = −λ2 x + + c1 , x ≤ 0 Ls+ = (x, y)|y = −ξ2 x −

bc1 + c2



a

generated by the eigenvalue −ξ2 (−λ2 ) associated with the eigenvectors (1, −ξ2 ) ((1, −λ2 )) of A+ (A− ). This half-line intersects the plane {x = 0} at the point ms+ =





ms− =

0,



bc1 + c2

ξ1

0, −

 − c1 .

Bc1 + c2

λ1

 + c1

.

S. Shui et al. / Nonlinear Analysis 73 (2010) 1277–1288

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− It is obvious that the equation x+ p (s) = 0 (resp. xp (s) = 0) has a unique positive solution s = sp for the initial values s s xp = 0, −c1 < yp < ym+ (resp. xp = 0, ym− < yp < c1 ). Now, we define the following maps:

If p ∈ {x ≥ 0} and xp = 0, then define the map Π+ as Π+ (p) = (0, y+ p (sp )), where 0 < sp < +∞ for y < yms , sp = ∞ +

for y ≥ yms . +

If p ∈ {x ≤ 0} and xp = 0, then define the map Π− as Π− (p) = (0, y− p (sp )), where 0 < sp < +∞ for y > yms , sp = ∞ −

for y ≤ yms . −

Remark 1. In this paper, we only consider the case c2 > max{(ξ1 + ξ2 )c1 , (λ1 + λ2 )c1 } because there are no type II periodic solutions. 6. The Proof of Theorem 1 In this section, our main task is to prove Theorem 1. Firstly, we have: (i) The solution of differential equation (2) with initial value X + (0) = p is

−ξ2 bc1 − ξ2 c2 + ac1 + ay ξ1 s −ξ1 bc1 − ξ1 c2 − ac1 − ay −ξ2 s bc1 + c2 e + e + , a(ξ1 + ξ2 ) a(ξ1 + ξ2 ) a −ξ2 bc1 − ξ2 c2 + ac1 + ay ξ1 s −ξ1 bc1 − ξ1 c2 − ac1 − ay −ξ2 s y+ ξ1 e − ξ2 e − c1 . p (s) = a(ξ1 + ξ2 ) a(ξ1 + ξ2 ) x+ p (s) =

(5)

λ c +c

Since − 1ξ 2 − c1 < yΠ− ((0,−c )) < y < 1 λ1 2 and we have qualitative property of equilibrium of the half-plane {x > 0}, 1 2 2 we obtain that there exists a time s1 > 0 such that bc +c

x+ p (s1 ) = 0. By (5), we have y+ p (s1 ) =

−c1 ξ1 + ξ1 e−ξ2 s1 c1 + e−ξ2 s1 ξ2 y − c2 + e−ξ2 s1 c2 . ξ2

The proof of (ii) is similar to that of (i). Secondly, we give some assertions about the heteroclinic orbits. We note that if system (2) satisfies only one of the following conditions, then there exists a unique heteroclinic trajectory. If (2) satisfies both of the following conditions, then system (2) has a heteroclinic cycle: bc +c Bλ +c ¶ mu+ = ms− (i.e. − 1ξ 2 − c1 = − 1λ 2 + c1 ), 2

· ms+ = mu− (i.e.

bc1 +c2

ξ1

1

− c1 =

Bc1 +c2

λ2

+ c1 ).

Therefore the following result is obvious. c (−λ +ξ )

c (λ −ξ )

1 2 2 1 Proposition 6. If λ1 , λ2 , ξ1 , ξ2 > 0, and the system (2) satisfies condition ¶: c1 = ξ2 λ +λ or condition ·: c1 = ξ 2λ +λ , 2 2 1 ξ1 2 2 1 ξ1 c2 (λ2 −ξ1 ) then (2) has a unique heteroclinic trajectory. In particular, if ξ1 + ξ2 = λ2 + λ1 and c1 = ξ λ +λ ξ , (2) has a heteroclinic cycle, 2 2 1 1 which surrounds the sliding motion interval.

c (λ −ξ )

2 1 Next, we consider the existence of a closed trajectory when (2) has a heteroclinic cycle, i.e. ξ1 +ξ2 = λ2 +λ1 , c1 = ξ 2λ +λ . 2 2 1 ξ1 bc1 +c2 bc1 +c2 If the system (2) has a closed trajectory through the point p = (0, y) (where − ξ − c1 < y < ξ − c1 ), then 2

Π+ (p) = Π−−1 (p) must be satisfied. ¬ If c1 > 0, i.e. ξ2 > λ1 , λ2 > ξ1 , we note that     c2 (ξ1 + λ2 ) c2 (ξ2 + λ1 ) − −ξ2 s1 −λ1 s2 ξ1 s1 λ2 s2 y+ ( s ) − y (− s ) = ( e − e ) y + = ( e − e . ) y − 1 2 p p ξ1 λ1 + ξ2 λ2 ξ1 λ1 + ξ2 λ2

1

+λ2 ) +λ1 ) − c1 ), we have y + ξc12λ(ξ11+ξ > 0, y − ξc12λ(ξ12+ξ < 0. Hence, if (2) has a closed trajectory then it must 2 λ2 2 λ2 0 0 0 0 > 0 such that ξ2 s1 = λ1 s2 and ξ1 s1 = λ2 s2 . = λ1 s02 , ξ1 s01 = λ2 s02 and ξ1 + ξ2 = λ1 + λ2 , we have ξ1 = λ2 and ξ2 = λ1 , i.e. c1 = 0. But this gives rise to a

For any y ∈ (c1 ,

bc1 +c2

ξ1

have , By ξ contradiction. Hence, (2) doesn’t have any closed trajectory. λ1 c1 +c2 − + − ­ If c1 = 0, i.e. ξ2 = λ1 , ξ1 = λ2 , we have x+ )). This p (s) ≡ −xp (−s), yp (s) ≡ yp (−s) (where y ∈ (yΠ− ((0,−c1 )) , λ2 concludes the proof of case (a). s01

s02 0 2 s1

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S. Shui et al. / Nonlinear Analysis 73 (2010) 1277–1288

Case b. ¬ The solution of piecewise linear differential system (2) with initial value X (0) = mu− is x+ (s) = − mu −

y+ (s) = mu

(−ξ2 (c1 (λ2 ξ2 + ξ1 λ1 ) + c2 (ξ1 + λ2 ))eξ1 s + ξ1 (c1 (λ2 ξ1 + ξ2 λ1 ) + c2 (ξ2 − λ2 ))e−ξ2 s + (ξ1 + ξ2 )(λ2 c1 (−ξ1 + ξ2 ) − c2 λ2 )) , λ2 ξ1 (ξ1 + ξ2 )ξ2

eξ1 s (c1 (λ2 ξ2 + ξ1 λ1 ) + c2 (ξ1 + λ2 )) + e−ξ2 s (c1 (λ2 ξ1 + ξ2 λ1 ) + c2 (ξ2 − λ2 )) − λ2 (ξ1 c1 + c1 ξ2 )

λ2 (ξ1 + ξ2 )

−

.

(λ −ξ )c

(ξ −λ )c

Since c1 ∈ ( ξ λ2 +λ1 ξ2 , ξ λ2 +λ1 ξ2 ), we have yms > ymu and ymu < yms . By the qualitative properties of the equilibrium point 1 1 2 2 1 1 2 2 + − + − in the half-plane {x > 0}, we obtain that there is a time s1 > 0 such that

(s1 ) = 0, x+ mu −

(c1 (λ2 ξ1 + ξ2 λ1 ) + c2 (ξ2 − λ2 ))e−ξ2 s1 − (ξ1 c1 + c2 )λ2 . − ξ2 λ2 We note if Π+ (mu− ) = ms− , i.e.   (c1 (λ2 ξ1 + ξ2 λ1 ) + c2 (ξ2 − λ2 ))e−ξ2 s1 − (ξ1 c1 + c2 )λ2 Bc1 + c2 Bc1 + c2 + ymu (s1 ) − − + c1 = + − c1 = 0, (6) − λ1 ξ 2 λ2 λ1 (s1 ) = y+ mu

then (2) has a homoclinic trajectory.

λ (ξ λ c +ξ λ c +ξ c +λ c ) ln λ1 (ξ1 c 2λ1 +c 2 λ 1 ξ1 +λ2 c2 −ξ2 c2 ) 2 1 1 1 1 2 2 1 2 2 2

By (6), we obtain s1 = Similarly, we can get ­.

ξ2

1 )c2 > 0, where c1 ∈ ( ξ(λ1 λ21−ξ , (ξ2 −λ1 )c2 ). +λ2 ξ2 ξ1 λ1 +λ2 ξ2

Case c. By analogy to (i) and (ii), we have yΠ+ Π− (p) − y = −

eξ1 s1 − e−ξ2 s1

ξ1 ξ2 λ2

(A2 c2 + A1 c1 ),

where A2 = A21 + A22 + A23 ,

A1 = ξ1 A21 + λ1 A22 − λ2 A23 ,

A21 = λ2 ξ1 (1 − eξ1 s1 −λ2 s2 )(1 − e−ξ2 s1 ), A22 = ξ2 ξ1 (eξ1 s1 − e−λ2 s2 )(1 − e−λ2 s2 ), A23 = λ2 ξ2 (1 − e−ξ2 s1 −λ2 s2 )(1 − eξ1 s1 ). Set f (s1 , s2 ) = −

eξ1 s1 − e−ξ2 s1

ξ1 ξ2 λ2

(A2 c2 + A1 c1 ).

We have fs1 (s1 , s2 ) =

(ξ2 + ξ1 )ξ22 (−bc1 − c2 )g1 (s1 ) , (eξ1 s1 − e−ξ2 s1 )2

where g (s1 ) = ξ2 e2ξ1 s1 e−ξ2 s1 e−λ2 s2 + ξ2 e−ξ2 s1 − ξ2 e−ξ2 s1 eξ1 s1 − ξ2 eξ1 s1 e−ξ2 s1 e−λ2 s2 + ξ1 eξ1 s1

− eξ1 s1 e−ξ2 s1 e−λ2 s2 ξ1 − ξ1 eξ1 s1 e−ξ2 s1 + eξ1 s1 e−2ξ2 s1 e−λ2 s2 ξ1 .

If g (s1 ) = 0, then we obtain e−λ2 s2 =

(−ξ2 e−ξ2 s1 + ξ2 e−ξ2 s1 eξ1 s1 − ξ1 eξ1 s1 + ξ1 eξ1 s1 e−ξ2 s1 ) . eξ1 s1 e−ξ2 s1 (ξ2 eξ1 s1 − ξ2 − ξ1 + e−ξ2 s1 ξ1 )

Since 0 < e−λ2 s2 < 1 and ξ2 eξ1 s1 − ξ2 − ξ1 + e−ξ2 s1 ξ1 > 0, we have −ξ2 e−ξ2 s1 + ξ2 e−ξ2 s1 eξ1 s1 − ξ1 eξ1 s1 + ξ1 eξ1 s1 e−ξ2 s1 > 0. Let h(s) = −ξ2 e−ξ2 s + ξ2 e−ξ2 s eξ1 s − ξ1 eξ1 s + ξ1 eξ1 s e−ξ2 s . Since h(0) = 0 and h0 (s) = ξ12 eξ1 s (e−ξ2 s − 1) + ξ22 e−ξ2 s (1 − eξ1 s ) < 0 for any s > 0, then h(s) < 0 for any s > 0. This is a contradiction. Hence g (s1 ) 6= 0 and g (s1 ) > 0. Because f with respect to s1 or s2 is continuous and fs1 < 0, we know that f with respect to (s1 , s2 ) is a continuous function. ¬ By (0, −c1 ) and (0, c1 ) being tangential points, and the non-interaction property of the trajectory in {x > 0} and {x < 0}, respectively, we obtain that ymu < yΠ+ ((0,y)) < −c1 for any point p = (0, y), where c1 ≤ y < yms and c1 < yΠ− ((0,y)) < ymu +

+

for any point p = (0, y) where ms− < y ≤ −c1 . According to the condition ¬ and the results of (b), we have y+u (s01 ) > yms = − m−

−

Bc1 + c2

λ1

λ (ξ λ c +ξ λ c +ξ c +λ c )

+ c1 as s01 <

ln λ1 (ξ1 c 2λ1 +c2 λ1 ξ1 +λ2 c2 −ξ2 c2 ) 2 1 1 1 1 2 2 1 2 2 2

ξ2

,

−

S. Shui et al. / Nonlinear Analysis 73 (2010) 1277–1288

where s01 satisfy x+u (s01 ) = 0. Hence, ymu − y m−

−

of the trajectory in {x > 0}, we obtain that yΠ

> yΠ

u

+ Π− Π+ (m− )

u

+ (m− )

Π− ((0,y+u (s0 ))) m− 1

=

Bc1 +c2

λ2

− c1 − y

Π− ((0,y+u (s0 ))) m− 2

1285

> 0. By the non-interaction property

,

i.e.

[yΠ+ Π− (p) − y]|p=Π+ (mu− ) = f (s1 , s2 )|p=Π+ (mu− ) > 0. If p1 = (0, −c1 ), we have yΠ− (0,−c

1)

> c1 , so yΠ+ Π− ((0,−c1 )) + c1 < 0, i.e. [yΠ+ Π− (p) − y]|p=(0,−c1 ) = f (s1 , s2 )|p=(0,−c1 ) < 0.

Since f (s1 , s2 ) is a continuous function, i.e. yΠ+ Π− (p) − y is a continuous function, then there exist at least s01 , s02 > 0 such that yΠ+ Π− (p) − y = 0. ­ According to the condition of ­ and the results of (b), we have y−u (s02 ) > yms = m+

ξ (λ c ξ −c ξ +λ c ξ −λ c )

bc1 + c2

− c1 as s02 <

ξ1

+

2 2 1 2 1 1 1 2 ln ξ 2(−ξ1 c1 +λ 1 2 2 1 c1 ξ1 +c1 ξ2 λ2 +λ1 c2 )

λ1

,

where s02 satisfies x−u (s02 ) = 0. Hence, m+

ymu − y

Π+ ((0,y−u (s0 ))) m+ 2

+

=−

bc1 + c2

ξ2

− c1 − y

Π+ ((0,y−u (s0 ))) m+ 2

< 0,

i.e.

[yΠ+ Π− (p) − y]|p=mu+ = f (s1 , s2 )|p=mu+ > 0. If p1 = (0, −c1 ), we have yΠ− (0,−c

1)

> c1 , so yΠ+ Π− ((0,−c1 )) + c1 < 0, i.e. [yΠ+ Π− (p) − y]|p=(0,−c1 ) = f (s1 , s2 )|p=(0,−c1 ) < 0.

Since f (s1 , s2 ) is a continuous function, i.e. yΠ+ Π− (p) − y is a continuous function, then there exist at least s01 , s02 > 0 such that yΠ+ Π− (p) − y = 0. The proof of ® is similar to that of ¬.

Case d. In this case, we use the Filippov transformations to discuss the number of closed trajectories of system (2). In [24–27], the numbers of closed trajectories of the discontinuous Liénard system and continuous Liénard system, respectively, were studied. Now, we use our method to study system (2):



x˙ = P (x, y) y˙ = Q (x, y)

where



P (x, y) = y − c1 x < 0, Q (x, y) = Ax + By + c2



P (x, y) = y + c1 x > 0. Q (x, y) = ax + by − c2

We consider that the number of closed trajectories of system (2) in the region of D1 D2 , where D1 = {(x, y)|y ≥ ξ1 (x − bc1a+c2 )− c1 , y ≤ −ξ2 (x − bc1a+c2 )− c1 , x ≥ 0}, D2 = {(x, y)|y ≤ λ1 (x + Bc1A+c2 )+ c1 , y ≥ −λ2 (x + Bc1A+c2 )+ c1 , x ≤ 0} (see Fig. 2). Firstly, we conclude that

S

Ax + By + c2 ≥ max {λ1 (y − c1 ), −λ2 (y − c1 )} > 0, (x,y)∈D2

ax + by − c2 ≤ min {ξ1 (y + c1 ), −ξ2 (y + c1 )} < 0. (x,y)∈D1

Then, we set x1 = min(x,y)∈D1 { 1a (ax + by − c2 )} = min{− a(˜x+x )+by+c

1

ξ22

(bc1 + c2 ), − ξ12 (bc1 + c2 )} < 0 and let a(˜x + x1 ) = ax + by − c2 , 1

1 2 i.e. x = , where x˜ ≥ 0. a For x ≥ 0, system (2) changes into the following form:



x˙˜ = b(˜x + x1 ) + y + c1 x˜ ≥ 0. y˙ = a(˜x + x1 )

√

a2 x2 −2az

R x˜

1 Let z = 0 −a(˜x + x1 )dx˜ = − 2a x˜ 2 − ax1 x˜ , i.e. x˜ = −x1 − , where z ≥ 0. a Hence, system (2) changes into the following system as x ≥ 0:

dy dz

=

1 F1 (z ) − y

where F1 (z ) = −c1 +

,

0 ≤ z ≤ z01 =

√ b

a2 x21 −2az a

.

ax21 2

,

((7)1 )

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S. Shui et al. / Nonlinear Analysis 73 (2010) 1277–1288

Fig. 2. Phase portrait of system (2) with c1 , bc1 + c2 , Bc1 + c2 > 0.

Similarly, for x ≤ 0, we set x0 = max(x,y)∈D2 { A1 (Ax + By− + c2 )} = max{ 12 (Bc1 + c2 ), 12 (Bc1 + c2 )} > 0. λ1 λ2 A(˜x+x )−By−c

0 2 Let A(˜x + x0 ) = Ax + By + c2 , i.e. x = , where x˜ ≤ 0. A For x ≤ 0, system (2) changes into the following form:



x˙˜ = B(˜x + x0 ) + y − c1 x˜ ≤ 0. y˙ = A(˜x + x0 ) q

R x˜

A2 x20 −2Az

Let z = 0 −A(˜x + x0 )dx˜ = − ˜ − Ax0 x˜ , i.e. x˜ = −x0 + , where z ≥ 0. A Hence, system (2) changes into the following system as x ≤ 0: dy dz

=

A 2 x 2

1 F2 (z ) − y

,

0 ≤ z ≤ z02 =

Ax20 2

,

((7)2 )

q

where F2 (z ) = c1 −

B A2 x20 −2Az A

.

Theorem 7. If system (2) has a unique pair z ∗ , z0 , 0 < z0 < z ∗ < z01 such that the following conditions hold: (1) F20 (z ) < 0(F20 (z ) > 0) as 0 < z < z02 , (2) if z 6= z0 , 0 < z < z01 , then (z0 − z )F10 (z ) > 0((z0 − z )F10 (z ) < 0), and F1 (z ∗ ) = F2 (z ∗ ), (3) F100 (z ) < 0(F100 (z ) > 0) as 0 < z < z01 , (4)

0

0

F −F (F¯1 (y) − F¯2 (y)) > 0 (<0), i.e. F2 0 F 0 1 > 0 (<0) as β < y < F1 (z ∗ ) (F1 (z ∗ ) < y < β ), where β = maxi=1,2 1 2 {limz →z0i (Fi (z ))}, z = F¯i (y) are the inverse functions of y = Fi (z ), i = 1, 2, d dy

then system (2) has at most one closed trajectory. This theorem is from [25] (or see [26]). We will use it to investigate the uniqueness of system (2). Now, we begin the proof for case d. S Assume that L is a closed orbit of system (2) and L = L1 L2 , L1 = L ∩ {(x, y)|x ≥ 0}, L2 = L ∩ {(x, y)|x ≤ 0}. We set clockwise as the positive orientation; then

I  L

∂P ∂Q + ∂x ∂y



Z  dt = L1

∂ P+ ∂ Q+ + ∂x ∂y



Z  dt + L2

∂ P− ∂ Q− + ∂x ∂y



Z dt = L1

F10 (z )dy −

Z

F20 (z )dy. L2

S. Shui et al. / Nonlinear Analysis 73 (2010) 1277–1288

Since F10 (z ) = √ 2−2b

a x1 −2az

I  L

∂P ∂Q + ∂x ∂y

and F20 (z ) = q



Z

B A2 x20 −2Az

, we have

Z

−b

dt =

a2 x21 − 2az

L1

B

dy −

q

1287

L2

q

A2 x20 − 2Az

dy.

Hence, ¬ if bB ≥ 0 and b + B 6= 0, then L ( ∂∂Px + ∂∂Qy )dt is positive or negative. We complete the proof of (d) ¬. Next we consider the case ­. Without loss of generality, we set b > 0 > B. Because of b > 0 > B, we have ξ1 > ξ2 , λ1 < λ2 . Then x0 = 12 (Bc1 + c2 ) and x1 = − 12 (bc1 + c2 ). Since

H

λ1

F2 (z ) − F1 (z ) = 2c1 − of F2 (z ) − F1 (z ) that

b a

q

a2 x21

− 2az −

q

B A

A2 x20

ξ2

− 2Az , where 0 ≤ z ≤ min{

F20 (z ) − F10 (z ) = − q

a2 x21 − 2az

−q

B

=−

A2 x20 − 2Az

2

,

ax21 2

}, it holds for the first derivative

q

q

b

Ax20

b A2 x20 − 2Az + B a2 x21 − 2az

q

A2 x20 − 2Az

q

a2 x21 − 2az

.

Hence, F20 (z ) > F10 (z ) is equivalent to 2(Ab2 − aB2 )z > b2 A2 x20 − B2 a2 x21 . (I) If Ab2 < aB2 , i.e. λ1 ξ1 < ξ2 λ2 , then F20 (z ) > F10 (z ) ⇔ z < Ax20

By the condition (I) of ­, we have b2 A2 x20 2 Ab2

(

B2 a2 x21 aB2

− −

)

 − min

Ax20 2

Hence, F20 (z ) > F10 (z ) for any z ∈ [0,

,

ax21 2

Ax20 2

< 

ax21 .

=

b2 A2 x20 −B2 a2 x21 2(Ab2 −aB2 ) 2

b2 A2 x20 2 Ab2

(

Ax20 aB2 (Ax20 − ax21 ) − B2 a2 x21 − = > 0. − aB2 ) 2 2(Ab2 − aB2 )

] as Ax20 < ax21 and λ1 ξ1 < λ2 ξ2 .

­ If Ab2 > aB2 , i.e. λ1 ξ1 > ξ2 λ2 , then F20 (z ) > F10 (z ) ⇔ z >

b2 A2 x20 −B2 a2 x21 2(Ab2 −aB2 )

By the condition (II) of ­, we have bAx0 < Bax1 . Therefore,

[0, min{

Ax20

ax21

.

b2 A2 x20 −B2 a2 x21 2(Ab2 −aB2 )

< 0 and F20 (z ) > F10 (z ) for any z ∈

}] as bAx0 < Bax1 and λ1 ξ1 < λ2 ξ2 . q q By the condition (2c1 + bx1 − Bx0 )(2c1 − ba a2 x21 − 2Aax20 ) < 0 (or (2c1 − AB A2 x20 − 2Aax21 )(2c1 + bx1 − Bx0 ) < 0), we know that there exists a unique z such that F1 (z ) = F2 (z ). 2

,

.

Since Ab < aB , we have 2

2

It is obvious that the other conditions of Theorem 7 are valid. We have completed the proof of (d).  Ax2

ax2

Remark 2. We know that if F1 (z ) 6= F2 (z ), z ∈ [0, min{ 20 , 21 }], then the system (2) doesn’t have any closed trajectory. Therefore, the following statements are valid. If the system (2) with Bb < 0 satisfies the following conditions: (I) (2c1 + bx1 − Bx0 )(2c1 −

b a

q

a2 x21 − 2Aax20 ) > 0 or (2c1 + bx1 − Bx0 )(2c1 −

B A

q

A2 x20 − 2Aax21 ) > 0,

(II) if one of the following conditions holds: (i) b > B, (λ1 ξ1 − ξ2 λ2 )(Ax20 − ax21 ) > 0 or (λ1 ξ1 − ξ2 λ2 )(bAx0 − Bax1 ) < 0, (ii) b < B, (λ1 ξ1 − ξ2 λ2 )(Ax20 − ax21 ) < 0 or (λ1 ξ1 − ξ2 λ2 )(bAx0 + Bax1 ) < 0, then system (2) doesn’t have any closed trajectory. Finally, with regard to Theorem 1, we give a numerical example. 7. Conclusion In this paper we have studied the qualitative properties of the general case of the two-dimensional discontinuous piecewise linear differential system (2) of saddle type with two zones. The system (2) with right-hand sides discontinuous

(b+B)c1 +2c2 > c1 , we have provided a complete qualitative analysis including the b −B

depends on seven parameters. For

existence and uniqueness of the initial value problem and the existence and uniqueness of the stationary point, and we have demonstrated the existence of homoclinic orbits and heteroclinic orbits. Finally, we obtain that the system (2) doesn’t have any type II closed trajectory, and preliminary results for the existence and uniqueness of the type I closed trajectories. According to the results for type I closed trajectories and Fig. 3, we conjecture that if system (2) with b + B 6= 0 has closed trajectories, then the number of closed trajectories is 1.

1288

S. Shui et al. / Nonlinear Analysis 73 (2010) 1277–1288

2

2

x

x –1.5

–1

–0.5

0.5

1

1.5

–1.5 –1

–0.5

0.5 0

0

–2

–2 y

y

–4

2

–4

x 1

1.5

–1.5

–1

0.5

–0.5

1

1.5

0

–2 y

–4

Fig. 3. The trajectories of system (2) (a = 4, b = 0, A = 3, B = −2, c1 = 1, and c2 = 9) with the initial values (0, y) = (0, 3.3200), (0, y) = (0, 3.3290), (0, y) = (0, 3.3320), respectively.

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