An Invariance Principle for Discontinuous
Righthand Sides Dynamical Systems
Sara Derivière1 and M.A. Aziz-Alaoui2
1
Université de Rouen, France, et Université de Sherbroooke (Qc), Canada
[email protected]
2
Université du Havre, France, [email protected]
Abstract
In this paper, we are concerned with discontinuous righthand sides
dynamical systems (called Filippov systems). We present an extension of
LaSalle invariance principle to locate chaotic attractors of such systems.
For this aim, we use locally Lipschitz continuous and regular Lyapunov
functions, as well as Filippov theory. The obtained results settled down
in the general context of differential inclusions. We use this theoretical
result to locate the attractor of a three-dimensional differential system for
which numerical evidence of the chaotic behavior is given.
1
Introduction
The stability properties of equilibria for continuous differential systems have
been developped by Lyapunov theorems and classical LaSalle invariance principle [5] (1960). In [6], an extension of LaSalle’s principle to locate chaotic
attractors has been done. We have recently adapted this result to find holes
within the attractor, that are regions in the phase space in which this chaotic
solution does not enter, see [3]. Generalizations of LaSalle’s invariance principle
for discontinuous systems are presented in [1, 7]. However, these results only
provide information about the stability of equilibria. In this paper, we present
an extension of LaSalle’s invariance principle for discontinuous differential systems that requires less restrictive conditions than those of theorems in [1, 7].
Our result allows to locate, besides trivial solutions, chaotic attractors of discontinuous systems.
To study differential equations with discontinuous righthand sides, we use
Filippov theory, see [4]. In this theory, in order to define the notion of solution
and the study of equilibria of discontinuous differential equations, the notion of
differential inclusions exetnd the differential equations. To do that, the convex
regularization (or Filippov regularization) is used. Indeed, discontinuous differential systems do not need to have classical solutions. The terms of set-valued
1
functions, Clarke generalized directional derivatives and gradients, are defined
in [2], and shortly summarized below.
Let a domain G ⊂ IRn be separated by a smooth surface Σ into domains ν−
and ν+ , so G = ν− ∪ Σ ∪ ν+ . Consider the following differential equation,
dx(t)
= f (x),
dt
(1)
where f : G → IRn is a continuous function in domains ν− and ν+ . The
discontinuity hyper-surface Σ is defined by an equation h(x(t)) = 0. Let η~ be
the normal of Σ directed from ν− towards ν+ : η~ = ~η (x(t)) = grad(h(x(t)).
Then, we have,
ν− = {x ∈ G : h(x(t)) < 0}, Σ = {x ∈ G : h(x(t)) = 0}, ν+ = {x ∈ G : h(x(t)) > 0}.
n
Let F : G → 2IR be a set-valued function. On ν− and ν+ f is continuous.
In this case we define,
F (x) = {f (x)},
x ∈ ν − ∪ ν+ .
We note f − (x) (resp. f + (x)), instead of f (x), when x ∈ ν− (resp. x ∈ ν+ ). On
Σ, we define,
f − (x) = lim f (x∗ ), and f + (x) = lim f (x∗ ),
x∗ ∈ν+
x∗ →x
x∗ ∈ν−
x∗ →x
x ∈ Σ.
Finally, the convex regularization of (1), given by Filippov in [4], consists
in defining F (x) as follows :

si x ∈ ν−
 {f − (x)}
{αf − (x) + (1 − α)f + (x), α ∈ [0, 1]} if x ∈ Σ
F (x) =
(2)

{f + (x)}
si x ∈ ν+
and the differential inclusion associated to (1) is
dx(t)
∈ F (x).
dt
2
(3)
Some definitions and results, Filippov’s guidelines
To state existence and unicity theorems for the solutions of differential inclusions, we need some definitions and reminders.
n
Let F : IRn → 2IR be a set-valued function.
2
Definition 2.1 (Basic conditions) A multivalued function F satisfies the basic conditions if, for all x ∈ IRn , F (x) is non-empty, bounded, closed and convex,
and if F is upper semi-continuous.
Definition 2.2 (Filippov solution) A Filippov solution of a discontinuous
dx(t)
differential equation
= f (x) is an absolutely continuous function x :
dt
dx(t)
∈ F (x), whith F (x) defined
[0, τ [→ IRn so that, for almost all t ∈ [0, τ [,
dt
by (2).
Theorem 2.1 (Existence of solutions for a differential inclusion) If F satisfies the basic conditions, then, for all initial condition x0 ∈ IRn , there exists a
Filippov solution x of the differential inclusion defined on [0, τ [, τ > 0 :
dx(t)
∈ F (x).
dt
We define the projections of f + and f − on the normal ~η , that are, fη+ =
f + .~
η and fη− = f − .~
η respectively (the dot represents the usual scalar product).
Theorem 2.2 (Unicity of the solution) If the assumptions given in the previous theorem are true and if, for all x ∈ Σ, fη+ (x) < 0 or fη− (x) > 0, then
the solution of the differential inclusion (3) is uniquely determined.
Let us consider the differential inclusion associated with the autonomous differdx(t)
= f (x), that is :
dt
ential equation
dx(t)
∈ F (x).
dt
(4)
In the next section, we will formulate and prove an invariance principle. This
is why we have decided to define the following notions.
Definition 2.3 A Lyapunov function for (4) is a continuous, positive definite
function, V : IRn → IR, so that, for all solution ϕ of (4) defined on an interval
I ⊂ IR and for all t1 , t2 ∈ I :
t1 ≤ t2 =⇒ V (ϕ(t2 )) ≤ V (ϕ(t1 )).
3
(5)
Definition 2.4 The Clarke upper generalized derivative of a function V at x
in the direction of v is :
V 0 (x, v) = lim sup
y→x h↓0
V (y + hv) − V (y)
h
Definition 2.5 A function V : IRn → IR is regular at x ∈ IRn if, for all
v ∈ IRn ,
(i) the usual right directional derivative V+0 (x, v) exists and
(ii) V+0 (x, v) = V 0 (x, v).
Definition 2.6 The Clarke generalized gradient of V in x is the set
∂V (x) = co
lim ∇V (xi ) : (xi ) → (x), (xi ) ∈
/ ΩV ,
i→+∞
with ΩV a set of zero measure in IRn on which the gradient of V is not defined,
and co(A) is the smallest convex and bounded set containing A.
Definition 2.7 The set-valued derivative of V with respect to (4) is given by
V˙ (x) = {a ∈ IR : ∃v ∈ F (x) so that p.v = a, ∀p ∈ ∂V (x)}.
3
Theoretical Results : Invariance Principle for
(chaotic) Filippov Systems
In this section, we will present the main result of this paper, an invariance
principle for discontinuous differential systems.
The following lemma is needed as the proof of the main theorem of this
section, see [1]. We denote by Sx0 the set of all solutions starting at x0 .
Lemma 3.1 Let ϕ be a solution of the differential inclusion (4) and let V :
d
IRn → IR be a locally Lipschitz continuous and regular function. Then dt
V (ϕ(t))
˙
d
exists almost everywhere and V (ϕ(t)) ∈ V (x) almost everywhere.
dt
The following theorem, given in [1], is a version of the LaSalle invariance principle for differential inclusions.
Theorem 3.2 Let V : IRn → IR be a locally Lipschitz continuous and regular
function for (4). Let us assume that for some l > 0, the connected component
Ll of the level set {x ∈ IRn : V (x) ≤ l} so that 0 ∈ Ll is bounded. Let x0 ∈ Ll ,
ϕ ∈ Sx0 and ZV = {x ∈ IRn : 0 ∈ V˙ (x)} and let M the largest weakly invariant
subset of ZV ∪ Ll .
Then dist(ϕ(t), M ) → 0 as t → ∞.
4
Now, we will present an extension of this theorem that requires less restrictive
conditions and that further allows to determine a region in the phase space
containing (chaotic) attractors for discontinuous differential systems, that is, an
estimate of the (chaotic) attractor of studied dynamical systems.
Theorem 3.3 Consider a discontinuous differential system (4). Let V : IR n →
IR be a locally Lipschitz continuous and regular function and let c : IR n → IR be
a continuous function so that, for all x ∈ IRn , max V˙ (x) ≤ −c(x). We define :
C := {x ∈ IRn : c(x) < 0}, E := {x ∈ IRn : c(x) = 0}, and l := supx∈C V (x).
Assume that Ωl := {x ∈ IRn : V (x) ≤ l} is bounded.
If x0 ∈ Ωl , then ϕ(.) ∈ Ωl , for all ϕ(.) ∈ Sx0 .
Moreover, if ϕ(.) is a bounded solution and M the largest weakly invariant
subset of Ωl ∪ E, then dist(ϕ(t), M ) → 0 as t → ∞.
Proof : First of all, let us note that if x ∈
/ Ωl , then x ∈
/ C and c(x) ≥ 0. So,
˙
˙
max V (x) ≤ 0, i.e. v ≤ 0, ∀v ∈ V (x).
• Let us prove the theorem for the first case.
We consider x0 ∈ Ωl and ϕ ∈ Sx0 . Then, by definition of Ωl , V (x0 ) = V (ϕ(0)) ≤
l. Assume that there exists t̃ > 0 so that ϕ(t̃) ∈
/ Ωl . Then, V (ϕ(t̃)) > l. Since V
is continuous, there exists t ∈ [0, t̃[ so that : V (ϕ(t)) = l and V (ϕ(t)) > l, ∀t ∈
d
]t, t̃]. It follows that there exists ]t, t+ε] on which V (ϕ(t)) is positive, which is
dt
d
impossible, because (since V is locally Lipschitz and regular) V (ϕ(t)) ∈ V˙ (x)
dt
(see Lemma (3.1)) and because we are outside of Ω , v ≤ 0, ∀v ∈ V˙ (x). It
l
follows that ϕ(t) ∈ Ωl , ∀t ≥ 0.
• Let us prove the theorem for a bounded solution ϕ.
We can suppose that ϕ(t) ∈
/ Ωl , ∀t ≥ 0 (otherwise, we consider the first part
of the proof).
Since the solution is not in Ωl , it follows that max V˙ (x) ≤ 0, and therefore V ◦ ϕ
is decreasing.
Moreover, V is continuous and ϕ is bounded, so V ◦ ϕ is bounded.
We denote by c the value c = limt→∞ V ◦ ϕ(t).
Let us consider y in the ω-limit set ω(x0 ) : by definition of ω(x0 ), ∃(tk ) %
∞ : ϕ(tk ) →k→∞ y, and by continuity of V : V (ϕ(tk )) → V (y) = c, for all
y ∈ ω(x0 ).
Let ψ be in the set Sy , with y ∈ ω(x0 ). By definition of the ω-limit set (invariant) ψ(t) ∈ ω(x0 ) (∀t > 0), and we have V (ϕ(t)) = c, ∀t > 0. So,
d
V (ϕ(t)) = 0, ∀t > 0.
dt
d
Furthermore, V (ϕ(t)) = 0 ∈ V˙ (x), so 0 ≤ max V˙ (x) ≤ −c(ψ(t)) and c(ψ(t)) ≤
dt
0, for all ψ ∈ Sy and all y ∈ ω(x0 ).
5
Moreover, ω(x0 ) ∈
/ Ωl , so ψ(t) ∈
/ Ωl , ∀t > 0 and then c(ψ(t)) ≥ 0, ∀t > 0.
In conclusion, c(ψ(t)) = 0, ∀t > 0, for all ψ ∈ Sy and all y ∈ ω(x0 ).
Since ω(x0 ) is invariant, we have proved that ω(x0 ) ⊂ {x ∈ IRn : c(x) = 0} = E.
4
Example and Application
Now, let us give an example that use the last theorem.
We consider the new discontinuous righthand sides dynamical system :

 ẋ = −σx + σy
ẏ = rx − y − sgn(y).|x|z

ż = −bz + |xy|
(6)
that numerically shows a chaotic attractor for σ = 10, r = 28.5 and b = 2.5, see
Fig.1.
b)
a)
20
40
15
30
y
z
10
20
5
0
0
5
x
10
10
15
0
5
x
10
15
Figure 1: Projection of the attractor of system (6) a) on plane xy and b) on plane
xz.
T
The differential inclusion associated with this discontinuous system is (ẋ, ẏ, ż) ∈
F (x, y, z) where F (x, y, z) is the convex regularization of f (x, y, z) given by the
set-valued function
{−σx + σy} × {rx − y − |x|zsgn(y)} × {−bz + |xy|} if y 6= 0,
F (x, y, z) = {−σx} × {[rx − |xz|, rx + |xz|])} × {−bz}
if y = 0.
To estimate the domain of existence of the chaotic attractor of the previous
system (6), we use Theorem 3.3 with
V (x, y, z) = α(δx + ξ)2 + β(y + ρ)2 + γ(µz + τ )2 ,
where α > 0, β > 0, γ > 0, δ, , µ, ξ, ρ and τ are parameters to determine. The
generalized gradient of V is given by
n
o n
o n
o
1
∂V (x, y, z) = αδ (δx + ξ) × β (y + ρ) × γµ (µz + τ ) ,
2
6
and the set-valued deriviative of V if y 6= 0 is :
1 ˙
V̄ (x, y, z) =
2
n
αδ (δx + ξ) (−σx + σy) + β (y + ρ) (rx − y − |x|z.sgn(y))
o
+γµ (µz + τ ) (|xy| − bz) ,
and,
n
o
1 ˙
V̄ (x, 0, z) = αδ (δx + ξ) (−σx)+βρ[rx−|xz], rx+|xz|]+γµ (µz + τ ) (−bz) .
2
So, if y 6= 0,
1 ˙
V̄ (x, y, z) =
2
n
− ασδ 2 x2 + ασδ 2 xy − ασδξx + ασδξy + β2 rxy
−β2 y 2 − β2 |xy|z + βρrx − βρy − βρ|xz|sgn(y)
o
+γµ2 |xy|z − γbµ2 z 2 + γµτ |xy| − γµbτ z ,
and,
1 ˙
V̄ (x, 0, z) ≤ −ασδ 2 x2 − ασδξx + βρrx + βρ|xz|
2
−γµ2 bz 2 − γµbτ z.
In both cases, for all (x, y, z, ) ∈ IR3 , the result is :
1 ˙
max
V̄ (x, y, z) ≤ −ασδ 2 x2 − β2 y 2 − γµ2 bz 2
2
max
+ασδ 2 xy + β2 r|xy| + γµτ |xy| + βρ|xz|
+(γµ2 − β2 )|xy|z − (ασδξ − βrρ)x
−(βρ − ασδξ)y − γµbτ z
≤
−ασδ 2 x2 − β2 y 2 − γbµ2 z 2
+ασδ 2 xy + (β2 r + γµτ )|xy| + βρ|xz|
+(γµ2 − β2 )|xy|z − (ασδξ − βrρ)x
−(βρ − ασδξ)y − γµbτ z
:= −c(x, y, z),
where the last equality obviously defines function c(x, y, z).
β2 r
and γµ2 = β2 , and then,
We choose τ = −
γµ
max
1 ˙
V̄ (x, y, z) ≤ −ασδ 2 x2 − β2 y 2 − γbµ2 z 2 + ασδ 2 xy + βρ|xz|
2
−(ασδξ − βrρ)x − (βρ − ασδξ)y − γµbτ z.
7
The function c(x, y, z) of Theorem 3.3 is then :
1
c(x, y, z) = ασδ 2 x2 + β2 y 2 + γµ2 bz 2 − ασδ 2 xy − βρ|xz|
2
+(ασδξ − βrρ)x + (βρ − ασδξ)y + γµbτ z.
(7)
We want values for parameters α, β, γ, δ, , µ, ξ, ρ and τ so that the above equation defines a bounded and convex manifold in order to simplify the computation
of the sup in Theorem 3.3. To do this, we reduce the equation of c to its canonical form, as follows.
Given the general equation of a quadric,
a11 x2 + a22 y 2 + a33 z 2 + 2a12 xy + 2a13 xz + 2a23 yz
2a10 x + 2a20 y + 2a30 z + a00 = 0,
composed of three terms :
• the quadratic part composed of terms of higher degree,
a11 x2 + a22 y 2 + a33 z 2 + 2a12 xy + 2a13 xz + 2a23 yz,
• the linear part composed of term of degree one,
2a10 x + 2a20 y + 2a30 z,
• and the constant term a00 ,
we follow the following steps :
i) we bring back the quadratic part written under matricial form (change of
basis) to its canonical form :


x
(x y z) ∆  y  ,
z
with

a11
∆ =  a12
a13
a12
a22
a23

a13
a23  ,
a33
ii) we include in the new basis the linear part of the quadric,
iii) in the new axis directed by the eigenvalues of ∆, the equation of the
quadric is :
λ1 x02 + λ2 y 02 + λ3 z 02 + 2µ1 x0 + 2µ2 y 0 + 2µ3 z 0 + a00 = 0,
8
with λ1 , λ2 and λ3 the eigenvalues of ∆.
µ1
µ2
µ3
, y 00 = y 0 +
, z 00 = z 0 +
, the
λ1
λ2
λ3
equation of the quadric is brought back to its canonical form :
iv) By the translation x00 = x0 +
x002
y 002
z 002
+
+
= 1,
B/λ1
B/λ2
B/λ3
(8)
µ2
µ2
µ21
+ 2 + 3 − a00 . B is a constant so, to have an ellipsoid for
λ1
λ2
λ3
equation (8), it is enough to take B, λ1 , λ2 and λ3 of the same sign.
with B =
Let us get back to the equation of c(x, y, z) and define c1 (x, y, z) = c+.+ (x, y, z),
c2 (x, y, z) = c+.− (x, y, z), c3 (x, y, z) = c−.+ (x, y, z), c4 (x, y, z) = c−.− (x, y, z).
In that case, the triple index indicates in which quadrant lies (x, y, z), for example, c+.+ (x, y, z) = c(x, y, z)|x>0,z>0 , c+.− (x, y, z) = c(x, y, z)|x>0,z<0 , etc...
We have to determine values to the parameters, so that c1 is an ellipsoid.
Its quadratic part is :
ασδ 2 x2 + β2 y 2 + γbµ2 z 2 − ασδ 2 xy − βρxz,
and its canonical form,


x
(x y z) ∆  y  ,
z
with

A
−A/2 −Bρ/2

B
0
∆ =  −A/2
−Bρ/2
0
C

and A = ασδ 2 > 0, B = β2 > 0 and C = γbµ2 > 0.
Choosing ρ = 0, we get :

A
−A/2 0
B
0 .
∆ =  −A/2
0
0
C

We decide to take values of parameters A, B and C so that the eigenvalues of
matrix ∆ are real and positive. That will ensure that equation c defines an
ellipsoid.
9
The eigenvalues of ∆ are positive if :


 det(A)
=a>0

A
−A/2
= AB − A2 /4 > 0 ⇐⇒ B > A/4
det
−A/2
B



det(∆) = ABC − A2 C/4 > 0 ⇐⇒ C > 0
⇐⇒ A > 0, B > A/4, C > 0.
Remember the constraints that we have already given :
ρ=0
(simplifies the calculus of the eigenvalues),
β2 r
τ =−
γµ
γµ2 = β2
(annulment of the large terms in xy),
(annulment ofthe term in |xy|z).
Let us take (for example),
A = B = 1, C = b,
in order to obtain an ellipsoid for c1 . The equation of c1 is now :
y 002
z 002
x002
+
+
= 1,
B/λ1
B/λ2
B/λ3
with B > 0, λ1 > 0, λ2 > 0, λ3 > 0.
Finally :
α = σ, β = 1, γ = 1, ξ = 0, ρ = 0, τ = −r,
= 1, δ = 1/σ, µ = 1.
Then,
c1 (x, y, z) = x2 + y 2 + bz 2 − xy − rbz.
To apply Theorem 3.3, we have to determine
sup
V (x, y, z).
{(x,y,z):c(x,y,z)<0}
To overcome this difficulty, we use the following lemma which can be found
in [6], but to make it clear for all readers, we give its proof here.
Lemma 4.1 Let V, c, ci : IRn −→ IR, i = 1, .., k be continuous functions so
that
c(x) ≥ inf{ci (x), i = 1, · · · , k}, ∀x ∈ IRn .
Let us consider
Ci := {x ∈ IRn : ci (x) < 0},
and C := {x ∈ IRn : c(x) < 0}.
Then, we have the following result :
10
• C⊂
n
[
Ci and sup c(x) ≤
x∈C
i=1
sup
x∈
Sk
i=1
c(x).
Ci
• Moreover, if Ci is bounded for all i, and if there exists a sequence of homeomorphisms Si : IRn −→ IRn , i = 1, · · · , k so that Cj = Sj−1 (Cj−1 ), ∀j =
2, · · · , k, C1 = Sk (Ck ), and V (Si (x)) = V (x), ∀x ∈ IRn , ∀i = 1, · · · , k.
Then
sup V (x) ≤ sup V (x), ∀j = 1 · · · k.
x∈C
x∈Cj
Proof : If x ∈ C, then inf{c1 (x), c2 (x), · · · , cn (x)} ≤ c(x) < 0.
n
[
So there exists j so that cj (x) < 0 and then, x ∈ Cj ⊂
Ci . Consequently,
C⊂
n
[
i=1
Ci and sup c(x) ≤
x∈C
i=1
c(x).
sup
x∈
Sk
i=1
Ci
We still need to show that sup V (x) ≤
sup V (x). Let y ∈ Ci+1 , then
x∈Ci
x∈Ci+1
there exists z ∈ Ci so that y = Si (z), so, V (y) = V (Si (z)) = V (z) ≤ sup V (x)
x∈Ci
and sup V (x) ≤ sup V (x), and consequently :
x∈Ci
x∈Ci+1
sup V (x) ≤ sup V (x) ≤ · · · ≤ sup V (x) ≤ sup V (x).
x∈Ck
x∈Ck−1
x∈C2
x∈C1
So,
sup V (x) ≤
x∈C
V (x) = sup V (x),
sup
x∈
Sk
i=1
∀j = 1, · · · , k
x∈Cj
Ci
Using the same notations of this lemma for the set C and Ci , we get
sup
V (x, y, z) ≤
(x,y,z)∈C
sup
V (x, y, z).
(x,y,z)∈C1
Here, C1 set (an ellipsoid) is bounded and convex and V (x, y, z) is a convex
function. That is why,
sup V (x, y, z) is reached on the boundary of C. So,
(x,y,z)∈C1
it can be computed using the Lagrange multipliers technique.
To do that, we consider the Lagrange function L,
L(x, y, z) = V (x, y, z) + ` c1 (x, y, z)
1 2
x + y 2 + (z − r)2 + ` x2 + y 2 + bz 2 − xy − rbz ,
=
σ
11
and we solve the system
∂L
∂L
∂L
∂L
=
=
=
= 0, namely,
∂x
∂y
∂z
∂`
 2

x + 2`x − `u



σ

 2y + 2`y − `x

 2z + 2b`z



 2
x + y 2 + bz 2 − xy
=
0
=
0
= 2r + `rb
=
rbz
We obtain two solutions (0, 0, 0) and (0, 0, r). As V (0, 0, 0) = r 2 and V (0, 0, r) =
0, we can conclude that sup V (x, y, z) is reached in (0, 0, 0) and
(x,y,z)∈C
sup
V (x, y, z) = r2
(x,y,z)∈C
Consequently, using Theorem 3.3, we have theoretically estimated the domain
of existence of the chaotic attractor of system (6) by the manifold defined by
1 2
x + y 2 + (z − r)2 ≤ r2 , i.e.,
σ
0.1x2 + y 2 + (z − 28.5)2 ≤ 28.52
This result is represented in Fig. 2.
a)
b)
25
45
20
40
15
35
10
30
5
y
z
0
-5
25
20
-10
15
-15
10
-20
5
-25
-40
-30
-20
-10
0
10
20
30
0
-40
40
-30
-20
-10
0
10
20
30
40
x
x
Figure 2: The chaotic attractor of system (6) inside its estimation, in projection a)
on plane xy and b) on plane xz.
5
Conclusion
In this paper, we have studied discontinuous righthand sides differential systems
(called Filippov systems), as well as a new theorem to estimate their (chaotic)
12
attractors. This theorem is an extension of the invariance principle written
in the case of discontinuous systems. In this study, a numerical evidence of
the chaotic behaviour of a new dicontinuous differential system has been given.
Our theorem only gives an estimation of these attractors. Our present work (in
progress) is to theoretically prove the existence of chaos in such discontinuous
systems. One way of doing so is to use the Conley index theory and to compute
algebraic invariants of particular spaces to understand the topology and the
dynamics of invariant sets and attractors.
References
[1] A. Bacciotti & F. Ceragioli, Stability and Stabilization of Discontinuous
Systems and Nonsmooth Lyapunov Functions, Esaim-COCV 4 (1999) 361376.
[2] F.H. Clarke, Optimization and Nonsmooth Analysis, Wiley and Sons, 1983.
[3] S. Derivière & M.A. Aziz-Alaoui, Estimation of Attractors and Synchronization of Generalized Lorenz Systems, Dynamics of Continuous, Discrete
and Impulsive Systems, Series B: Applications and Algorithms, Volume 10,
Number 6 (2003) 833-852.
[4] A.F. Filippov, Differential Equations with Discontinuous Righthand Sides,
Kluwer Academic Publishers, 1988.
[5] J.P. LaSalle : Some Extension of Lyapunov’s Second Method, IRE Trans.
Circuit Theory, vol. CT-7 (1960) 520-527.
[6] H.M. Rodrigues, L.F.C.Alberto & N.G. Bretas : Uniform Invariance Principle and Synchronization, Robustness with Respect to Parameter Variation, JDE 169 (2001) 228-254.
[7] D. Shevitz & B. Paden, Lyapunov Stability Theory of Nonsmooth Systems,
IEEE Trans. on Automatic Control, Vol. 39, No. 9 (1994) 1910-1914.
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