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An Extension of the Invariance Principle for a Class of Differential Equations with Finite Delay
Advances in Difference Equations volume 2010, Article number: 496936 (2010)
Abstract
An extension of the uniform invariance principle for ordinary differential equations with finite delay is developed. The uniform invariance principle allows the derivative of the auxiliary scalar function 
to be positive in some bounded sets of the state space while the classical invariance principle assumes that 
. As a consequence, the uniform invariance principle can deal with a larger class of problems. The main difficulty to prove an invariance principle for functional differential equations is the fact that flows are defined on an infinite dimensional space and, in such spaces, bounded solutions may not be precompact. This difficulty is overcome by imposing the vector field taking bounded sets into bounded sets.
1. Introduction
The invariance principle is one of the most important tools to study the asymptotic behavior of differential equations. The first effort to establish invariance principle results for ODEs was likely made by Krasovskiĭ; see [1]. Later, other authors have made important contributions to the development of this theory; in particular, the work of LaSalle is of great importance [2, 3]. Since then, many versions of the classical invariance principle have been given. For instance, this principle has been successfully extended to differential equations on infinite dimensional spaces, [4–7], including functional differential equations (FDEs) and, in particular, retarded functional differential equations (RFDEs). The great advantage of this principle is the possibility of studying the asymptotic behavior of solutions of differential equations without the explicit knowledge of solutions. For this purpose, the invariance principle supposes the existence of a scalar auxiliary function 
satisfying 
and studies the implication of the existence of such function on the 
-limit of solutions.
More recently, the invariance principle was successfully extended to allow the derivative of the scalar function 
to be positive in some bounded regions and also to take into account parameter uncertainties. For ordinary differential equations, see [8, 9] and for discrete differential systems, see [10]. The main advantage of these extensions is the possibility of applying the invariance theory for a larger class of systems, that is, systems for which one may have difficulties to find a scalar function satisfying 
.
The next step along this line of advance is to consider functional differential equations. Let 
be the space of continuous functions defined on 
with values in 
, and 
a compact subset of 
. In this paper, an extension of the invariance principle for the following class of autonomous retarded functional differential equations
is proved.
The main difficulty to prove an invariance principle for functional differential equations is the fact that flows are defined on an infinite dimensional space. It is well known that, in such spaces, boundedness of solutions does not guarantee precompactness of solutions. In order to overcome this difficulty, we will impose conditions on function 
to guarantee that solutions of (1.1) belong to a compact set.
The extended invariance principle is useful to obtain uniform estimates of the attracting sets and their basins of attraction, including attractors of chaotic systems. These estimates are obtained as level sets of the auxiliary scalar function 
. Despite 
is defined on the state space 
, we explore the boundedness of time delay to obtain estimates of the attractor in 
, which are relevant in practical applications.
This paper is organized as follows. Some preliminary results are discussed in Section 2; an extended invariance principle for functional differential equation with finite delay is proved in Section 3. In Section 4, we present some applications of our results in concrete examples, such as a retarded version of Lorenz system and a retarded version of Rössler system.
2. Preliminary Results
In what follows, 
will denote the Euclidean 
-dimensional vector space, with norm 
, and 
will denote the space of continuous functions defined on 
into 
, endowed with the norm 
.
Let 
, be a continuous function, and, for each 
, let 
be one element of 
defined as 
. The element 
is a segment of the graph of 
, which is obtained by letting 
vary from 
to 
. Let 
, be a continuous function. For a fixed 
and 
, consider the following initial value problem:
Definition 2.1.
A solution of (2.1)-(2.2) is a function 
defined and continuous on an interval 
, such that (2.2) holds and (2.1) is satisfied for all 
.
If for each 
and for a fixed 
, the initial value problem (2.1)-(2.2) has a unique solution 
, then we will denote by 
the orbit through 
, which is defined as 
. Function 
belongs to the 
-limit set of 
, denoted by 
, if there exists a sequence of real numbers 
, with 
as 
, such that 
, with respect to the norm of 
, as 
.
Generally, on infinite dimensional spaces, such as the space 
, the boundedness property of solutions is not sufficient to guarantee compactness of the flow 
. The compactness of the orbit will be important in the development of our invariance results. In order to guarantee the relatively compactness of set 
and, at the same time, the uniqueness of solutions of (2.1)-(2.2), the following assumptions regarding function 
are made.
-
(A 1) For each
, there exists a real number 
such that, 
for all 
and for all 
. -
(A 2)For each
, there exists a real number 
, such that
(2.3) -
for all
, 
and all 
.
, there exists a real number 
such that, 
for all 
and for all 
.
, there exists a real number 
, such that
, 
and all 
.Under conditions (A 1)-(A 2), the problem (2.1)-(2.2) has a unique solution that depends continuously upon 
, see [4]. Moreover, one has the following result.
Lemma 2.2 (compacity of solutions [4]).
If 
is a solution of (2.1)-(2.2) such that 
is bounded, with respect to the norm of 
, for 
and assumptions (A1)-(A2) are satisfied, then 
is the unique solution of (2.1)-(2.2). Moreover, the flow 
through 
belongs to a compact subset of 
for all 
.
Lemma 2.2 guarantees, under assumptions (A 1) and (A 2), that bounded solutions are unique and the orbit is contained in a compact subset of 
.
Let 
, be a solution of problem (2.1)-(2.2) and suppose the existence of a positive constant 
in such a manner that
for every 
and 
. The following lemma is a well-known result regarding the properties of 
-limit sets of compact orbits 
[6].
Lemma 2.3 (limit set properties).
Let 
be a solution of (2.1)-(2.2) and suppose that (2.4) is satisfied. Then, the 
-limit set of 
is a nonempty, compact, connected, invariant set and 
, as 
.
Lyapunov-like functions may provide important information regarding limit-sets of solutions and also provide estimates of attracting sets and their basins of attraction. Thus, it is important to consider the concept of derivative of a function along the solutions of (2.1).
Definition 2.4.
Let 
be a continuous scalar function. The derivative of 
along solutions of (2.1), which will be denoted by 
, is given by
Remark 2.5.
Function 
is well defined even when solutions of (2.1) are not unique. In order to be more specific, suppose 
and 
are solutions of (2.1), satisfying the same initial condition, then it is possible to show [11] that
Remark 2.6.
Generally, if 
is continuously differentiable and 
is a solution of (2.1), then the scalar function 
is differentiable in the usual sense for 
. In spite of that, it is possible to guarantee the existence of 
for 
assuming weaker conditions; for example, if 
is locally Lipschitzian, it is possible to show that 
is well defined. For more details, we refer the reader to [12].
3. Main Result
In this section, we will prove the main result of this work, the uniform invariance principle for differential equations with finite delay. But first, we review a version of the classical invariance principle for differential equations with delay, which has been stated and proven in [4]. Consider the functional differential equation
Theorem 3.1 (the invariance principle).
Let 
be a function satisfying assumptions (A1) and (A2) and 
a continuous scalar function on 
. Suppose the existence of positive constants 
and 
such that 
for all 
. Suppose also that 
for all 
. If 
is the set of all points in 
where 
and 
is the largest invariant set in 
, then every solution of (3.1), with initial value in 
approaches 
as 
.
In Theorem 3.1, constants 
and 
are chosen in such a manner that the level set, 
, that is, the set formed by all functions 
such that 
, is a bounded set in 
. Using this assumption, it is possible to show that the solution 
, starting at 
with initial condition 
, is bounded on 
for 
. Now, we are in a position to establish an extension of Theorem 3.1. This extension is uniform with respect to parameters and allows the derivative of 
be positive in some bounded sets of 
. Since in practical applications it is convenient to get information about the behavior of solutions in 
, our setting is slightly different from that used in Theorem 3.1.
In what follows, let 
, 
, and 
be continuous functions and consider, for each 
, the sets
Moreover, we assume that the following assumptions are satisfied:
-
(i)
is a bounded set in 
, -
(ii)
, for all 
, -
(iii)
, for all 
, -
(iv)
there is a real number
such that 
.
is a bounded set in 
,
, for all 
,
, for all 
,
such that 
.Under these assumptions, a version of the invariance principle, which is uniform with respect to parameter 
, is proposed in Theorem 3.2.
Theorem 3.2 (uniform invariance principle for retarded functional differential equations).
Suppose function 
satisfies assumptions (A1)-(A2). Assume the existence of a locally Lipschitzian function 
and continuous functions 
. In addition, assume that function 
takes bounded sets into bounded sets. If conditions (i)–(iv) are satisfied, then, for each fixed 
, we have the following.
-
(I)
If
. Then,-
(1)
the solution
of (1.1) is defined for all 
, -
(2)
, for 
, where 
, -
(3)
, where 
, for all 
, -
(4)
tends to the largest collection 
of invariant sets of (1.1) contained in 
as 
.
-
(1)
-
(II)
If
. Then,-
(1)
is defined for all 
, -
(2)
belongs to 
for all 
, -
(3)
belongs to 
for all 
, -
(4)
tends to the largest collection 
of invariant sets of (1.1) contained in 
.
-
(1)
. Then,
of (1.1) is defined for all 
,
, for 
, where 
,
, where 
, for all 
,
tends to the largest collection 
of invariant sets of (1.1) contained in 
as 
.
. Then,
is defined for all 
,
belongs to 
for all 
,
belongs to 
for all 
,
tends to the largest collection 
of invariant sets of (1.1) contained in 
.Proof.
In order to show (I), we first have to prove that 
, for all 
. Let 
be a solution of (2.1), satisfying the initial condition 
and suppose the existence of 
such that 
, that is, 
. By assumption, we have 
and 
. Using the Intermediate Value Theorem [13] and continuity of 
with respect to 
, it is possible to show the existence of 
such that 
and 
for all 
. On the other hand, since function 
is nonincreasing on 
we have 
, but this leads to a contradiction, because 
. Therefore, 
, for all 
, which implies that 
is bounded and defined for all 
. By definition of 
, we have that 
for all 
. Since 
is a bounded set, according to Lemma 2.2, the orbit 
belongs to a compact set. As a consequence of Lemma 2.3, the 
-limit set, 
of (2.1)-(2.2) is a nonempty invariant subset of 
. Hence, 
tends to the largest collection 
of invariant sets of (2.1) contained in 
.
In order to prove (II), we can suppose that 
, for all 
. On the contrary, if for some 
, then the result follows trivially from (I). Since 
, for all 
, 
is a nonincreasing function of 
, which implies that 
, for all 
. As a consequence, solution 
for all 
. This implies that 
, for all 
, which means that 
for some positive constant 
, since set 
is bounded by hypothesis. Therefore, the solution 
is bounded in 
, which allows us to conclude the existence of a real number 
such that 
.
By conditions (A 1)-(A 2) and Lemma 2.2, the orbit 
lies inside a compact subset of 
. Then, by Lemma 2.3, the 
is a nonempty, compact, and connected invariant set.
Next we prove that 
is a constant function on the 
-limit set of (2.1)-(2.2). To this end, let 
be an arbitrary element of 
. So, there exists a sequence of real numbers 
, 
, 
, as 
such that 
as 
. By continuity of 
, we have that 
. As a consequence, 
is a constant function on 
. Since 
is an invariant set, then 
for all 
. Since 
for 
, we have for 
,
which implies that 
and thus 
. The proof is complete.
Remark 3.3.
Theorem 3.2 provides estimates on both 
and 
. For this purpose, we explore the fact that boundeness of 
implies boundedness of 
in 
.
Remark 3.4.
If for each 
, or if for all 
, the solution, 
of (2.1) leaves the set 
for sufficiently small 
and if all conditions of Theorem 3.2 are verified, then we can conclude that solutions of (2.1), with initial condition in 
tend to the largest collection of invariant sets contained in 
. In this case, 
is an estimate of the attracting set in 
, in the sense that the attracting set is contained in 
, and 
is an estimate of the basin of attraction or stability region [8] in 
, while 
and 
are estimates of the attractor and basin of attraction, respectively, in 
.
Next, theorem is a global version of Theorem 3.2 that is useful to obtain estimates of global attractors.
Theorem 3.5 (the global uniform invariance principle for functional differential equations).
Suppose function 
satisfies assumptions (A1)-(A2). Assume the existence of Lipschitzian function 
and continuous functions 
. If conditions (ii)-(iii) are satisfied, the following condition
holds and the set 
is a bounded subset of 
, then for each fixed 
one has the following
-
(I)
If
, then-
(1)
the solution
of (1.1) is defined for all 
, -
(2)
, for 
, -
(3)
for all 
, -
(4)
tends to the largest collection 
of invariant sets of (1.1) contained in 
as 
.
-
(1)
-
(II)
If the solution of (2.1)-(2.2) with initial condition
satisfies 
, then 
tends to the largest collection of invariant sets contained in 
, as 
.
, then
of (1.1) is defined for all 
,
, for 
,
for all 
,
tends to the largest collection 
of invariant sets of (1.1) contained in 
as 
.
satisfies 
, then 
tends to the largest collection of invariant sets contained in 
, as 
.Proof.
The proof of (I) is equal to the proof of (I) of Theorem 3.2. If for some 
,
, then the proof of (II) follows immediately from the proof of item (I). Suppose that 
for all 
. So, from (3.4), we have that 
and 
, for all 
. Then, we can conclude that function 
is nonincreasing on 
and bounded from below. Therefore, there exists 
such that 
. Using assumptions (A 1)-(A 2) and Lemma 2.3, we can infer that the 
-limit set 
of (2.1)-(2.2) is a nonempty, compact, connected, and invariant set. Let 
, then there exists an increasing sequence 
, 
, 
, as 
such that 
as 
. Using the continuity of 
, we have 
, for all 
. This fact and the invariance of the 
-limit set allow us to conclude that
Therefore, 
for all 
, which implies 
.
Theorem 3.5 provides information about the location of a global attracting set. More precisely, if the same conditions of Remark 3.4 apply, then 
is an estimate of the attracting set.
In order to provide estimates of the attractor and the basin of attraction via Theorems 3.2 and 3.5, we have to calculate the maximum of function 
on the set 
. This is a nonlinear programming problem in 
. In our applications, functions 
and 
are usually convex functions, which allows us to use the next result that simplifies the calculation of the maximum of 
in practical problems.
Lemma 3.6 (see [14]).
Let 
be a Banach space with norm 
and let 
be a continuous convex function. Suppose that 
is a bounded, closed, and convex subset in 
such that 
attains the maximum at some point 
. Then, 
attains the maximum on 
, the boundary of the set 
.
4. Applications
Example 4.1 (a retarded version of Lorenz system).
In this example, we will find a uniform estimate of the chaotic attractor of the following retarded version of the Lorenz system:
For 
, the term with retard disappears and the problem is reduced to the original ODE Lorenz system model.
Parameters 
, 
, and 
are considered unknown. The expected values of these parameters are 
, 
, and 
. For these nominal parameters, simulations indicate that system (4.1) has a global attracting chaotic set. We assume an uncertainty of 
in these parameters. More precisely, we assume parameters belong to the following compact set:
where 
, 
, 
, 
, 
, and 
. The time delay and parameter 
are considered constants. In this example, we have assumed 
and 
.
Consider the following change of variables:
In these new variables, system (4.1) becomes
In order to write system (4.4) into the form of (1.1), consider 
defined as 
,
, 
, where
Thus, system (4.1) becomes
with initial condition given by 
.
On
consider the Lyapunov-like functional 
given by
Consider 
as being
and 
as being
Then, we have
and as a consequence, condition (ii) of Theorem 3.2 is satisfied. Function 
is radially unbounded. So, condition (i) is satisfied for any real number 
. Calculating the derivative of 
along the solution we get,
Rewriting the previous inequation in a matrix form, one obtains
where 
, 
, and 
.
We can make the quadratic term positive definite with an appropriate choice of parameter 
. Using Sylvester's criterion, we obtain the following estimates for the parameters:
The previous inequalities permit us to infer that if we choose 
sufficiently small, then the matrix that appears in (4.13) becomes positive definite. This was expected because system (4.1) becomes the classical Lorenz system for 
. If 
satisfies the previous inequality, then condition (iii) is satisfied.
In order to estimate the supremum of function 
in the set 
, consider the set
It can be proved that the supremum of 
calculated over set 
is equal to the supremum of function 
over set 
. Since 
is a closed and convex set in 
, Lemma 3.6 guarantees that the maximum of function 
is attained on the boundary of 
. With that in mind, consider the following Lagrangian functional:
We get the following conditions to calculate the maximum of function 
in the set 
:
The maximum is attained at 
, 
, 
and
which implies that solutions of problem (4.1) are ultimately bounded and enter in the set
in finite time. So, by Theorem 3.2, the set
is an estimate for the basin of attraction and the set
is an estimate for the attraction set for the system (4.5).
Example 4.2 (generalized Rössler circuit).
Consider the classical Rössler system of ordinary differential equation with bounded time delay in the variable 
Let 
be the space formed by functions 
, such that 
, 
, and 
and consider function 
defined by
The nominal values of parameters are 
and 
and an uncertainty of 
is admitted in the determination of these parameters. In this case, we have that 
, 
, 
, 
. In addition, we consider the following set of parameters 
:
Consider the scalar functions 
, 
, 
, and 
as being, respectively,
Then, we have
and condition (ii) of Theorem 3.2 is satisfied. We also have
which implies that
If we take 
, then we have
for all 
.
Next, we will make the association 
, 
, 
with 
and considering the Lagrangian function
then we have the following extreme conditions:
After some calculation, we have 
, 
, and 
, and this implies
As consequence of Theorem 3.2, the generalized Rössler attractor tends to the largest invariant set contained in the set
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Acknowledgment
This research was supported by FAPESP Grant no. 07/54247-6.
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Rabelo, M., Alberto, L.F.C. An Extension of the Invariance Principle for a Class of Differential Equations with Finite Delay. Adv Differ Equ 2010, 496936 (2010). https://doi.org/10.1155/2010/496936
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DOI: https://doi.org/10.1155/2010/496936