An Extension of the Invariance Principle for a Class of Differential Equations with Finite Delay
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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 Open image in new window to be positive in some bounded sets of the state space while the classical invariance principle assumes that Open image in new window . 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.
Keywords
Lipschitzian Function Chaotic Attractor Global Attractor Functional Differential Equation Invariance Principle1. 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, 5, 6, 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 Open image in new window satisfying Open image in new window and studies the implication of the existence of such function on the Open image in new window limit of solutions.
More recently, the invariance principle was successfully extended to allow the derivative of the scalar function Open image in new window 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 Open image in new window .
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 Open image in new window 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 Open image in new window . Despite Open image in new window is defined on the state space Open image in new window , we explore the boundedness of time delay to obtain estimates of the attractor in Open image in new window , 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, Open image in new window will denote the Euclidean Open image in new window dimensional vector space, with norm Open image in new window , and Open image in new window will denote the space of continuous functions defined on Open image in new window into Open image in new window , endowed with the norm Open image in new window .
Definition 2.1.
A solution of (2.1)(2.2) is a function Open image in new window defined and continuous on an interval Open image in new window , such that (2.2) holds and (2.1) is satisfied for all Open image in new window .
If for each Open image in new window and for a fixed Open image in new window , the initial value problem (2.1)(2.2) has a unique solution Open image in new window , then we will denote by Open image in new window the orbit through Open image in new window , which is defined as Open image in new window . Function Open image in new window belongs to the Open image in new window limit set of Open image in new window , denoted by Open image in new window , if there exists a sequence of real numbers Open image in new window , with Open image in new window as Open image in new window , such that Open image in new window , with respect to the norm of Open image in new window , as Open image in new window .

(A 1) For each Open image in new window , there exists a real number Open image in new window such that, Open image in new window for all Open image in new window and for all Open image in new window .
 (A 2)For each Open image in new window , there exists a real number Open image in new window , such that(2.3)

for all Open image in new window , Open image in new window and all Open image in new window .
Under conditions (A 1)(A 2), the problem (2.1)(2.2) has a unique solution that depends continuously upon Open image in new window , see [4]. Moreover, one has the following result.
Lemma 2.2 (compacity of solutions [4]).
If Open image in new window is a solution of (2.1)(2.2) such that Open image in new window is bounded, with respect to the norm of Open image in new window , for Open image in new window and assumptions (A1)(A2) are satisfied, then Open image in new window is the unique solution of (2.1)(2.2). Moreover, the flow Open image in new window through Open image in new window belongs to a compact subset of Open image in new window for all Open image in new window .
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 Open image in new window .
for every Open image in new window and Open image in new window . The following lemma is a wellknown result regarding the properties of Open image in new window limit sets of compact orbits Open image in new window [6].
Lemma 2.3 (limit set properties).
Let Open image in new window be a solution of (2.1)(2.2) and suppose that (2.4) is satisfied. Then, the Open image in new window limit set of Open image in new window is a nonempty, compact, connected, invariant set and Open image in new window , as Open image in new window .
Lyapunovlike functions may provide important information regarding limitsets 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.
Remark 2.5.
Remark 2.6.
Generally, if Open image in new window is continuously differentiable and Open image in new window is a solution of (2.1), then the scalar function Open image in new window is differentiable in the usual sense for Open image in new window . In spite of that, it is possible to guarantee the existence of Open image in new window for Open image in new window assuming weaker conditions; for example, if Open image in new window is locally Lipschitzian, it is possible to show that Open image in new window is well defined. For more details, we refer the reader to [12].
3. Main Result
Theorem 3.1 (the invariance principle).
Let Open image in new window be a function satisfying assumptions (A1) and (A2) and Open image in new window a continuous scalar function on Open image in new window . Suppose the existence of positive constants Open image in new window and Open image in new window such that Open image in new window for all Open image in new window . Suppose also that Open image in new window for all Open image in new window . If Open image in new window is the set of all points in Open image in new window where Open image in new window and Open image in new window is the largest invariant set in Open image in new window , then every solution of (3.1), with initial value in Open image in new window approaches Open image in new window as Open image in new window .
In Theorem 3.1, constants Open image in new window and Open image in new window are chosen in such a manner that the level set, Open image in new window , that is, the set formed by all functions Open image in new window such that Open image in new window , is a bounded set in Open image in new window . Using this assumption, it is possible to show that the solution Open image in new window , starting at Open image in new window with initial condition Open image in new window , is bounded on Open image in new window for Open image in new window . 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 Open image in new window be positive in some bounded sets of Open image in new window . Since in practical applications it is convenient to get information about the behavior of solutions in Open image in new window , our setting is slightly different from that used in Theorem 3.1.
 (i)
Open image in new window is a bounded set in Open image in new window ,
 (ii)
Open image in new window , for all Open image in new window ,
 (iii)
Open image in new window , for all Open image in new window ,
 (iv)
there is a real number Open image in new window such that Open image in new window .
Under these assumptions, a version of the invariance principle, which is uniform with respect to parameter Open image in new window , is proposed in Theorem 3.2.
Theorem 3.2 (uniform invariance principle for retarded functional differential equations).
 (I)If Open image in new window . Then,
 (1)
the solution Open image in new window of (1.1) is defined for all Open image in new window ,
 (2)
Open image in new window , for Open image in new window , where Open image in new window ,
 (3)
Open image in new window , where Open image in new window , for all Open image in new window ,
 (4)
Open image in new window tends to the largest collection Open image in new window of invariant sets of (1.1) contained in Open image in new window as Open image in new window .
 (1)
 (II)If Open image in new window . Then,
 (1)
Open image in new window is defined for all Open image in new window ,
 (2)
Open image in new window belongs to Open image in new window for all Open image in new window ,
 (3)
Open image in new window belongs to Open image in new window for all Open image in new window ,
 (4)
Open image in new window tends to the largest collection Open image in new window of invariant sets of (1.1) contained in Open image in new window .
 (1)
Proof.
In order to show (I), we first have to prove that Open image in new window , for all Open image in new window . Let Open image in new window be a solution of (2.1), satisfying the initial condition Open image in new window and suppose the existence of Open image in new window such that Open image in new window , that is, Open image in new window . By assumption, we have Open image in new window and Open image in new window . Using the Intermediate Value Theorem [13] and continuity of Open image in new window with respect to Open image in new window , it is possible to show the existence of Open image in new window such that Open image in new window and Open image in new window for all Open image in new window . On the other hand, since function Open image in new window is nonincreasing on Open image in new window we have Open image in new window , but this leads to a contradiction, because Open image in new window . Therefore, Open image in new window , for all Open image in new window , which implies that Open image in new window is bounded and defined for all Open image in new window . By definition of Open image in new window , we have that Open image in new window for all Open image in new window . Since Open image in new window is a bounded set, according to Lemma 2.2, the orbit Open image in new window belongs to a compact set. As a consequence of Lemma 2.3, the Open image in new window limit set, Open image in new window of (2.1)(2.2) is a nonempty invariant subset of Open image in new window . Hence, Open image in new window tends to the largest collection Open image in new window of invariant sets of (2.1) contained in Open image in new window .
In order to prove (II), we can suppose that Open image in new window , for all Open image in new window . On the contrary, if for some Open image in new window , then the result follows trivially from (I). Since Open image in new window , for all Open image in new window , Open image in new window is a nonincreasing function of Open image in new window , which implies that Open image in new window , for all Open image in new window . As a consequence, solution Open image in new window for all Open image in new window . This implies that Open image in new window , for all Open image in new window , which means that Open image in new window for some positive constant Open image in new window , since set Open image in new window is bounded by hypothesis. Therefore, the solution Open image in new window is bounded in Open image in new window , which allows us to conclude the existence of a real number Open image in new window such that Open image in new window .
By conditions (A 1)(A 2) and Lemma 2.2, the orbit Open image in new window lies inside a compact subset of Open image in new window . Then, by Lemma 2.3, the Open image in new window is a nonempty, compact, and connected invariant set.
which implies that Open image in new window and thus Open image in new window . The proof is complete.
Remark 3.3.
Theorem 3.2 provides estimates on both Open image in new window and Open image in new window . For this purpose, we explore the fact that boundeness of Open image in new window implies boundedness of Open image in new window in Open image in new window .
Remark 3.4.
If for each Open image in new window , or if for all Open image in new window , the solution, Open image in new window of (2.1) leaves the set Open image in new window for sufficiently small Open image in new window and if all conditions of Theorem 3.2 are verified, then we can conclude that solutions of (2.1), with initial condition in Open image in new window tend to the largest collection of invariant sets contained in Open image in new window . In this case, Open image in new window is an estimate of the attracting set in Open image in new window , in the sense that the attracting set is contained in Open image in new window , and Open image in new window is an estimate of the basin of attraction or stability region [8] in Open image in new window , while Open image in new window and Open image in new window are estimates of the attractor and basin of attraction, respectively, in Open image in new window .
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).
 (I)If Open image in new window , then
 (1)
the solution Open image in new window of (1.1) is defined for all Open image in new window ,
 (2)
 (3)
 (4)
Open image in new window tends to the largest collection Open image in new window of invariant sets of (1.1) contained in Open image in new window as Open image in new window .
 (1)
 (II)
If the solution of (2.1)(2.2) with initial condition Open image in new window satisfies Open image in new window , then Open image in new window tends to the largest collection of invariant sets contained in Open image in new window , as Open image in new window .
Proof.
Therefore, Open image in new window for all Open image in new window , which implies Open image in new window .
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 Open image in new window 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 Open image in new window on the set Open image in new window . This is a nonlinear programming problem in Open image in new window . In our applications, functions Open image in new window and Open image in new window are usually convex functions, which allows us to use the next result that simplifies the calculation of the maximum of Open image in new window in practical problems.
Lemma 3.6 (see [14]).
Let Open image in new window be a Banach space with norm Open image in new window and let Open image in new window be a continuous convex function. Suppose that Open image in new window is a bounded, closed, and convex subset in Open image in new window such that Open image in new window attains the maximum at some point Open image in new window . Then, Open image in new window attains the maximum on Open image in new window , the boundary of the set Open image in new window .
4. Applications
Example 4.1 (a retarded version of Lorenz system).
For Open image in new window , the term with retard disappears and the problem is reduced to the original ODE Lorenz system model.
where Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , and Open image in new window . The time delay and parameter Open image in new window are considered constants. In this example, we have assumed Open image in new window and Open image in new window .
with initial condition given by Open image in new window .
where Open image in new window , Open image in new window , and Open image in new window .
The previous inequalities permit us to infer that if we choose Open image in new window 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 Open image in new window . If Open image in new window satisfies the previous inequality, then condition (iii) is satisfied.
is an estimate for the attraction set for the system (4.5).
Example 4.2 (generalized Rössler circuit).
for all Open image in new window .
Notes
Acknowledgment
This research was supported by FAPESP Grant no. 07/542476.
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