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Advances in Difference Equations

, 2010:496936 | Cite as

An Extension of the Invariance Principle for a Class of Differential Equations with Finite Delay

  • Marcos Rabelo
  • L. F. C. Alberto
Open Access
Research Article

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 Principle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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, 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 .

The next step along this line of advance is to consider functional differential equations. Let Open image in new window be the space of continuous functions defined on Open image in new window with values in Open image in new window , and Open image in new window a compact subset of Open image in new window . 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 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 .

Let Open image in new window , be a continuous function, and, for each Open image in new window , let Open image in new window be one element of Open image in new window defined as Open image in new window . The element Open image in new window is a segment of the graph of Open image in new window , which is obtained by letting Open image in new window vary from Open image in new window to Open image in new window . Let Open image in new window , be a continuous function. For a fixed Open image in new window and Open image in new window , consider the following initial value problem:

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 .

Generally, on infinite dimensional spaces, such as the space Open image in new window , the boundedness property of solutions is not sufficient to guarantee compactness of the flow Open image in new window . The compactness of the orbit will be important in the development of our invariance results. In order to guarantee the relatively compactness of set Open image in new window and, at the same time, the uniqueness of solutions of (2.1)-(2.2), the following assumptions regarding function Open image in new window are made.

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 .

Let Open image in new window , be a solution of problem (2.1)-(2.2) and suppose the existence of a positive constant Open image in new window in such a manner that

for every Open image in new window and Open image in new window . The following lemma is a well-known 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 .

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 Open image in new window be a continuous scalar function. The derivative of Open image in new window along solutions of (2.1), which will be denoted by Open image in new window , is given by

Remark 2.5.

Function Open image in new window is well defined even when solutions of (2.1) are not unique. In order to be more specific, suppose Open image in new window and Open image in new window are solutions of (2.1), satisfying the same initial condition, then it is possible to show [11] that

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

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 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.

In what follows, let Open image in new window , Open image in new window , and Open image in new window be continuous functions and consider, for each Open image in new window , the sets
Moreover, we assume that the following assumptions are satisfied:

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).

Suppose function Open image in new window satisfies assumptions (A1)-(A2). Assume the existence of a locally Lipschitzian function Open image in new window and continuous functions Open image in new window . In addition, assume that function Open image in new window takes bounded sets into bounded sets. If conditions (i)–(iv) are satisfied, then, for each fixed Open image in new window , we have the following.
  1. (I)
     
  2. (II)
     

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.

Next we prove that Open image in new window is a constant function on the Open image in new window -limit set of (2.1)-(2.2). To this end, let Open image in new window be an arbitrary element of Open image in new window . So, there exists a sequence of real numbers Open image in new window , Open image in new window , Open image in new window , as Open image in new window such that Open image in new window as Open image in new window . By continuity of Open image in new window , we have that Open image in new window . As a consequence, Open image in new window is a constant function on Open image in new window . Since Open image in new window is an invariant set, then Open image in new window for all Open image in new window . Since Open image in new window for Open image in new window , we have for Open image in new window ,

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).

Suppose function Open image in new window satisfies assumptions (A1)-(A2). Assume the existence of Lipschitzian function Open image in new window and continuous functions Open image in new window . If conditions (ii)-(iii) are satisfied, the following condition
holds and the set Open image in new window is a bounded subset of Open image in new window , then for each fixed Open image in new window one has the following
  1. (I)
    If Open image in new window , then
    1. (1)

      the solution Open image in new window of (1.1) is defined for all Open image in new window ,

       
    2. (2)
       
    3. (3)
       
    4. (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 .

       
     
  2. (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.

The proof of (I) is equal to the proof of (I) of Theorem 3.2. If for some Open image in new window , Open image in new window , then the proof of (II) follows immediately from the proof of item (I). Suppose that Open image in new window for all Open image in new window . So, from (3.4), we have that Open image in new window and Open image in new window , for all Open image in new window . Then, we can conclude that function Open image in new window is nonincreasing on Open image in new window and bounded from below. Therefore, there exists Open image in new window such that Open image in new window . Using assumptions (A 1)-(A 2) and Lemma 2.3, we can infer that the Open image in new window -limit set Open image in new window of (2.1)-(2.2) is a nonempty, compact, connected, and invariant set. Let Open image in new window , then there exists an increasing sequence Open image in new window , Open image in new window , Open image in new window , as Open image in new window such that Open image in new window as Open image in new window . Using the continuity of Open image in new window , we have Open image in new window , for all Open image in new window . This fact and the invariance of the Open image in new window -limit set allow us to conclude that

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).

In this example, we will find a uniform estimate of the chaotic attractor of the following retarded version of the 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.

Parameters Open image in new window , Open image in new window , and Open image in new window are considered unknown. The expected values of these parameters are Open image in new window , Open image in new window , and Open image in new window . For these nominal parameters, simulations indicate that system (4.1) has a global attracting chaotic set. We assume an uncertainty of Open image in new window in these parameters. More precisely, we assume parameters belong to the following compact set:

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 .

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 Open image in new window defined as Open image in new window , Open image in new window , Open image in new window , where
Thus, system (4.1) becomes

with initial condition given by Open image in new window .

consider the Lyapunov-like functional Open image in new window given by
Then, we have
and as a consequence, condition (ii) of Theorem 3.2 is satisfied. Function Open image in new window is radially unbounded. So, condition (i) is satisfied for any real number Open image in new window . Calculating the derivative of Open image in new window along the solution we get,
Rewriting the previous inequation in a matrix form, one obtains

where Open image in new window , Open image in new window , and Open image in new window .

We can make the quadratic term positive definite with an appropriate choice of parameter Open image in new window . Using Sylvester's criterion, we obtain the following estimates for the parameters:

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.

In order to estimate the supremum of function Open image in new window in the set Open image in new window , consider the set
It can be proved that the supremum of Open image in new window calculated over set Open image in new window is equal to the supremum of function Open image in new window over set Open image in new window . Since Open image in new window is a closed and convex set in Open image in new window , Lemma 3.6 guarantees that the maximum of function Open image in new window is attained on the boundary of Open image in new window . With that in mind, consider the following Lagrangian functional:
We get the following conditions to calculate the maximum of function Open image in new window in the set Open image in new window :
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 Open image in new window
The nominal values of parameters are Open image in new window and Open image in new window and an uncertainty of Open image in new window is admitted in the determination of these parameters. In this case, we have that Open image in new window , Open image in new window , Open image in new window , Open image in new window . In addition, we consider the following set of parameters Open image in new window :
Then, we have
and condition (ii) of Theorem 3.2 is satisfied. We also have
which implies that
If we take Open image in new window , then we have

for all Open image in new window .

Next, we will make the association Open image in new window , Open image in new window , Open image in new window with Open image in new window and considering the Lagrangian function
then we have the following extreme conditions:
After some calculation, we have Open image in new window , Open image in new window , and Open image in new window , and this implies
As consequence of Theorem 3.2, the generalized Rössler attractor tends to the largest invariant set contained in the set

Notes

Acknowledgment

This research was supported by FAPESP Grant no. 07/54247-6.

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Copyright information

© Marcos Rabelo and L. F. C. Alberto. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidade Federal de Pernambuco, UFPERecifeBrazil
  2. 2.Departamento de Engenharia ElétricaUniversidade de São PauloSão CarlosBrazil

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