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In this paper, we provide a new approach to study the geometry of attractor. By applying category, we investigate the relationship between attractor and its attraction basin. In a complete metric space, we prove that the categories of attractor and its attraction basin are always equal. Then we apply this result to both autonomous and non-autonomous systems, and obtain a number of corresponding results.

Attractors of a given system are of crucial importance, this is because that much of longtime dynamics is represented by the dynamics on and near the attractors. It is well known that the global attractors of dynamical systems can be very complicated. The geometry can be very pathological, even in the finite dimensional situation. To have a better understanding on the dynamics of a system, it is quite necessary for us to study the topology and geometry of the attractors. In the past few decades, there appeared many studies. In [

In this paper, by using Ljusternik-Schnirelmann category (category for short), we try to provide a new approach to studying the geometry of the global attractor. Category is a topological invariant, which often be used in the estimate of the lower bound of the number of critical points, see [

We will prove the main results in Section 3 and give some applications in Section 4. Before that we provide some preliminaries and results in Section 2.

We recall some basic definitions and facts in the theory of dynamical systems for semiflows on complete metric spaces. Let

Definition 2.1 A semiflow (semidynamical system) on

We usually write

From now on, we will always assume that there has been given a semidynamical system

The asymptotic compactness property (A) is fulfilled by a large number of infinite dimensional semiflows generated by PDEs in application [

Let

The attraction basin of

The set

Definition 2.2 A compact set

Let

Definition 2.3 A function

In order to prove our result, we need following theorem (see Theorem 3.5 in [

Theorem 2.4 ([

where

Remark 2.5 We emphasize that the

In the following, we recall some basic results on the Ljusternik-Schnirelmann category (category for short).

Definition 2.6 Let

A set

The category defined above has properties as follows.

Lemma 2.7 Properties for the category:

1)

2) (Monotonicity)

3) (Subadditivity)

4) (Deformation nondecreasing) If

5) (Continuity) If

6) (Normality)

For the proof of this lemma, we refer readers to [

Remark 2.8 By (2) and (5), we can easily obtain that if

Just by the definition of category, we can prove the following lemma:

Lemma 2.9 Let

The main results can be stated as follows:

Theorem 3.1 Let

Proof. Since

Since

If we find a set

by using monotonicity again and (3.2}), we have

Then combine (3.1}) and (3.4), we will obtain the result

Now the rest of the work in this proof is in finding the appropriate set

By the Remark 2.5,

Hence, let

We use the method in [

Here

Then

Since

Now we just let

Now to extend our result to non-autonomous case, we consider a skew-product system, which consists of a base semiflow, and a semiflow on the phase space that is in some sense driven by the base semiflow. More precisely, the base semiflow consists of the base space

The dynamics on the phase space

satisfy the cocycle property

1)

2)

3)

Then we can define an autonomous semigroup

If we assume that the autonomous semigroup

Corollary 3.2 Let

In this section, we further apply our results to some special metric space

Example 1. Assume

Proof. Suppose the contrary. Then there exist at least one point

Note that

Thus, we have

On the other hand, by virtue of Theorem, we have

Using similar arguments, one can prove the case of

Example 2. In skew-product flow case, we assume

Proof. Suppose the contrary. Then there exist at least one point

Note that

By Lemma 2.9,

Thus, we have

On the other hand, by Virtue of Theorem 3.1, we have

Remark 3.3 If

we can obtain the same result.

Remark 3.4 By Theorem 15.7 in [

This work was supported by the National Nature Science Foundation of China (11261027) and Longyuan youth innovative talents support programs of 2014, and by the innovation Funds of principal (LZCU-XZ2014-05).

This work was supported by the National Nature Science Foundation of China (11261027) and Longyuan youth innovative talents support programs of 2014, and by the innovation Funds of principal (LZCU-XZ2014-05).

Jinying Wei,Yongjun Li,Mansheng Li, (2015) Category of Attractor and Its Application. Journal of Applied Mathematics and Physics,03,725-729. doi: 10.4236/jamp.2015.37086