On the Topological Entropy of Nonautonomous Differential Equations

The purpose of this paper is to extend the concept topological entropy to nonautonomous linear systems. Next, we shall give estimation of the topological entropy for the class of bounded linear equations on n  . Finally, we are about to investigate the invariant properties of one through the transformations such as topological conjugacy, topological equivalence and kinematically similar and then show that topological entropy of one is equal to sum of positive Lyapunov characteristic exponents.


Introduction and Preliminaries
Topological entropy, which describes the complexity of a system, plays an important role in topological dynamical systems.It was first induced by Adler et al. [1] as an invariant of topological conjugacy.Later, Bowen [2] gave equivalent definition of topological entropy which measures for how fast the solutions of dynamical system move part.For a linear map on n  , topological entropy is given by the sum of the logarithms of the operator's eigenvalues with absolute value greater than 1, see [2].Recently, Hoock generates for certain infinite-dimensional linear systems, see [3].In particular, he also showed that topological entropy of a strongly continuous semigroup is given by sum of real parts of the unstable eigenvalues of the infinitesimal generator, if the unstable part is finite-dimensional.The main result of present paper is a generalization of several results for nonautonomous linear systems in the finite-dimensional case.Now we introduce some basic notations for nonautonomous differential equations.Let the linear equation ( ) , where ( ) A t is the real matrix function which is uniformly bounded on +  .In this paper, we consider ( ) X t is a fundamental matrix solution of (1).For the basic theory of this Equation (1), we refer to the book of Dalecki et al. [4].In order to describe topological entropy for (1), we introduce the concept of spanning and separated sets following [2].For  Analogously, a set S K ⊂ is said to be an ( ) Proof.Suppose S is the ( ) (since the definition of ( )
is a constant matrix for all t ∈  then the definition above coincide the definition of A.-M. Hoock (see [3]), i.e. ( ) The sum is taken over all eigenvalues i λ of A with 0 Y t is other fundamental matrix solution of (1) then

( ) ( )
. Indeed, by [5] there is a converse matrix C such that ( ) ( ) where X is some fundamental matrix solutions.Remark 1.3.Since all norms on n  are equivalent so ( ) h A does not depend on the norm chosen.
We now give an outline of the contents of this paper.In Section 2, one gives the upper estimation for topological entropy for the class of bounded equations.
In particular, we are going to show that one is less than nM where n is dimension of space and M upper bounded of ( ) A t for all t + ∈  .In Section 3, we concentrate the invariant property of topological entropy.As consequence, one shall prove that topological entropy of the periodic equations is equal to the sum of all positive Lyapunov characteristic exponents of them.Finally, Section 4, we shall show that topological entropy of ( 1) is equal to sum of positive Lyapunov characteristic exponents.

Estimation of Topological Entropy for Bounded Linear Equation
In this section we shall give the estimation of topological entropy for bounded linear equation.We shall begin with the following technique lemma. Assuming Proof.It is clear that . Converse, we know that for any , , , .
δ and hence

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The following theorem is the main theorem in this section.
Theorem 2.1.Assume the Equation ( 1) has matrix function ( ) where n is a dimension of matrix ( ) X t a fundamental matrix solutions of (1).First of all, we is proving the following claim ( ) ( ) ( ) where we denote ( ) , B a r is the ball whose centre at a with radius r.Indeed, let K is a compact subset of n inf log 0, lim sup log , , , .

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The last relation is true for all the compact sets K in n  .It means ( ) ( ) ( ) To prove the converse inequality, suppose K δ ∈    with δ is a arbi- trary number such that 1 δ < .Suppose S is an ( ) where Γ is Euler's gamma function, is the volume for ball of radius n.We have Because the last inequality hold for all ( ) , by Lemma 2.1, we obtain ( ) ( ) ( ) From ( 4) and ( 6), the desired our claim hold.For any

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Compare with the claim (3), the desired inequality hold.

Topological Entropy and the Transformations
Let the equation where B is the real matrix function which is also uniformly bounded on +  .Let ( ) X t , ( ) Y t are fundamental matrix solutions of ( 1) and (7), respectively.The solutions of the Equations ( 1) and ( 7) are said to be topological conjugate if there is a homeomorphism : To start this section, we give the question: Is topological entropy invariant property with the topological conjugacy?The first, one considers the simple example.Let the two equations, namely A, B, corresponding, 0 0 and , 0 0 where , , , 0 a b c d > and a c ≠ .As in [3], A.-M. Hoock shown that

( ) ( )
h A a h B c = ≠ = .On the other hand, by Theorem 2.50 in [5] two the equa- tions above are topological conjugacy.Hence, topological entropy is not invariant property with the topological conjugacy.The following, we shall give a compare critical of topological entropy in term of homeomorphism h and a sufficient condition of homeomorphism h such that topological entropy is invariant.

( ) ( )
Proof.Without lost the generation, we suppose that h is nonexpanding map.
In other word, ( ) ( ) For the case of discrete, topological entropy is invariant to topological conjugacy, but it is no longer true for continuous case, even for the coefficient matrix is constant.This implies that topological entropy becomes more complex in continuous case.In other words, topological conjugacy cannot preserve the speed of the lose information for nonautonomous linear equations.
The following, we are going to consider property of topological entropy with topological equivalence.

4) If ( )
x t is a solution of (1) then is a solution of (7).
Remark 3.2.Condition (4) implies the equality A straightforward verification shows that topological equivalence is an equivalence relation in the class of nonautonomous equations.
The Equations ( 1) and ( 7) are said to be kinematically similar if there exists a continuous differential invertible matrix function ( ) x S t v = takes the solutions of (1) on to the solutions of (2).
Remark 3.4.If the Equations ( 1) and ( 7) are kinematically similar, then they are topological equivalence.Indeed, in the definition of topological equivalence it suffices to set ( ) ( ) where ( ) S t is the function realizing the kinematically similarity.
The following theorem presents the sufficient condition of topological equivalence which prevents topological entropy.Proposition 3.2.Let ( 1) and ( 7) are topological equivalence with the homeomorphism , h g satisfy , and , , , h t x h t y t x y g t x g t y t x y , t t γ γ are scalar bounded function on n  and 1 2 , α α positive constants.Then ( ) ( ) Proof.Let ( ) X t , ( ) Y t are fundamental matrix solutions of ( 1) and ( 7), re- spectively.Suppose , by definition of spanning set, there ex- where ( ) lim lim sup , , , , lim lim sup , , 0, , , By the similar proof above, we also have ( ) ( ) It is clear that if (1) and ( 7) are kinematically similar then they satisfy all hypothesis of previous proposition with ( ) ( ) (where ( ) S t is kinematic similarity).Therefore the class of kinematically similar nonautonomous equations is invariant topological entropy.
Corollary 3.2.If (1) is periodic equation then ( ) where the sum take all the positive Lyapunov characteristic exponents of that equation.
Proof.By Theorem 2.3.1 in [5] and from previous remark, we obtain

( ) ( ) h A h B =
where ( ) B t are a constant matrix.On the other hand, by [3], where the sum takes all the positive eigenvalues of B. Using [5]   again, we have the complete proof.

Topological Entropy and Lyapunov Exponents
In this section, we show that topological entropy of the Equation ( 1) is equal to sum of positive Lyapunov characteristic exponents.
Given a fundamental matrix solution X of (1), consider the quantities ( ) where i e denotes the ith standard unit vector.When In this section, we can always work with a normal basis ( ) X t which has or- dered Lyapunov exponents With these definitions we get the following theorem.( ) fixed.Assume that we can choose a fix point K x K ∈ such that K is covered by a box ( ) ( ) where i e is the i th unit vectors.Suppose the fundamental matrix solution is arranged in the order a increase of the Lyapunov exponents.For each 0, 0 t β ≥ > , we consider the finite subset of ( ) x X e j e e e λ β λ β λ β ε γ β For any x K ∈ then x can be written the form ( ) From the last equation and definition of Lyapunov characteristic exponents, one obtain   To order the reverse inequality, let ( ) and hence the second inequality is proved. By previous lemma, the following definition of topological entropy makes sense

−
S t (called a kinematic si- milarity) such that ( ) are bounded and such that the transformation ( ) all possible fundamental matrix solutions, then the i λ are called the Lyapunov exponents, or Lyapunov characteristic numbers, and the corresponding fundamental matrix solution is called a normal basic.
and (9), we conclude the proof.

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any t

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Journal of Applied Mathematics and PhysicsWe would prove the following claim.Claim 2. The subset ( )