HORSESHOE IN A CLASS OF PLANAR MAPPINGS

Bursting dynamics of mappings is investigated in this paper. We first present stability analysis of the mappings' equilibria with various parameters. Then for three mappings P, P¯, and P^ with different parameters, we study their powers P4, P¯6, and P^4. We show that the mappings thus obtained are chaotic by giving a rigorous verification of existence of horseshoes in these mappings. Precisely, we prove that the mapping P¯6 is semiconjugate to the 3-shift mapping; the mappings P4 and P^4 are semiconjugate to the 4-shift mapping.


Introduction
Bursting is ubiquitous in physical and biological systems, especially in neural systems [3].Studying networks of bursters in neural systems poses challenging mathematical problems.Even simulating such networks is a computational challenge, since thousands of stiff nonlinear ordinary differential equations (ODEs) may be involved.
Bursting dynamics of mappings has recently been investigated by physicists [1,5,6,2].Using a discrete-time system, say x n+1 = f (x n ), instead of a system of ODEs, provides one with a number of theoretical and computational advantages.For example, it is possible to explore collective behavior of millions of coupled discrete-time bursters with only modest computational effort.
In order to study the phenomenon of bursting, a simple discrete-time model of spiking neurons was proposed in [3], which is of the following form: In this paper, we study a more general mapping as follows.
Let l = 0.04, n = 6, Ī = 41, (1.1) can be rewritten as follows: Depending on the values of the parameters, this mapping can produce a variety of bursting patterns [3].In addition, this simple model can reproduce 20 most fundamental neuron-computational properties of biological spiking neurons summarized in [3].

The stability of equilibria with various parameters
We first study in this section the stability of equilibria of (1.2) for different parameters.
Equation In the next section, we recall a result on horseshoes theory developed in [8,9], which is essential for rigorous verification of existence of chaos in the systems discussed in this paper.

Review of a topological horseshoe theorem
Let X be a metric space, D is a compact subset of X, and f : D → X is mapping satisfying the assumption that there exist m mutually disjoint subsets D 1 ,..., and D m of D, the restriction of f to each D i , that is, f |D i is continuous.Definition 3.1.Let γ be a compact subset of D, such that for each 1 ≤ i ≤ m, γ i = γ ∩ D i is nonempty and compact, then γ is called a connection with respect to D 1 ,...,D m−1 and D m .
Let F be a family of connections γ's with respect to D 1 ,...,D m−1 and D m satisfying the following property: Then F is said to be an f -connected family with respect to D 1 ,...,D m−1 and D m .Theorem 3.2.Suppose that there exists an f -connected family F with respect to D 1 ,..., D m−1 and D m .Then there exists a compact invariant set For the proof of this theorem, see [9].Here the "semiconjugate to the m-shift" is conventionally defined in the following sense.If there exists a continuous and onto mapping such that h • f = σ • h, then f is said to be semiconjugate to σ, where σ is the m-shift (mapping) and m is the space of symbolic sequences to be defined below.Let S m = {1, ...,m} be the set of nonnegative successive integer from 1 to m.Let m be the collection of all one-infinite sequences with their elements of S m , that is, every element s of m is of the following form: Now consider another sequence si ∈ S m .The distance between s and s is defined as with the distance defined as (3.4), m is a metric space, and the m-shift mapping σ : m → m is defined as follows [7]: For the concept of topological entropy, the reader can refer to [4].We just recall the result stated in Lemma 3.3, which will be used in this paper.Lemma 3.3.Let X be a compact metric space, and f : X → X a continuous mapping.If there exists an invariant set where h( f ) denotes the entropy of the mapping f .In addition, for every positive integer k,

.7)
A well-known fact is that if the entropy of continuous mapping is positive, then the mapping is chaotic [4].In Figure 4.2, dl i , are the left sides of d i , dr i are the right sides of d i , i = 1,2,3,4.P 4 (dl i ) and P 4 (dr j ) lie on the left side of dl 1 , i = 1,3, j = 2,4; P 4 (dl i ) and P 4 (dr j ) lie on the right side of dr 4 , i = 2,4, j = 1,3.
It is easy to see from connects dl 1 and dr 4 from the above arguments, P 4 (l ∩ d i ) also connects dl 1 and dr 4 , i = 2,3,4.Therefore, it is easy to see, in view of Definition 3.1, that there exists a P 4family with respect to these four subsets d 1 , d 2 , d 3 and d 4 for the mapping P 4 .It follows from Theorem 3.2 that there exists an invariant set K 1 of D, such that P 4 restricted to K 1 is semiconjugated to 4-shift dynamics.Let h(P 4 ) be the entropy of the mapping P 4 , it can be concluded from Lemma 3.3 that h(P 4 ) ≥ h(σ) = log 4, consequently the entropy of the mapping P is not less than (1/2)log2.From above facts, it can be concluded that the mapping P 4 is chaotic.
Because K 1 is an invariant set, every point in K 1 cannot skip out of the quadrangle D under iteration of the mapping P.Then, the resetting has nothing to do with the points in K 1 .Figure 4.3 is the quadrangle D and its image under the sixth iteration of the mapping P, that is P6 .The image of D under P6 has discontinuous part because that not all of the points on the edge of D are reset at the same time.Three disjoint quadrangles as defined in Definition 3.1 can also been constructed, they are h 1 , h 2 , h 3 .The vertexes' coordinates of h 1 , h 2 , h 3 are listed in Table 4.2.

Horseshoes for the iteration of the mapping
In Figure 4.4, hl i , are the left sides of h i , hr i are the right sides of h i , i = 1,2,3.P6 (hl 1 ), P6 (hr 2 ) and P6 (hr 3 ) lie on the left side of hl 1 , P6 (hr 1 ), P6 (hl 2 ) and P6 (hl 3 ) lie on the right side of hr 3 .It is easy to see from Figure 4.4 that every line l lying in D and connecting the side hl 1 and hr 3 has nonempty connections with h 1 , h 2 , and h 3 .Furthermore, P6 (l ∩ h i ) connects hl 1 and hr 3 from the above arguments, i = 1,2,3.In view of Definition 3.1, there exists a P6 -family with respect to these three subsets h 1 , h 2 , and h 3 for the mapping P6 .It follows from Theorem 3.2 that there exists an invariant set K 2 of D, such that P6 restricted to K 2   is semiconjugated to 3-shift dynamics.Let h( P6 ) be the entropy of the mapping P6 , it can be concluded from Lemma 3.3 that h( P6 ) ≥ h(σ) = log3, consequently the entropy of the mapping P is not less than (1/6)log3.The entropy of the mapping P is positive, then it can be concluded that the mapping P6 is chaotic.Because K 2 is an invariant set, every point in K 2 cannot skip out of the quadrangle D under iteration of the mapping P.Then, the resetting has nothing to do with the points in K 2 .

4.3.
Horseshoes for the iteration of the mapping I.The mapping I in [3] is a typical mapping with two saddle points.The mapping can produce a variety of bursting patterns, including those corresponding to IB (intrinsically bursting) and CH (chattering) neocortical neurons.A discussion on chaotic property in this mapping is given in this section.
Figure 4.5 is the image of the quadrangle D and its image under the fourth iteration of the mapping P, that is P 4 .It can be seen from Figure 4.5 that there exist discontinuous points.The reason is the same as that in Section 4.1: the resetting (1.3) is responsible for this phenomenon.Four disjoint quadrangles as defined in Definition 3.1 can also been constructed, they are k 1 , k 2 , k 3 , k 4 .The vertexes' coordinates of k 1 , k 2 , k 3 , k 4 are listed in Table 4.3.Their images under the fourth iteration of the mapping P are shown in Figure 4.6.
In Figure 4.6, the four quadrangles are k 1 , k 2 , k 3 , k 4 from left to right in turn.kl i is the left side of k i , kr i is the right side of k i , i = 1,2,3,4.P 4 (kl i ) and P 4 (kr j ) lie on the left side of kl 1 , i = 2,4, j = 1,3; P 4 (kl i ) and P 4 (kr j ) lie on the right side of kr 4 , i = 1,3, j = 2,4.With the same analysis of the fourth paragraph in Sections 4.1 and 4.2, it can be concluded that there exists a P 4 -family with respect to these four subsets d 1 , d 2 , d 3 and d 4 for the mapping P 4 , and an invariant set K 3 of D, such that P 4 restricted to K 3 is Because K 3 is an invariant set, every point in K 3 cannot skip out of the quadrangle D under iteration of the mapping P.Then, the resetting has nothing to do with the points in K 3 .

Conclusion
Studying networks of bursters in neural systems poses challenging mathematical problems.A bursting dynamics of mappings has been investigated in this paper.Based on the stability analysis of the system's equilibria with various parameters, we study three mappings of the system, that is P 4 , P6 and P 4 .We show that the mappings are chaotic by giving a rigorous verification for existence of horseshoes in these mappings.We prove that the mapping P6 is semiconjugate to the 3-shift mapping, the mapping P 4 and P 4 are semiconjugate to the 4-shift mapping.Then, the mappings P 4 , P6 , and P 4 are chaotic.

(1. 1 )
Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2006, Article ID 67908, Pages 1-10 DOI 10.1155/DDNS/2006/67908 (1.2) has two equilibria, let A and B denote them.Let the modules of the eigenvalues of A be m 11 and m 12 , the modules of the eigenvalues of B be m 21 and m 22 .m 11 , m 12 , m 21 , and m 22 vary with the parameters a and b as shown in Figure 2.1, m i, j = m i, j (a,b), i, j = 1,2.Denote by M the plane z = 1 in Figure 2.1.With the reference plane M, it can be concluded that A is a saddle point when a and b vary from 0 to 0.99, respectively, and B may be a saddle point, a stable equilibrium, or an unstable equilibrium when a and b varies from 0 to 0.99.Typical values for c and d are c = −56, d = −16.When a = 0.24 and b = 0.6, let P denote the mapping II.The mapping P has two saddle points (−10.2786,−6.1672)and (−99.7214,−59.8328).When a = 0.64 and b = 0.2, let P denote the mapping II, the mapping P has a saddle point (−10.8917,−8.7134)and a stable equilibrium (−94.1083,−75.2866).
m 11 as a function of the parameters a and b. m 12 as a function of the parameters a and b. m 21 as a function of the parameters a and b.
m 22 as a function of the parameters a and b.
Figure 4.1 is the quadrangle D and its image under the fourth iteration of the mapping P(that is P 4 ).It can be seen from Figure 4.1 that there exist discontinuous points.The resetting (1.3) is responsible for this phenomenon, because not all of the points on the edge of D are reset at the same time.Four disjoint quadrangles as defined in Definition 3.1 can been constructed, they are d 1 , d 2 , d 3 , d 4 .The vertexes' coordinates of d 1 , d 2 , d 3 , d 4 are listed in Table 4.1.Their images under the mapping P 4 are shown in Figure 4.4.
P with a saddle point and a stable equilibrium.Let P denote the mapping corresponding to the parameters a = 0.64 and b = 0.2, the mapping P has a saddle point and a stable equilibrium.With these parameters, we find a quadrangle D with its vertexes (−81.4747,−72.2222),(−39.4470,−84.9708), (−41.2903,−91.8713),and (−83.3180,−83.5673)by means of careful computation.

− 4 P 4 (
130 −120 −110 −100 −90 −80 −70 −60 −50 −40 −dr 1 ) (a) The four disjoint subsets of D and the image of d 1 under the mapping P 4 .The four disjoint subsets of D and the image of d 2 under the mapping P 4 .The four disjoint subsets of D and the image of d 3 under the mapping P 4 .The four disjoint subsets of D and the image of d 4 under the mapping P 4 .

Figure 4 . 2 .
Figure 4.2.The four disjoint subsets of D and their images under the mapping P 4 .

Figure 4 . 3 .
Figure 4.3.The quadrangle D and its image under the mapping P6 .

Figure 4 . 4 .
Figure 4.4.The three disjoint subsets of D and their image under the mapping P6 .