Flows of Continuous-Time Dynamical Systems with No Periodic Orbit As an Equivalence Class Under Topological Conjugacy Relation

Home , Flow (mathematics)

Flows of Continuous-Time Dynamical Systems with No Periodic Orbit as an Equivalence Class under Topological Conjugacy Relation

Tahir Ahmad and Tan Lit Ken Department of Mathematics, Faculty of Science, Nanotechnology Research Alliance, Theoretical and Computational Modeling for Complex Systems, University Technology Malaysia 81310 Skudai, Johor, Malaysia

Abstract: Problem statement: Flows of continuous-time dynamical systems with the same number of equilibrium points and trajectories, and which has no periodic orbit form an equivalence class under the topological conjugacy relation. Approach: Arbitrarily, two trajectories resulting from two distinct flows of this type of dynamical systems were written as a set of points (orbit). A homeomorphism which maps between these two sets is then built. Using the notion of topological conjugacy, they were shown to conjugate topologically. By the arbitrariness in selection of flows and their respective initial states, the results were extended to all the flows of dynamical system of that type. Results: Any two flows of such dynamical systems were shown to share the same dynamics temporally along with other properties such as order isomorphic and homeomorphic. Conclusion: Topological conjugacy serves as an equivalence relation in the set of flows of continuous-time dynamical systems which have same number of equilibrium points and trajectories, and has no periodic orbit.

Key words: Dynamical system, equilibrium points, trajectories, periodic orbit, equivalence class, topological conjugacy, order isomorphic, Flat Electroencephalography (Flat EEG), dynamical systems

INTRODUCTION models. Nandhakumar et al . (2009) for example, the dynamics of robot arm is studied. On the other hand, Dynamical system is a system where its temporal (Krishan et al ., 2010) shows that by adopting neuro- evolution from some initial state is dictated by a set of fuzzy system, the design of robust controllers for rules (Eduard et al ., 1999). Another way to understand uncertain non-linear dynamical systems can be done this is, it consists of a set of variables that describe its without resorting to system model simplifications and state and a law that describe the evolution of the state linearization and without imposing structural conditions variables with time (i.e., how the state of the system in on the system uncertainties. This shows that the use of the next moment of time depends on the input and its concept of dynamical system is vast. Mark and Marc state in the previous moment of time) (Eugene, 2007). (2005), the importance of dynamical systems as an There exist various classifications of dynamical systems, approach in understanding development was discussed. e.g. continuous-time (flow or semiflow) or discrete-time, One power of dynamical systems approach is that we continuous state or discrete state and linear or non linear. can tell something or many things, about a system In order to study the system of interest more richly, some without knowing all the details that govern the system authors require the state space to possess certain specific evolution (Eugene, 2007). With the use of this structure such as compact Hausdorff space as in (Tim mathematical theory, it allows one to talk about and Justin, 1989), separable metric space (Flytzanis, stability, equilibrium, bifurcations. 1976), or even smooth manifold as define in (Artur, This study is motivated by the study in (Tahir and 1979). For some survey on the definition of dynamical Tan, 2010), where the authors shows that the dynamics system we direct the readers to (Chen, 2000; Ling and of an epilepsy patients who is having seizure, modeled Anthony, 2007; Sekhar et al ., 1999). as a continuous dynamical system can be transported to Dynamical systems have been used in many areas Flat Electroencephalography (Flat EEG) that are of research, e.g. fluid flow analysis, economic modeled in the same way. This study basically processes (stock market models), physics, medicine, generalizes the theorems obtained from (Tahir and Tan, meteorology, astronomy, and population growth 2010). Corresponding Author: Tahir Ahmad, Department of Mathematics, Faculty of Science, Nanotechnology Research Alliance, Theoretical and Computational Modeling for Complex Systems, University Technology Malaysia 81310 Skudai, Johor, Malaysia 207 J. Math. & Stat., 7 (3): 207-215, 2011

Literature review: Some dynamical systems are Suppose {φt(x 0):t ∈ℜ} is an orbit starting at x0∈X, governed by differential equations and this is the type of then this orbit is periodic if for each x∈{φt(x 0):t ∈ℜ}, dynamical systems that will be discussed in this study. A there exist a time T∈ℜ such that φt(x) = x. In our system of differential equations is relations between its discussion, we restrict our dynamical system to those functions and its derivatives (James, 2007). Sometimes that has no periodic orbit (since it is not possible to they are referred to as vector field because they assign a construct a homeomorphism from a loop to ℜ), those vector (direction and magnitude) to each point in the with the same number of equilibrium points (since a state space (also known as phase space). Two general single point cannot be homeomorphic to ℜ) and with classes of system of differential equations are the same dimension in terms of their state space. We autonomous and non-autonomous. Systems that depend also assume that the number of trajectories of the explicitly on time are referred to as non-autonomous or systems discussed to be same so that our time-dependent vector fields and those which do not homeomorphism will be constructible. For some survey depend explicitly on time are referred to as autonomous. on periodic system, readers may read (Baryarama et al. , Every non-autonomous vector fields can be made into 2005) where the periodicity of the HIV/AIDS epidemic autonomous by redefining the time as a new dependent in a mathematical model that incorporates complacency variable. Thus, for simplification purposes we will just is discussed or (Ibrahim et al ., 2007) for the periodic speak of continuous-time dynamical system from this and non periodic (complex) behavior of a model of point onwards. bioreactor with cell recycling. The type of dynamical system that we will be discussing in this study will be of continuous-time MATERIALS AND METHODS dynamical system non-periodic orbit where the evolution rule is a flow. An example usage on the flow Let x0∈X, be a initial state such that φ(x) = x , concept in different area of research is in (Adeqbie and 0 0 0 Alao, 2007), where the temperature-dependent viscous then the flow will traces out an orbit or trajectory. =φ{ ( ) ∈ℜ } fluid between parallel heated walls is modeled as a Denote this trajectory as Oφ ()x t x:t o . Note φ t o flow. According to (James, 2007), a flow t(x) is a one- that the set ℜ is linearly ordered by the usual less than parameter differentiable mapping φ:ℜ × X → X, such or equal relation, ≤. Taking account of this property, it that it fulfills two properties, which are: is not difficult to see that elements in the set Oφ is t()x o φ( ) = ∀∈ 0 x x xX also linearly ordered, with the ordering relation be defined as follows ∈ℜ And for all t and S :

( ) ≺ ( ) ⇔ ≤ φ φ =φ x,tiiφ ()x x,t jj t ij t t s ts+ t o

φ φ Therefore, the pair (Oφ , ≺ φ ) is linearly where the composition symbol, ° means t° s (x) = to()()x to x φ φ t( s(x)). ordered and we have lemma 1. As an example, see ∈ φ For each x X, t(x) defines a curve in X as t varies (Tahir et al ., 2005), where the state space trajectory of ℜ over . This curve is known as orbit or trajectory. A seizure is augmented and shown to exhibit linear consequence from the property (b) which is also known ordering properties. The construction is however a little as group property is that two distinct trajectories will different from the one we have just presented above not cross. A property of flow is that it is differentiable, because the formulation of dynamical system they started therefore there is an associated ordinary differential off from is slightly different. Nevertheless, they actually equations or more precisely a vector field which assigns meant the same as the augmented trajectory is simply the a vector (magnitude and direction) to each of the points in the state space (James, 2007). To visualize this, evolution of states in seizure over an interval of time.

consider a point chosen from the state space, this point Lemma 1: (Oφ , ≺ φ ) is linearly ordered. will generate a curve in the state space. The movement to()()x to x of this point from one state to another is according to the evolution rule (flow) and the tangent vector to each Now, according to (Steve, 2008), every linear point on this curve is exactly the vectors computed by ≤ using the vector field: ordering, can induce a strict linear ordering,< which can be written as: d fx() = φ() x ∋= t0 dt t x

Thus, we can certainly “transform” the linearly Consequently, the trajectory is composed into a

ordered set (O φt(xo) , ≺ φt(xo) ) into a strictly linearly topological space (O φt(xo) ,τoφt(xo) ), namely linearly ordered set (O φt(xo) , ≺ ⋅φt(xo) ), with the ordering relation ordered topological space (LOTS). LOTS are also a defined as: Generalized Ordered space (GO-space) (Bennet and Lutzer, 1996). In fact, the class of GO-spaces is exactly • (x,t) ≺ φ ( x,t ) the class of all subspaces of LOTS (Bennet et al ., iit()x o jj 2001). Hence from (Kopperman et al ., 1998), which If: says that a GO-space is Hausdorff, we have

• (O , τ ) as a Hausdorff LOTS. φ ()x Oφ () ( ) t o tx o x,tii≺ φ () ( x,t jj ) tx o

Lemma 3: (O φ ,τ φ ) is a Hausdorff LOTS. And: (xo) o t(xo) Now consider two distinct flows from two (x,t) ≠ ( x,t ) continuous-time dynamical systems with no periodic ii jj orbit, φ:ℜ×X →X and ψ:ℜ×Y →Y such that the

Or simply: dimension of the state space of X and Y are same. Using the above argument, we then have two strictly • ⇔ < • • (x,t) ≺ φ ( x,t) t t linearly ordered set (Oφ , ≺ φ ) and (Oψ , ≺ ψ ) iit()x o jj ij to()()x to x to()()y to y and this two strictly linearly ordered set can be And this can be stated formally as. composed into LOTS (Table 1). From Table 1 we notice the existence of the • following lemma. Lemma 2: (Oφ()() , ≺ φ ) is strictly linearly ordered. tox to x Lemma 4: Let (x i, ti) represents a point on a trajectory

Therefore, we have formulated a strictly linearly Oφ at time ti. Similarly, let (y i,t i) represents another t()x o ordered set from a particular trajectory which resulted point on another trajectory Oψ () at time ti. Then, for from a flow of an arbitrary continuous-time dynamical ty o ≤ system. Before we proceed further, let us emphasize ti tj, a point (x i,t i) on the trajectory Oφ precede t()x o once that, whenever we speak continuous-time another point (x j,t j) if and only if a point (y i,t i) on dynamical system from this point onwards, we meant another trajectory Oψ () precede another point (y j,t j). those with the properties stated earlier. In the following, ty o we explain how the trajectory can be composed into a In other words: topological space. ( ) ( ) ⇔ ( ) ( ) x,t11≺φ () x,t 22 y,t 11 ≺ ψ () y,t 22 Using the definition of interval topology in tx o ty o (Kopperman et al ., 1998) one can define a topology on a strictly linearly ordered set by using a subbasis which Proof: From the linear order of continuous-time dynamical system 1 (Table 1) we have: consists of the collection of all order open rays that can be defined as follow: ( ) ⇔ ≤ x,tii≺ φ () ( x,t jj) t ij t tx o ∈ •  {()x,t Oφ() |x,t() ≺ φ () () a,t } tox to x  And from the linear order of continuous-time   S= ,x,t{}() ∈ O |b,t() ≺• () x,t (1) dynamical system 2 (Table 1) we have: Oφ () φ()x φ () x  tx o to to  ϕ ∀()() ∈ ( ) ≺ ( ) ⇔ ≤ ,,Oφ() : a,t,b,t O φ ()  y,tiiψ ()y y,t jj t ij t tx o t x o  t o

Combining these two, we will have: Equation 1 will generate a topology called interval topology. The elements of the topology are the union of ⇔ (x,t) ≺φ ( x,t) ( y,t) ≺ ψ ( y,t ) 11t()x o 22 11t()y o 22 any finite intersections of elements of Eq. 1. Note that each element in Eq. 1 is itself open in the topology As desired. generated by it. We denote this topology as: θ → Next, define a function : Oφ O ψ as to()()x to y n   τ =αα= ∈  O|∪ ∩ s:sS jjO   φ ()x   φ ()x t o = = t o θ(( )) = ( ) i1 j1  i  x,t y,t 209 J. Math. & Stat., 7 (3): 207-215, 2011

Table 1: Two continuous-time dynamical system Continuous-time dynamical system 1 Type Continuous-time dynamical system 2 φℜ×: X → X ∋ X =ℜ n Flow ψℜ×: Y → Y ∋ Y =ℜ n =φ ∈ℜ =ψ ∈ℜ Oφ { () x:t } Orbit Oψ { () y:t } t()x o t o t()y o t o ⇔ ≤ ⇔ ≤ ()x,t≺ φ ( x,t) t t Linear order relation ()y,t≺ ψ ( y,t) t t iit()x o jj ij iit()y o jj ij

(Oφ()() , ≺ φ ) Linearly ordered set (Oψ()() , ≺ ψ ) tox to x toy to y • ⇔ < • ⇔ < ()x,t≺ φ ( x,t) t t Strict linear order ()y,t≺ ψ ( y,t) t t iit()x o jj ij iit()y o jj ij • • (Oφ()() , ≺ φ ) Strictly linearly ordered set (Oψ()() , ≺ ψ ) tox to x toy to y ∈ •  ∈ •  {()x,t Oφ() |x,t() ≺ φ () () a,t,} {()y,t Oψ() |y,t() ≺ ψ () () c,t,} tox to x  toy to y  ∈ •  ∈ •  {}()x,t Oφ() |b,t() ≺ φ () () x,t,  {}()y,t Oψ() |d,t() ≺ ψ () () y,t,  S = tox to x Subbasis S = toy to y Oφ ()   Oψ ()   tx o ϕ ty o ϕ ,O φ ()  ,O ψ ()  tx o  ty o  ∀ ∈ ∀ ∈ :()() a,t,b,t O φ ()  :()() c,t,d,t O ψ ()  tx o  ty o  n   n   τ =αα= ∈ τ =ββ= ∈ O|∪ ∩ s:sS jjO   Interval topology O|∪ ∩ s:sS jjO   φ ()x   φ ()x ψ ()y   ψ ()y t o = = t o t o = = t o i1 j1  i  i1 j1  i  (O , τ ) Hausdorff LOTS (O , τ ) φ ()x Oφ ψ ()y Oψ t o t()x o t o t()y o

∈ θ((x,t)) = ( y,t ) There exist (x,t) O φ Theorem 1: is a bijective function. t()x o Such that θ((x,t)) = ( y,t ) Proof: Therefore, θ((x,t)) = ( y,t ) is surjective Function: Suppose (x,t) = ( x,t ) ii jj Since θ((x,t)) = ( y,t ) is injective and surjective, thus it ( ) = ( ) Then by lemma 4, y,tii y,t jj is bijective. ■ Next, using the bijective function we show that ≺ ≺ θ() = θ (Oφ()()x , φ x ) and (Oψ()()y , ψ y ) are order or ( x,tii) (( x,t jj )) to to to to isomorphic.

Thus, θ((x,t)) = ( y,t ) is indeed a function. ( ≺ ) Theorem 2: Oφ()()x , φ x is order isomorphic to to to (Oψ()() , ≺ ψ ) θ() = θ toy to y Injective: Suppose ( x,tii) (( x,t jj ))

Proof: From lemma 4 we have: By the defined function, this implies:

⇔ (x,t) ≺φ ( x,t) ( y,t) ≺ ψ ( y,t ) 11t()x o 22 11t()y o 22 (y,t) = ( y,t ) ii jj Substituting the function θ((x,t)) = ( y,t ) into this Now by lemma 4, we then have: lemma, we then have

(x,t) = ( x,t ) ii jj ⇔ θ θ (x,t) ≺φ ( x,t) ( x,t) ≺ ψ ( x,t ) 11t()x o 22 11t()y o 22

Thus, θ((x,t)) = ( y,t ) is injective. Therefore, (Oφ()() , ≺ φ ) is order isomorphic to tox to x ( )∈ ( ≺ ) Surjective: For all y,t O ψ Oψ()()y , ψ y . t()y o to to 210 J. Math. & Stat., 7 (3): 207-215, 2011

θ(( )) = ( ) ∈ • We now proceed to show that x,t y,t is a Since {()x,t Oφ |b,t() ≺ φ () x,t } is an order-open to()x to() x homeomorphism. ray.

∈ • Thus, {()x,t Oφ() |b,t() ≺ φ () () x,t } is an open Theorem 3: The function θ((x,t)) = ( y,t ) is tox to x

set in Oφ () . continuous. tx o

Proof: Consider the set Case 3: The set φ, i.e., empty set. θ−1 (φ)=φ ϕ∈ Clearly, (trivial). Since SO φt()x o ∈ •  {()y,t Oψ |y,t() ≺ ψ () c,t } to()y to() y φ   Thus, is an open set in Oφ ()x .   t o =() ∈ () • () SO  ,y,t{} Oψ() |d,t≺ ψ () y,t  ψt()y o toy to y   Case 4: The set Oψ () , i.e. the whole set. ϕ ∀()() ∈ ty o ,,Oψ()y : a,t,b,t O ψ () y  t o t o  θ−1 ( ) = θ−1 Clearly, Oψ()y O φ () x since is surjective. to to ∈ Since Oφ () S O Thus, Oφ () is an open set in Case 1: Sets of the form: tx o φt()x o tx o

Oφ ()x . ∈ • t o {()y,t Oψ() |y,t() ≺ ψ () () c,t } toy to y Combining these four cases, we have the inverse

Clearly: image of each element in the subbasis SO is open in ψt()y o

(O , τ ) . Thus, θ((x,t)) = ( y,t ) is a continuous −1 • φ ()x Oφ () θ ∈ t o tx o ({()y,t Oψ |y,t() ≺ ψ () c,t }) to()y to() y function that maps from (O , τ ) to • φ ()x Oφ () = ∈ t o tx o {}()x,t Oφ |x,t() ≺ φ () a,t to()x to() x τ (Oψ () , O ) . ty o ψt()y o ∈ θ−1 For some (a,t) O φ () , since is bijective. tx o −1 ∈ • θ (( Notice that the set {()x,t Oφ() |x,t() ≺ φ () () a,t } Theorem 4: The function y,t))=(x,t) is continuous. tox to x is an order-open ray in Oφ () , i.e. element of subbasis. Proof: Use similar argument in the proof for theorem 3. tx o −1 In that case, the function θ (( y,t))=(x,t) is bijective, ∈ • Since {()x,t Oφ() |x,t() ≺ φ () () a,t } is an order-open tox to x continuous and open. Thus, it is a homeomorphism ∈ • τ τ ray Thus, {()x,t Oφ() |x,t() ≺ φ () () a,t } is an open set from (Oφ () , O ) to (Oψ () , O ) . Next we show tox to x tx o φt()x o ty o ψt()y o in Oφ . that the flows of the two continuous-time dynamical t()x o system are topologically conjugated temporally. The topological notion of conjugacy used will be the one Case 2: Sets of the form: introduced in (James, 2007). This topological notion of conjugacy is one of the usual ways to relate two ∈ • {()y,t Oψ |d,t() ≺ ψ () y,t } dynamical systems (Erik and Joseph, 2010). In fact, it is to()y to() y one of the most useful and interesting among the Clearly: different possibilities of introducing an equivalence relation to classify dynamical systems (Nguyen, 1996). To achieve this, we conjugate two trajectories resulting θ−1 ∈ • ({()y,t Oψ() |d,t() ≺ ψ () () y,t }) toy to y from two different flows, and then by the fact that the

= ∈ • initial state chosen is arbitrary, this result were {}()x,t Oφ() |b,t() ≺ φ () () x,t tox to x extended to the rest pair of trajectories in the state spaces. Start off by defining two functions that acts on ( )∈ θ−1 For some b,t O φ ()x , since is bijective. two different set of points (trajectories) which results t o from two different flows, respectively as: ∈ • Notice that the set {()x,t Oφ() |b,t() ≺ φ () () x,t } tox to x φ → ψ → is an order-open ray in Oφ () , i.e., element of subbasis. : Oφ O φ and : Oψ O ψ tx o x0 to()()x to x y0 to()()y to y 211 J. Math. & Stat., 7 (3): 207-215, 2011

Where: For all (x,t)∈ X ×ℜ ,y,t( ) ∈ Y ×ℜ , and m∈N.

φ( ) =( ) ∀∈( ) ∈ℜ Notice that our claim that the homeomorphism, h xiix,t x,t i1i1+ + x,t ii Oφ () and t i 0 tx o defined above conjugate the two flows topologically

must be true. This is clear from the properties of flow And: given earlier where two distinct trajectories will not ψ( ) =( ) ∀∈( ) ∈ℜ intersect and that the property (a) for flow is defined for yiiy,t y,t i1i1+ + y,t ii Oψ () andt i 0 ty o all x∈X. Thus, we can imagine the state space of the dynamical system as a space filled with a set of Then it will be these two functions which we will trajectories. In fact, the union of all these trajectories show they are topologically conjugated. form the state space set. For example, if the state space 2 of a flow is the Euclidean space ℜ , then for each point φ → Theorem 5: x : Oφ()()x O φ x and on the state space, there will be a trajectory, and the 0 to to 2 ψ → union of all this trajectories will be the set ℜ . This can y : Oψ()() O ψ is topologically conjugated. 0 toy to y also be visualized as the flow partitions the state space

( )∈ into classes where each class represents a trajectory and Proof: For any xi1− ,t i1 − O φ () tx o our method in conjugating the two flows is by Composition of functions θ and φ is conjugating each pair of these trajectories. As the two x0 flows were chosen arbitrary, thus we conclude all the above results in the following theorems. θ[ φ (x− ,t − )] x0 i1 i1 = θ () xi ,t i Theorem 6: Any two continuous-time dynamical = ()y ,t systems with no periodic orbit are: i i • Order-isomorphic Composition of functions ψ and θ is • y0 Homeomorphic • Topologically conjugated temporally

ψ[ θ ( x− ,t − ) ] y0 i1i1 Proof: Let φ: ℜ× X → X and ψ: ℜ× Y → Y be two = ψ ()y− ,t − y0 i1 i1 flows of continuous-time dynamical systems with non- = () ( )∈ × ℜ yi ,t i periodic orbit. For all xi1− ,t i1 − X and h:X×ℜ→ Y ×ℜ be defined as (2): This shows that θφ =ψ θ x0 y 0 h φ( x ,t ) Thus, the two trajectories are topologically i1− i1 − = () conjugated. h x,ti i In constructing this proof, the point chosen as an = () yi ,t i initial state in both trajectories is arbitrary, as such for the rest of pair of points, their trajectories can also be ψ h( x ,t ) conjugated topologically using similar method i1− i1 − = ψ () presented above. In that case, the two flows, yi1− ,t i1 − φ: ℜ× X → X and ψ: ℜ× Y → Y can be conjugated = () xi ,t i topologically by the homeomorphism h:X×ℜ→ Y ×ℜ , which can be defined as: As such, hφ = ψ h . Therefore, the two flows are topologically conjugated. θ()() ∈ 0 x,t if x,t Oφ () where  tx o θ → θφ=ψθ RESULTS  0:Oφ()() O ψ and 0xy0  tox to y 0 0 θ()() ∈ In this study, we have shown that the flows of any  1 x,t if x,t O φ ()  tx 1 two continuous-time dynamical systems with the same θ → θφ=ψθ  where1 :Oφ()() O ψ and 1xy1  t1x t1 y 1 1 number of equilibrium points and trajectories, and has no h() x,t =  (2) θ()() ∈ periodic orbit can be conjugated topologically  2 x,t if x,t O φ () tx 2 temporally. Besides, their trajectories are linearly ordered  θ → θφ=ψθ where2 :Oφ()() O ψ and 2xy2 ....  t2x t2 y 2 2 and order isomorphic to each other by the relation θ()() ∈ induced from their flow Fig. 1. By endowing the interval  m x,t if x,t O φ ()x t m topology, both the LOTS are proven to be Hausdorff and  θ → θ φ =ψ θ wherem :Oφ()() O ψ and mx y m  tmx tm y m m homeomorphic to each other. 212 J. Math. & Stat., 7 (3): 207-215, 2011

DISCUSSION disorder as well as assisting neurologist in classifying seizures. EEG has been used extensively to record the In the theory of dynamical systems, classification abnormal brain activity associated with epileptic problem is important as it provides a method to simplify seizures (Fig. 2). It is recorded on the surface of the objects under investigation and gives us an insight into scalp using electrodes, thus the signal is retrievable the structure of dynamical system (Nguyen, 1996). The results obtained in this study are therefore crucial as it non-invasively. The type of activity and the area of the provides us another viewpoint in observing a dynamical brain that is recorded from EEG will assist the system of our interests. One example can be found in physician in prescribing the correct medication for (Tahir et al ., 2010) whereby the dynamics of patient certain type of epilepsy. Patients with epilepsy that having epileptic seizure is showed transported to a visual cannot be controlled by medication will often have platform namely, Flat EEG. Since they are topologically surgery in order to remove the damaged tissue. Thus the conjugate, they have the same topological properties EEG plays an important role in localizing this tissue. (Robbin, 1972). Thus, we can study the topological On the other hand, Flat EEG is a method for properties of seizure from Flat EEG. mapping high dimensional signal, namely EEG into a Epilepsy is a general term used for a group of low dimensional space (MC) developed in (Tahir et disorders that cause disturbances in electrical signal of al ., 2006). The whole process of this novel model is the brain. In epilepsy there is a miniature brainstorm of consists of three main parts. The first part is certain groups of brain cells and this is often associated flattening the EEG where the transformation of three with a sudden and involuntary contraction of a group of dimensional space into two dimensional space that muscles and loss of consciousness. It can happen in a involved location of sensor in patients head with small area of the brain or the whole brain. Depending on EEG signal (Fig. 3). This flattening process can the part of the brain that is affected, it causes involuntary preserves magnitude and orientation of the surface changes in body movement or function, sensation, (Tahir et al ., 2006). Secondly, the EEG is processed awareness, or behavior where these changes are known using Fuzzy C-Means (FCM) to obtain the number as epileptic seizure (Fig. 4a). of cluster centers. Finally, the optimal number of Electroencephalography (EEG) is the recording of clusters is determined using cluster validity analysis. electrical activity originating from the brain in contrast to Magnetoencephalography (MEG) which records the magnetic fields. It plays an important diagnostic role in epilepsy and provides supporting evidence of a seizure

Fig. 2: EEG signal

Fig. 1: Framework Fig. 3: EEG projection 213 J. Math. & Stat., 7 (3): 207-215, 2011

Fig. 4: Topological conjugacy between seizure and Flat EEG

This new model enables tracking of brainstorm during REFERENCES seizure (Tahir et al ., 2006). Figure 4b contains examples of Flat EEG. Red dots represent the Adeqbie, K.S. and F.O. Alao, 2007. Flow of electrodes while green dots represent the cluster centers temperature-dependent viscous fluid between after the transformation from the scalp of the patients. parallel heated walls: exact analytical in the By using Theorem 6, the dynamics of epileptic seizure solutions presence of viscous dissipation. J. Math. can be portrayed on Flat EEG (Fig. 4). For more details, Stat., 3: 12-14. DOI: 10.3844/jmssp.2007.12.14 see (Tahir et al ., 2010). Artur, O. L., 1979. Stuctural stability and hyperbolic

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