Stability Analysis of Periodic Solutions of Bonhoeffer–Van Der Pol System with Applied Impulse

In this paper, the stability analysis of periodic solutions of Bonhoeffer–van der Pol system with applied impulse was investigated using Lyapunov direct method. Through the use of appropriate values of the control parameters, three equilibria points and periodic solutions of the system were obtained. A Lyapunov candidate which depended on the parameters was constructed. Hence, we concluded that the equilibria points have different regions of stability and instability of the system in which the two regions were found to be stable and the other region was unstable. Furthermore, Mathcad software was used to analyze the behavior of the system, thereby improving known results in literature.


Introduction
Consider the system ẋ c(x + Y − x 3  3 + z) ẏ transformations y Y + z, a A + bz (2) where y and a are dependent variables for the transformation equation.Equation (1) becomes ẋ c(x + y − x 3 3 ) ẏ 1 c (−x − by + a) (3) with periodic initial conditions x(0) x(2π) ẋ(0) ẋ(2π) Bonhoeffer-van der Pol system is a nonlinear system of differential equations with two or more control parameters.The system was introduced by [1] as a simplified version (twodimensional) representation of the four-dimensional Hodgen-Huxley system.This system describes how nerve impulse travel down the axon of a nerve cell.The Bonhoeffer-van der Pol system is a physiological model which describes the electrical activities of certain nerve cells.However, from the mathematical perspective, Bonhoeffer-van der Pol system is a class of periodically forced nonlinear relaxation oscillators.The equation is used in the modeling of cardiac pulses from electrocardiographic signals, normal cardiac pulses which account for the adaptive response for every single heartbeat, manipulation of cell division from inactive state to dormancy state and formation of a multicellular fruiting body.In [2], Bonhoeffer-van der Pol equation was used to simulate neuron cells and control the dynamics of the neuron cells from chaotic to periodic state.Under proper parameters, Bonhoeffer-van der Pol equation describes regulation of heart rate pulse.It is highly preferred over others because of its possibility to facilitate the classification of normal and pathological heart beats for diagnosing purposes.
Furthermore, its applications are seen in the modeling of a cell cycle which describes the lipid peroxidation in an open membrane and the corresponding production of free radicals.Due to the wide range of applications of Bonhoeffer-van der Pol equation in different fields of endeavor, many researchers have worked on Bonhoeffer-van der Pol equation using different methods.See [3, 4, 5, 6 and 7].
Stability is a qualitative property of a differential equation that is important in linear and nonlinear system.It describes the behavior of the system when the system undergoes small changes.Analytically, stability is determined by the interval placed on the total derivative of the system formed by the given differential equation.For linear system with one parameter, stability is assessed only at one equilibrium point.However, for a nonlinear system with more than two parameters, the search for the stability analysis of the equilibrium point becomes an issue.Emphasis on stability has been discussed by many authors.For instance, see [8][9][10][11][12][13][14][15] and their references therein.Several methods have been used by some authors to study stability of nonlinear system which are Lyapunov method [16] and [17], contraction mapping principle [18], topological index method [19], new maximum principle [20], upper and lower solution method [21] and derivative expansion method [22].However, few works have been recorded on the stability analysis of Bonhoeffer-van der Pol system with applied impulse.
The novelty of this research work is based on the application of impulse to Bonhoeffer-van der Pol equation which described the process of heartbeat.The impulse controls the rate of heartbeat which is set by the SA (sinoatrial node).The SA fires an impulse which spread through the walls of the right and left arteria causing them to contract.The impulse travels to the AV (atrioventricular node) where the electrical signal is delayed before it enters the ventricles.This delay gives the arteria time to contract.The impulse further travel through a pathway of fibers called the His-Purkinje Network.This network sends the impulse into the ventricles and causes them to contract.These processes force blood out of the heart to the lungs and body.The process repeats itself again which results to a complete cycle.This cycle is a self-repeating process that must obey the rule of periodicity otherwise will results to irregular heartbeat or bradycardia.The complete cycle explained the normal heartbeat which must be asymptotically stable for the system to work effectively.The normal heartbeat is often called sinus rhythm because the SA node fires regularly.However certain conditions like exercises, excitations and medical conditions can alter the equilibrium of the body.These conditions will be explained using the numerical behavior of Bonhoeffer-van der Pol equation.
This paper is motivated by [3] and [16] where similar works were discussed but the major differences stem from the application of impulse to the stability analysis of periodic solutions of Bonhoeffer-van der Pol system.The objective of this paper is to investigate the stability analysis of periodic solution of of system (1).The paper will further investigate the effect of the parameters on the system and analyze the behavior of the system using Mathcad software.

Preliminaries
Definition 2.1 Consider a real valued function V : R n → R which is continuously differentiable.Then the continuous function V (x) is said to be positive definite if V (x) > 0 and V (0) 0 for all x 0 and otherwise it is negative definite.

Definition 2.3 (Periodic Solution) It is the solution y
f (x) of a differential equation with the property that there exist a positive real number k 0 such that f (x + k) f (x).k is called the period of the function.Definition 2.4 (Stability) An equilibrium solution x e of an autonomous system is said to be stable if for every ε > 0 there exist δ > 0 such that every solution x(t) having an initial conditions within δ ie x(t 0 ) − x e < δ of the equilibrium remain within ε is x(t) − x e < ε for all t ≥ t 0 .Definition 2.5 (Impulse) It is a concept that involves an object momentum change when force is introduced for a period of time.

Stability of the Equilibrium Point
Consider the system defined by Eq. ( 3) where a, b, c are constant parameters and c 0 Equation (3) can be combined to give a Lienard equation of the form: where By setting c −1, x 0, b 0 and a 0, Eq. ( 4) reduces to Equation ( 5) is oscillatory in R. Hence, the solution obtained in Eq. ( 5) is periodic.
The equilibrium points are obtained by setting the right hand side of Eq. ( 3) to zero which gives; Eliminating y from Eq. ( 6) we obtain the cubic equation To solve the cubic equation in (7), we let a 0 and substitute for b 3. Equation ( 7) becomes The roots of Eq. ( 8) are x 1 0, x 2 √ 2 and x 3 − √ 2 Substituting x 1 , x 2 , x 3 into Eq.( 6) we have y 1 0, y 2 − 2 3 and y 3 2 3 Therefore, the equilibrium points are (0, 0), ( √ 2, − 2 3 ) and (− √ 2, 2 3 ) Linearized form of Eq. ( 4) is given by: where f and g are polynomials of degree two and three respectively.The Lyapunov function.is calculated as the total energy of the system which is given by where K E is the kinetic energy of the system, and P E is the potential energy of the system.V (x, y) is a function defined by V (x, y) : R 2 → R. Equation (10) can further be written as where Equation ( 12) is the Lyapunov function for the system.
Clearly, V (x, y) is continuous at (x 0 , y 0 ) since lim (x, y)→(x 0 , y 0 ) V (x, y) V (x 0 , y 0 ).(x 0 , y 0 ) is an element of the domain of V (x, y).For V (0, 0) we have Evaluating for other values of the equilibrium point we have This shows that V (x, y) > 0 for the equilibrium points except at the origin.Hence Therefore, V (x, y) is positive definite.Since V (x, y) > 0∀x, y 0, the time derivative of the energy of the system is given by With the condition placed on c in Eq. ( 1), V (x, y) < 0∀x, y 0. Hence, from theorem (2.2) the equilibrium point is asymptotically stable.

Stability of the Equilibria
The equilibria of system (3) which is determined by the following systems ).We shall now investigate the stability property of the above equilibria.The variational matrix of system (3) is Proof From Eq. ( 20), the Jacobian matrix of the system about the equilibrium point C 1 (0, 0) is given by.
The characteristics equation of Eq. ( 21) is given as The eigenvalues are λ 1 where α 3 c − c.Both eigenvalues are real and nonzero.Since the eigenvalues have opposite signs, C 1 (0, 0) is a saddle and thus unstable.
Proof From Eq. ( 20), the Jacobian matrix of the system about the equilibrium point.
The characteristics equation of Eq. ( 23) is given by Proof From Eq. ( 20), the Jacobian matrix of the system about the equilibrium point.
The characteristics equation of Eq. ( 25) is given by

Numerical Simulation of the Results
We illustrate the numerical simulation of the results using the MATHCAD software; Simulation-1 α : 0.7, β : 0.8, γ : 3 Define a function that determines a vector of derivative value at any solution point (t, Y )

Conclusions
From our results, Lyapunov direct method is very effective in constructing suitable Lyapunov functions that depended on the parameters.The advantages of this method are that it is general, used for varying an uncertain systems and can be used to estimate the region of attraction for an equilibrium point.Hence, we concluded that the stability analysis of Bonhoeffer-van der Pol system strongly depended on the parameters.Application of our results can be seen in the process of heartbeat where certain conditions can alter the equilibrium state of the body.The increase in the parameters will cause abnormal heartbeat while the effect of the applied impulse triggers the system to maintain a stable process of heartbeat.The numerical behaviors are explained as follows: In Fig. 1a, the trajectory profile of Bonhoeffer-van der Pol equation is unstable and not periodic.This could increase the impulse rate as the trajectory moves toward the positive axis.
In Fig. 1b, the trajectory shows that the solution is not periodic and unstable.This could lead to heart attack if the system is not controlled.
In Fig. 1c, the trajectory is highly unstable.This behavior describes an irregular heartbeat which may lead to stroke and sudden death.
In Fig. 2a and b, the trajectory is periodic in nature.This behavior describes a normal heartbeat with a usual sinus rhythm.
In Fig. 2c, the phase portrait describe a stable equilibrium point.This behavior describes a stable heartbeat in which some processes like contraction and relaxation do not operate outside the given boundary.
In Fig. 3a and b, the trajectory describe a normal heartbeat in which the increase or decrease in the impulse is regulated.This regular movement is effective with the help of the control parameters present in the Bonhoeffer-van der Pol system.This type of heartbeat can be seen in adults in which under condition of maximum work can increase their heart rate to over 200 beats per minute.
In Fig. 3c, stable equilibrium point of the system is illustrated.As the trajectory is far from the equilibrium point, certain conditions like exercises and excitements can alter the equilibrium state of the system thereby increasing the heartbeat.
In Fig. 4a and b, the trajectory describe a normal heartbeat for elderly people under condition of minimum work.The heart rate is below 150 beats per minute.
In Fig. 4c, stable equilibrium point of the system is illustrated.This portrait describes a normal heartbeat.
In Fig. 5a and b, the trajectory describe an irregular heartbeat which could cause by some factors like high blood pressure, diabetes etc.
In Fig. 5c, the trajectory describes a stable normal heartbeat.

6 c − 2 .
The eigenvalues are real, nonzero and negative.Consequently, C 2

Fig. 1 a
Fig. 1 a Graph of x against t showing the trajectory profile of the Bonhoeffer-van der Pol equation which starts out from the origin and moves away from the equilibrium point on the positive axis.This shows that the equilibrium point is unstable and the solution is not periodic.b The trajectory profile of the Bonhoeffer-van der Pol equation which moves away from the equilibrium point on the positive axis.This shows the instability and aperiodic nature of the system.c A phase diagram of Bonhoeffer-van der Pol equation showing the instability nature of the system

Fig. 2 aFig. 3 aFig. 4 aFig. 5 a
Fig. 2 a The trajectory profile of periodic solution of the Bonhoeffer-van der Pol equation.b The trajectory profile of the Bonhoeffer-van der Pol equation depicting the periodic nature of its solution.c: A phase portrait of Bonhoeffer-van der Pol equation depicting the stability of the equilibrium point within a bounded region 50 endpoints of solution interval.Define additional argument for the ODE solver.t0 : 0 Initial value of independent variable.