THE SOLUTION STABILITY OF VAN DER POL'S EQUATION .

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 United States License.

THE SOLUTION STABILITY OF VAN DER POL'S EQUATION .

THE SOLUTION STABILITY OF

VAN DER POL'S EQUATION .

by CO. H . TRAN .

HUI - NCU HCMC

Vietnam coth123@math.com & cohtran@math.com Copyright 2004 November 06 2004 ------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ** Abstract : The Van der Pol differential quation is solved by averaging method . ** Subjects: Vibration Mechanics , The Differential equations . ------------------------------------------------------------------------------------------------------------------------------------------------------------------------Introduction This worksheet demonstrates Maple's capabilities in finding the graphical solution and dealing with the stability of the steady state solution of Van der Pol 's differential equation . All rights reserved. Copying or transmitting of this material without the permission of the authors is not allowed . We consider the Van Der Pol differential equation :

[0] Two topics that we will be in discussion are - Finding the steady state solution of this equation by averaging method .- Estimating the stability of solution obtained . 1 .Define the model of problem : We examine the effect of non-linear system under external force caused by the AC generator ( see fig 1. ) ( fig 1. ) The differential equation of this model is given in the form + [1] After simplifying we obtain : [2]

2 . Construct the algorithm .

Generally the Van Der Pol differential equation can be expressed by : [3]In the special case let and the equation will be rewritten as : By using the transform [4]Then ω and substitute these relations to ( 2 ) it follows : ω Or [5] Thus we begin with : [6] To normalize this equation we find the solution which is expressed in the form x = acos ( t + γ )It is advantageous to write ϕ = t + γ then the solution will be x = acosϕ [7]From the transform x = acosϕ , x' = asin ϕ . We have x" = dx'/dt = a'sinϕ acosϕ . ϕ' By substituting to ( 6 ) , it gives a'sinϕ a cosϕ.ϕ '+a cosϕ μ (1 cosϕ ).(a sinϕ ) = 0 [8]In the other hand dx/dt = x' = a'cosϕ asinϕ .ϕ' = asinϕ [9] Obviously we reach to the following system of differential equations [10]

>

Warning, the name GramSchmidt has been rebound
Warning, the protected names norm and trace have been redefined and unprotected

>

(1)

>

(2)

>

(3)

>

	(4)

>

By using the symbols a(tt) = a'(t) , ϕ(tt) = ϕ'(t) [11] Execute the averaging method for a'(t) and ϕ'(t) , we have

>

(5)

>

(6)

>

The expressions of a' and ϕ' are calculated in the forms : a' = μ < > = [12]

ϕ' = μ < > = [13]

The steady state solution occurs when a0 = 0 or a0 = 2 If a0 = 0 then x = x' = 0 this is a trivial solution ( equilibrium ) . If a0 = 2 then x = 2cosϕ and x' = -2sinϕ . [14] The necessary and sufficient condition for solution stability includes a' = da/dt = Ψ(a) with Ψ(ao) = 0 and Ψ'(ao) <>

>

(7)

>

	(8)

>	subs(a=2,DaohamcuaPsi(a));

>	print("Gia tri cua a''(t) tai a = 2 la : a''(2) = ",subs(a=2,DaohamcuaPsi(a)));

>	subs(a=0,DaohamcuaPsi(a));

>	print("Gia tri cua a''(t) tai a = 0 la : a''(0) = ",subs(a=0,DaohamcuaPsi(a)));

	(9)

>

Thus if ao = 0 , a"(0) = then the solution is not stable .

If ao = 0 , a"(0) = then the solution is stable asymtotically.

Note : By solving the equation ( 12 ) for the vibration amplitude a(t) ( slowly varying coefficients ) then finding the solution expression x(t) of Van Der Pol differential equation , we get

>

a0(tt);

>	diff_eq:= diff(a(t),t)=-mu*a(t)*(-4+(a(t)^2))/8;

>	init_con:=a(0)=ao;biendo:=[dsolve({diff_eq,init_con}, {a(t)})];

>	x:=biendo[1]*cos(phi);

	(10)

>

with ϕ = t + γ . [15]

>

(11)

>

Warning, the previous bindings of the names RationalCanonicalForm and adjoint have been removed and they now have an assigned value

>	with(plots):a:=0.5;y:=t->2*cos(t + gamma)/sqrt((a^2-a^2*exp(-0.1*t)+4*exp(-0.1*t))/a^2);

>	plot(y(t),t=0..4*Pi);

>	animate(plot,[2*cos(x*t + gamma)/sqrt((a^2-a^2*exp(-0.1*t)+4*exp(-0.1*t))/a^2),x=0..4*t],t=0..Pi,frames=100 );

>

Use graphical method to consider the solution stability of Van Der Pol's differential equation :

>	restart;with(DEtools):mu:=0.5;

>	DEplot({(D@@2)(x)(t)+x(t)-mu*(1-x(t)^2)*D(x)(t)=0},{x(t)},t=-10..50,[[x(0)=1,D(x)(0)=1]],stepsize=0.05,title=`Nghiem on dinh cua pt Van Der Pol`);

>

3 . Conclusion . From graphical results , we reach to conclusion that the steady state solution stability of Van Der Pol's diffential equation must be precise and estimating the property of solution obtained is very necessary . As presented above , we might also use the normalization to ( 2 ) by determining the non-trivial solution in the form

[16]

with k = 1 Otherwise Substitute x ' to ( 6 ) after simplifying it follows :

>

>	f:=vector([[0], [mu*F]]);

>	V:=linsolve(A,f);

Warning, the name GramSchmidt has been rebound
Warning, the protected names norm and trace have been redefined and unprotected
	(12)

>

We rewrite the expressions of M' = N' = [17]

With

Use averaging method for ( 17 ) we get :

M ' = μ <- Fsint > =

N' = μ <> = [18]

The steady state solution exists when M = 0 and N = 0 ( trivial solution ) Or M = 2 or M = -2 and N = 0

Thus x = 2cost , x = 2cost If M = 0 and N = 0 or N = 4 Then x = 4sint The steady state solutions can be formed generally :

But these forms are equivalent to x = 2cosϕ [ see ( 14 ) ]

Next we begin with Krylov - Bogoliubov approximate method for the Van der Pol's differential equation

with ) x ' , and μ is a small constant .

The solution will be estimated by x = r(t) cosϕ(t)

By taking the first order approximate terms ( neglecting the second and third order errors of constant μ ) we can find the amplitude r(t) and the global phase function ϕ(t) from the following system

[19]

If the initial condition r(0) = ro is satisfied then the solution of ( 15 ) will be equivalent to the solution of second order linear differential equation

[20]

with the error of estimation based on the order of

The first order approximate is noticeable in the case of periodic vibration , because the equivalent linear differential equation gives us the accumulation and dissipation of energy based on vibration period which we might obtain from the given non-linear differential equation . Therefore it is useful to apply the equivalent second linear differential equation to observe the non-linear resonant phenomenon .REFERENCES

[1] Menh .C. Nguyen , Dao dong phi tuyen , Manuscript , Vien Co hoc , Hanoi 2002 . [2] Korn . G , Korn . T ., So tay toan hoc , Trans Vietnamese , NXB DH-THCN , Hanoi 1978

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THE SOLUTION STABILITY OF VAN DER POL’S EQUATION .

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Author : Co Hong Tran MMPC-VN.

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 United States License.

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 Toán học thuần túy, theo cách của riêng nó, là thi ca của tư duy logic. 
Pure mathematics is, in its way, the poetry of logical ideas. 

Albert Einstein .