Studies on Rayleigh Equation

Transcription

1 Johan Matheus Tuwankotta Studies on Rayleigh Equation Intisari Dalam makalah ini dipelajari persamaan oscilator tak linear yaitu persamaan Rayleigh : x x = µ ( ( x ) ) x. Masalah kestabilan lokal di sekitar titik tetap dan hubungannya dengan paramater µ di bahas. Juga akan dibuktikan eksistensi selesaian periodik yang dalam hal ini juga merupakan limit cycle dengan menggunakan teorema Poincare-Bendixson. Selesaian periodik tersebut akan dihampiri dengan menggunakan metode Poincare-Linsted. Sebagai perbandingan juga di hitung selesaian numerik dan dibandingkan dengan selesaian asimtotik Abstract In this paper we consider a type of nonlinear oscillator known as Rayleigh equation, i.e. x x = µ ( ( x ) ) x. We study local the stability of the fixed point in the presence of positive parameter µ. We are also looking at the existence of the periodic solution which is also a limit cycle in this equation. Using Poincare-Bendixson s theorem we proof that the limit cycle exists. For small values of the parameter, we use asymptotic analysis to approximate the periodic solution. The method we apply in this paper is Poincare-Linsted method. We calculate also the numerical solution for the system and compare the result with the asymptotic. INTEGRAL, vol. 5 no., April 000

2 I. Introduction Consider a system of differential equations u = f t, u,µ.. () where u R, f is a vector valued function in R and µ is a parameter. We assume that solution exists and is unique with given a initial value. This system is also known as a planar system. The system depended on the parameter µ. For a fixed µ, let u µ (t) = ϕ µ (t) be a solution of the system (the subscript shows the dependency on the parameter). This vector valued function ϕ µ parameterize by t defines a flow in the two dimensional plane u = (x, y). The curve of points (x, y) generated by ϕ µ (t) when we let t run from - to is called the trajectory. A collection of trajectories of different initial value is calling the phase portrait of the system. A simple question that arises is; how to know the phase portrait of a system. The best way to have the phase portrait of a system is to calculate the general expression of the solution of the system. Unfortunately, this is not an easy thing to do especially when f is non-linear. The second best way is to calculate the integral of the system. An integral (in this case) is a real valued, two variables function F(x, y) such that d dt ( F( x, y)) ϕ = 0, which is the derivative of F with respect to time evaluated at a particular solution zero. Consequently, F(x, y) = C define a curve in the (x, y) plane such that the flow of system () will map the curve to it self. Such a curve is called invariant curve of the system. In elementary courses on differential equation such a curve is called implicit solution. (Unfortunately not every system has an integral). We remark that if we can calculate one of those (either the solution or the integral) we will have a global picture of the flow. It means that the analysis is valid for all time (t) and everywhere in the (x, y) plane. That is why these two methods are also known as global analysis technique. Naturally, when the global picture cannot be achieved, we can try to have a local picture. We can restrict our analysis in a small area around some interesting location. The question is, where the interesting location is. Some of the locations, which are considered to be interesting, are around a fixed point or around a periodic orbit. A fixed point (also known as a constant of motion point, or critical point, or singular point) is a point (x 0, y 0 ) in the plane such that f(x 0, y 0 ) = 0. A periodic orbit is a solution ϕ µ (t) where a real number T such that ϕ µ (t T) = ϕ µ (t) exists. Now supposed we have a periodic orbit that start at a particular point. After T time it will be back at that starting point. It means the trajectory will be a closed curve. The converse is also true if we assume that T is finite. The most common tool to understand locally the flow of the system is linearization. We expand f to its Taylor series around a particular solution, i.e. u = f( ϕµ )( u ϕµ )! () where the dots represent the higher order terms and represent the partial derivative with respect to spatial variables. And then we can use linear analysis to obtain the information about the flow. See [], [4] or [5] for example. Our goal in this paper is to analyze the non-linear oscillator known as the Rayleigh equation, i.e. x x = µ ( ( x ) ) x.(3) where µ is a positive parameter. We give the analysis in the neighborhood of INTEGRAL, vol. 5 no., April 000

3 the fixed point. We will also show that there exists a periodic orbit and that it is stable. This stable periodic orbit is also known as the stable limit cycle. Due to the fact that the analytic calculation of the limit cycle is complicated, we will construct an approximation of it s for a small parameter using Poincare-Linsted method. For a large parameter we can construct the approximation using boundary layer technique or singular perturbation technique (see [3]). These technique is mathematically non trivial so we will skip them. In this paper we will also construct the numerical comparison of the analysis. of the solutions being π-periodic. Thus, the origin (0,0) is a center point if µ = 0. In general this statement is not always true. II. Fixed Point Analysis To calculate the fixed point of the Rayleigh oscillator, we transform the system into a system of first order differential equations. This is done by setting x = x and y = x'. The Rayleigh equation then becomes x = y..(4) y = x µ ( y ) y It is easy to see that the fixed point of (4) is only (0,0). In general, for a system like in () the fixed point is also depended on µ. The linearized system of (3) written in matrix form is: ' x 0 x =...(5) ' y µ y From linear analysis we know that the stability of the fixed point (0,0) dependeds on the eigen-values of the linearized system (5). The information on the stability also gives information on the flow it-self around the point. In this case we have the eigen values µ ± µ 4 λ, =...(6) For µ =0 we see that the eigen-values are purely imaginary λ, = ± i. This is clear since if µ = 0, what we have is just a linear harmonic oscillator with all Figure : Phase portrait for µ = 0. The fixed point in this case is a center point and all solutions are periodic of period π. Obviously the eigen-value of a non linear system should be non linearly dependent on the system. Thus if we linearize locally, we restrict the domain so that the non- linear effect can be considered as a small perturbation. If the real part of the eigen value is nonzero then we can choose the domain small enough so that the non-linear perturbance will be small enough. It implies that the real part will still be the same in sign (see [4],[5] for details). This is not the case for zero real part of the eigen-value. In the case of all the real parts of the eigen-values is zero we have to consider a higher order term on the expanded system. In the case that there is at least one is non zero, we can use Center Manifold Theorem (see [] for details). This case does not arise on this problem so we omit it. For 0 < µ < we can write the eigen µ ± ω i values as λ, =, ω < µ. In INTEGRAL, vol. 5 no., April 000 3

4 this case, the flow around the origin is spiraling outward the fixed point. Thus, if we start on a point close to the origin the trajectory will move around the origin with increasing radius. It is interesting that most of the periodic orbits of the case µ=0, break up. In fact there is only one periodic orbit that survives and forms a limit cycle. The next case is if µ =. In this case we have only one eigen-value. This eigen value corresponds to single a eigen-vector k. Thus the origin will look like a source which is an improper node. The phase flow is flowing out from the origin along one direction which can be approximate by the eigenvector. Figure 3: Phase portrait for µ =. It represents a improper node where the flow is going outward along one direction. For µ >, we have two different real eigen-values. Thus we will have two linearly independent eigen-vectors. Since the eigen values are different, then the origin will also be improper node sources. The phase flow is flowing out of the origin along two direction only. The picture in figure () until (4) are drawn using Maple V. It is clear that the analysis of the origin coincides with the numerical result using Maple V. Figure : Phase portrait for µ = 0.5. The flow around the fixed point is spiraling outward the fixed point. Figure 4: Phase flow for µ =.5. III. Existence of the Periodic Solution To proof the existence of the periodic orbit, we will apply a fundamental 4 INTEGRAL, vol. 5 no., April 000

5 theorem of Poincare-Bendixson. This theorem will be stated without proof. Theorem: Poincare-Bendixson If D is a closed bounded region of the (x, y)- plane and a solution u(t) of a nonsingular system () is such that u(t) D, then the solution either a closed path, approaches a closed path, or approaches a fixed point. The theorem in other word says that if we have a closed and bounded region which the flow of the system in () is flowing into the region, then there is a periodic solution or a fixed point in the region. We will apply this theorem to a general type of oscillator with damping by constructing a region which is invariant to the flow. After that we will relate the Rayleigh equation to the general equation we have. Consider now an oscillator equation known as Lienard equation, i.e. x f ( x) x x = 0..(7) We define x F( x) = f ( s) ds 0 and assume it to be an odd function. Futher assume that for x > β (a real positive number) F is positive, goes to infinity for x goes to infinity, and monotonically increasing, while for 0 < x < α F is negative. We transform (7) to a first order system using transformation (x, x ) (x, y = x F(x)). Using this transformation we have x = y F( x)..(8) y = x. Now define a transformation to polar coordinate by R = ( x y ). Consequently we have R = -xf(x). It implies that for -α < x < α, R 0. Thus the flow on the circle domain with radius α is flowing out of the domain. What we need to construct is a domain where the flow is flowing into the domain. See figure (5). Before we start constructing the domain, we first make some important remark on the gradient field of the flow. From (8) we can calculate dy x dx = y F ( x. It ) implies that on the y axis the tangent is horizontal and on the curve y = F(x) the tangent is vertical. One can clarify easily that if we start at an initial value say A we will have a trajectory as in figure (5) (this can be easily done by considering the negative or positiveness of x or y at certain location). The reflection symmetry of the system gives the negative trajectory. Our purpose is now to proof that R(A) > R(D). This can be done by considering the line integral RD RA = dr. We split the ABCD curve ABCD into three segments AB, curve ABCD into three segments AB, BC, CD. We also have two expressions for dr, i.e. Figure 5: Constructing the invariant domain. xf x dr = dx or dr = F( x) dy. y F( x) INTEGRAL, vol. 5 no., April 000 5

6 We first consider segment AB and CD. If A (= (0, y A )) is high enough then we know that y-f(x) is large while -xf(x) is bounded. Thus for y A going to infinity, the integrals dr and dr go to AB CD zero. It also means that the integral is dominated by the line integral in the segment CD. In segment CD we have BC C dr = F( x) dy. Since we assume B that for x > β, F is positive and monotonically increasing, we see that the integral will be negative and monotonically decreasing (it is clear since the integration is from positive values to the negative values of y). Also it is continuosly dependent on y A. Thus if y A goes to infinity the integral goes to infinity. It means that we can choose y A large enough such that R(A) > R(D). In the same way we can show R(-A) < R(-D). Now consider the domain in figure (5) which is bounded by the circle with radius α, the two trajectories and the line segment connecting the trajectories. This domain defines an invariant closed and bounded domain. Moreover, the flow is going into the domain. Hence the Poincare- Bendixson s theorem applies. We know that the only fixed point of the Lienard s system is the origin. The conclusion is that we have at least one periodic solution. Moreover, since F is monotonically increasing and consequently the integral in the segment BC is monotonically decreasing, the condition that if α=β then leads to precisely one periodic solution. The question arises is how to apply this result to the Rayleigh equation. If we differentiate The Rayleigh equation, we have x µ ( 3( x ) ) x x = 0....(9) Define z = 3 x and transform (7) into z µ z z z = 0...(0) The solution of (9) is just a solution of ordinary Rayleigh equation with an additional constant. Thus by setting the constant to be zero, we have the original solution. Since (0) satisfies the assumption we now find that there exists at least one periodic solution. We can proof that in the case of (0), α=β. Thus we have a unique periodic solution. IV. Asymptotic Approximation The next aim is to approximate asymptotically the periodic solution of Rayleigh equation. We will apply the Poincare-Linsted Method to approximate the periodic solution. We will first give a short introduction to the method. Details of the method can be found in [4]. Consider a second order differential equation of the form x x = εf( x, x, ε) () It is easy to see that for ε = 0, all solutions are π-periodic. For ε 0 we assume that there exists a periodic solution with period T(ε) starting at initial values x(0) = a(ε) and x'(0) = 0. The fact that the period and the location or the periodic orbit are dependent on ε is natural. Define a time transformation θ = ωt such that in this new time variable, the period of the periodic orbit is π. We write ω - = - εη(ε). Obviously, the periodic solution is also dependent continuesly on ε. Thus we assume that we can write the periodic solution as x( θ) = x0 εx ε x!...() With the new time variable we can write () in the form of (the dot represents the derivative with respect to θ) 6 INTEGRAL, vol. 5 no., April 000

7 x" x"" x = ε ηx ( ) f x,, εη εη ε. If we write the right hand side as εgxx (, ", ε) and transform it to its integral form, then the periodicity condition of the solution leads to two equations, i.e. π F ( a, η) = cos( τ ) g x, x", ε, ηdτ = 0 F ( a, η) = sin( τ ) g x,", x ε, ηdτ = 0 0..(3) π 0 Implicit function theorem gives the conditions for the existence of a nontrivial solution of (3) in the neighborhood of ε = 0, provided ( F, F) 0.(4) a, η For the case that the existence condition is satisfied we will have a unique periodic solution. This is in agreement with the previous analysis (we have proven that the Rayleigh system has a unique periodic solution). Note that if (4) fail to hold, it does not mean that there is no periodic solution. In many cases such as in hamiltonian system, the periodic solution is not isolated so that apriory we know that the condition fails to hold. It is instructive to apply the method. Note that this is an asymptotic method so that it is valid for µ 0. Thus we take µ equal to small parameter ε. The calculations are rather routine and lengthy. We write the result of the calculation up to order 4. The calculation is done using Maple release V. The periodic solution is approximated by () where x 0 ( θ ) = x ( θ ) = x ( θ ) = x 3 ( θ ) = 78 x 4 ( θ) = cos( θ ) 6 cos( θ ) 3 sin( 3 θ ) 3 cos( 3 θ ) 88 3 cos( 5θ ) 3 sin( 7 θ) cos( θ) 3 cos( θ) 6 3 cos( 7θ) sin( θ ) sin( θ ) 3 cos( 5 θ ) 3 sin( 3θ ) 3 cos( 5θ) 3 cos( 9θ) and θ = ωt. Obviously this is a nontrivial thing to do but nevertheless, as we noted above it is instructive to do it. Furthermore, we also calculate the period, i.e T = π πε πε O( ε ) The location of the periodic solution is a = ε 743 O ε ( ε ) We will check on this result with a numerical integration of the system using MatLab 5.. V. Numerical Result We will now check the approximation above using a numerical integration. We plot both of the result in one picture so that we can immediately compare the result. We use only the first approximation x(θ) = x 0 (θ) and O(ε 5 ) approximation for ω. This has the advantage of a very long time-scale. It means that the approximation is valid for quite a long time and in this case until ε -4. The numerical integration is done using build in integrator in MatLab 5. for non-stiff system, i.e. ODE45. This integrator uses Runge-Kutta scheme for INTEGRAL, vol. 5 no., April 000 7

8 integration. We plot the result in figure (6), (7), (8) and (9). The curve plotted by symbol o represent the numerical solution and the line curve is for the asymptotic approximation. In figure (6) and (7) we take ε=0.05. Thus the approximation is still good until 6,000 second. For numerical integration this long time integration is not recommendable. We have to worry also about the numerical error. Thus we integrate with initial values x(0) = a and x (0) =0 for 0 second only. Obviously we can expect a very good result in the comparison. In figure (8) and (9), we increase the value of ε to 0.5. We integrate with the same initial values for 80 second. This is already longer than the timescale of validity of the approximation. Thus we expect to see an O(ε) deviation on the picture. In figure (8) the deviation is not very clear but in figure (9) it is. Figure 6: The comparison between numerical result and asymptotic approximation for ε=0.05. This picture represents the time evolution of the periodic solution. Figure 8: The comparison of the time evolution for ε=0.5. Figure 9: The comparison in the (x, x )- plane for ε=0.5. Figure 7: The comparison between numerical result and asymptotic approximation for ε=0.05 on the (x, x )-plane. To make the error even clearer we plot the error defined as the difference between the numerical solution and the asymptotic solution in figure (0) and (). 8 INTEGRAL, vol. 5 no., April 000

9 Beside the method we have used here there is also a simpler method to get the approximation. The method is called averaging method. This is a very natural method introduced intuitively by Lagrange et. al.. Unlike the Poincare- Linsted method, this method is more general since it can be applied to approximate non-periodic solutions. The disadvantage of the averaging method is that, to get a higher order approximation is non-trivial, while for the Poincare-Linsted method it is routine. Also in extending the timescale, using averaging method is rather difficult. For a higher dimension case, where everything is restricted, averaging is easier to apply. Figure 0: The error on x. Figure : The error on x. VI. Remarks and Acknowledgment We have shown an example of asymptotic approximation of a periodic solution of a planar system. This Poincare-Linsted method can also be extended to higher dimension system. We note that it is not trivial to do so. We would like to express our gratitude to Santi Goenarso for her diversified contribution. We also like to thank our student Maynerd Tambunan for providing a comparison work for the Maple calculation. References: [] Boyce, W.E., Di Prima, R.C., Elementary Differential Equations and Boundary Value Problems, John Wiley & Sons, Inc., New York et. al., 99. [] Carr, J., Applications of Center Manifold Theorem, Applied Mathematical Science 35, Springer- Verlag, New York, 98. [3] O Mailley Jr., R. E., Singular Perturbation Methods for Ordinary Differential Equations, Applied Mathematical Science 89, Springer- Verlag, New York, 99. [4] Verhulst, F., Nonlinear Differential Equations and Dynamical Systems nd ed., Springer-Verlag, Berlin, 996. [5] Wiggins, S., Introduction to Nonlinear Dynamical System, Text on Applied Mathematics, Springer-Verlag, New York, 990. [6] Guckenheimer, J., Holmes, P., Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Applied Mathematical Science 4, Springer-Verlag, New York, 983. Author Johan Matheus Tuwankotta is a lecturer at the Mathematics Department ITB. address: tuwankotta@math.uu.nl Received August 5, 999; revised September INTEGRAL, vol. 5 no., April 000 9