THE SEPARATRIX FOR A SECOND ORDER ORDINARY DIFFERENTIAL EQUATION OR A 2 2 SYSTEM OF FIRST ORDER ODE WHICH ALLOWS A PHASE PLANE QUANTITATIVE ANALYSIS
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1 THE SEPARATRIX FOR A SECOND ORDER ORDINARY DIFFERENTIAL EQUATION OR A SYSTEM OF FIRST ORDER ODE WHICH ALLOWS A PHASE PLANE QUANTITATIVE ANALYSIS Maria P. Skhosana and Stephan V. Joubert, Tshwane University of Technology Department of Mathematics and Statistics Private Bag X680, Pretoria, 0001, FIN skhosanapm@tut.ac.za Abstract: This research letter will present the method of finding the separatri of the autonomous system of a second order Ordinary Differential Equation (ODE) by means of quantitative solutions on the phase plane. Keyword(s): Computer Algebra System (CAS), Saddle Point, Separatri 1. Introduction: The paper eamines first order systems of ordinary differential equations and shows how to determine phase plane portraits and identify separatrices when there is a saddle point. In order to do so we describe how to use a computer algebra system (CAS) to generate trajectories from contour plots when possible and from the numerical investigations. The choice of software is Derive 6, for its simplicity and easy to read mathematical epression. In many cases the equation of separatri can be determined. Generating a phase plane is useful, for at a glance one can observe what initial values give rise to bounded solutions, periodic solution and other important features. In a classroom environment the phase plane permit the instructor to concentrate on the qualitative aspects of the model under investigation.. Motivation of the paper. The CAS has been a paradigm shift on how to solve mathematics equations today, in this case a second order nonlinear differential equations or a system will be considered which allows a phase plane. We have chosen Derive 6 as software of choice to be able to eplore our investigation for simple reasons. Unlike many other mathematical tools, Derive 6 allows teacher or students to represent algebraic symbol in their correct form. Consider differentiating the epression + 1. Here Derive 6 has display the steps in the simplification of an epression along with the transformation rules applied, helping students to understand the principles behind the calculations. 1
2 Derive 6 also allows students to enter linear epressions in a two-dimensional manner and subsequently eperiment with the epressions to learn how to linearize epressions and understand the structure. Linearization in mathematics and its applications in general refers to finding the linear approimation to a function at a given point. We will use linearization for assessing the local stability of an equilibrium point of a system of nonlinear differential equations. This method is used in the field such as engineering, physics, economics, and ecology. The phase plane has been used as tool to analyze autonomous system. Our interest in this paper is to find a separatri of the autonomous system of ODE. We consider a system define by d f (, t) dt and linearized system can be written as d. Df ( 0, t). dt where 0 is the point of interest and Df ( 0 ) is a Jacobian matri of f () evaluated at 0. In stability analysis, one can use the eigenvalues of the Jacobian matri evaluated at an equilibrium point to determine the nature of the equilibrium. If all the eigenvalues are positive, the equilibrium is unstable; if they all negative the equilibrium is stable; and if the values are mied signs, the equilibrium is a saddle point. Any comple eigenvalues will appear in comple conjugate pairs and indicate spiral (or circular if the real components are zero) around the equilibrium. Then we can plot the trajectories to establish whether we have a separatri. Let us consider the following differential equation describing the motion of the pendulum: d g + sin θ 0. dt l The way we understand separatri in mathematics is the boundary separating two modes of behavior in a differential equation. In this eample marks the transition between the pendulum swinging back an forth and the pendulum making full circles.
3 . An outline how to find a separatri: Consider: An autonomous second order differential equation is the one of the form: Q (, ) (1) in which the independent variable does not appear eplicitly. If we introduce a new variable y, we obtain the equivalent system d dt dy dt y Q (, y) () This is the special case of the general autonomous system of two first order DE's called Plane Autonomous System. y P(, y) y Q(, y) () Let us assume the system in equation () is continuous and all its partial derivative eist and are continuous on a specific choose region R in the phase plane. This assumption applies to all differentials that will be discussed in this research letter. Suppose we consider main properties of the autonomous differential equation, as represented in the phase plane by equations () Q(, ), 1. We determine all the critical points of equation () by solving the following equations below, Py (, ) 0, Qy (, ) 0
4 using simultaneous equations. The resulting solutions are called equilibrium/constant solution situated at P (,0) 0, Q (,0) 0. We linearize the system in equation () at each critical point (and we consider only real values) to classify each one of the critical points according to eigenvalues and corresponding eigenvectors. The nature of the equilibrium points/ critical points Classification Basic Solution Set Improper node λ and λ ; real, distinct and same sign λ1t λt { ue ; ve } 1 Deficient node λ λ one dimensional eigenspace λ1 1 { t λt ue ;( w + tu) e } 1 ; Star or proper node (any basis of ) λ λ two dimensional eigenspace λ1t λt { ue ; ve } 1 ; Saddle point λ1t λt λ < 0 < λ ; { ue ; ve } 1 Center (or vorte) iβt iβt λ λ iβ real and nonzero {Re[ ue ];Im[ ue ]} 1 ; Spiral point (or focus) λ λ α + iβ real and nonzero ( ) 1 ; ( α+ β) ( α+ β) i t i t {Re[ ue ];Im[ ue ]}. We solve, if possible the equation of the phase paths given by dy Q(, y) ; y f( ) + C ; where C is the constant of integration. The resulting solution is d y called Solution curves or Orbit solution, [] in the y-plane which is considered as the phase plane. 4. This constant C is very important since, if it is determined by the critical point as the initial condition of the system in the specified region R or interval I, the resulting curve might be a SEPARATRIX.(Hypotheses) 4
5 5. To establish the supposed hypothesis above we construct the trajectories or solution paths, with the initial conditions on the given differential equation. The graph plotted is might be called a separatri function. NOTE: The graph is not plotted by numerical solvers e.g. Runge-Kutta in Derive6 / Mathematica. 6. Then we plot the Phase Portrait using other initial conditions in the surrounding neighborhood of the critical points inside the region R using curve established by line and line4 above. 4. Several models where separatri eist. Pendulum Spring models Competing species Predator-prey Modified predator-prey 5. Main results ( Eample of the outline given in section ) Consider the second order differential equation that model a soft spring. + 0 Its plane autonomous system given by P(, y) Q(, y) y y Determining all the critical points of the system y 0 then ( 0 1) 0 0, ± 1 Linearizing the system at each critical point and classifying these critical points according to stability and type (if possible) 5
6 0 y 1 0 y y (0,0) 1 0 y Eigenvalues and corresponding eigenvectors are given λ ± j eigenvalues which gives a centre point in the system according to table above. This is stable. Similarly: Linearizing the system at critical point 1,0) and ( 1,0) ( gives the same eigenvalues λ ± eigenvalues which gives a saddle point in the system and is unstable. and when λ then the corresponding eigenvector is y when λ then the corresponding eigenvector is y Determining the phase plane equation: y y dy d y ydy ( ) d y C This is the solution curve, where the constant C might give a separatri when calculated using the critical points values. At this stage the CAS is introduced to plot the trajectories or solution path and the choice of software is Derive 6. First the value of C must be determined and initially the critical point is used for and y. 6
7 The phase path / trajectories at point ( 0,0 ) which gives two separatrices The phase path / trajectories at points 1,0) and ( 1,0) the centre. ( gives no separatri but a different behavior from Plotted o the same system of ais: 7
8 Now using different initial conditions we can generate the phase portrait of the solution curve. We mentioned earlier that qualitative analysis give more insight to the differential equation. In this case the solution. In the center the circle enclosed within the separatrices says part of the soft spring differential equation gives a periodic solution. While around the saddle point displays a different behavior which is parabolic in nature. 6. Conclusion Is it possible to find the separatri equation analytically? The answer to this question is yes, provided: Phase plane autonomous system eists and is possible to linearize the system. One characteristic of a separatri is most likely to be found where one of the critical point is the saddle point. Most separatrices occur where there is an isolation point like a center as a critical point but more studies shows that if there is no isolation point it is most likely that there is no separatriri. Derive can be considered as the most easiest and most efficient software to eplorer the eistence of a separatri. We can conclude that using Computer Algebra System (CAS) made it easy to develop the theory of separatri fully. This can be a relevant study for undergraduate student, in order to introduce the concept of solving the second order ordinary differential equation as well as CAS at the same time. Mostly the study gives an opportunity to analyze the solution based on qualitative mean which gives far more interesting characteristic and lot of insight to the problem at hand. 8
9 7. References [1] ZILL D. G., CULLEN MR., 001 Differential Equations with Boundary-Value Problems, Brooks/Cole,. [] BORELLI R.L., COLEMAN C.S., 1998 Differential Equations: A Modeling Perspective, John Wiley & Sons, Inc.,. [] KOSTESILCH E. J., ARMBRUSTER D., 1996,Introductory Differential Equations from Linearity to Chaos, Addison-Wesley Publishing Company,. [4] EDWARD C. H., PENNY D. E., 199 Elementary Differential Equations with Boundary Value Problems, Prentice Hall,. [5] JOUBERT S.V., 00 TECHNIKON Mathematics III with DERIVE 5, Great White Publishers. 9
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