F 1 x 1,,x n. F n x 1,,x n

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1 Chapter Four Dynamical Systems 4. Introuction The previous chapter has been evote to the analysis of systems of first orer linear ODE s. In this chapter we will consier systems of first orer ODE s that are not linear but are autonomous; i.e., these are systems of the form t x t x n t i.e., X t F X t. F x,,x n F n x,,x n. Recall that a system is sai to be autonomous if F X t oes not epen explicitly on t. These systems are often referre to as ynamical systems. It will often be the case that the analytic solution for such systems can not be foun but a great eal about the solution for the system can be iscovere by analyzing the behavior of the solution in the neighborhoos of special points in the omain calle critical points. A point a in R n is a critical point for this system. if an only if F a 0. A critical point is also calle a stationary point because if the system is place at a stationary point, it will remain there since X t 0; i.e., the state is not changing. A curve C X t : t 0 is calle an orbit or trajectory for the system. if X t satisfies X t F X t. In the neighborhoo of a critical point, the orbits of a nonlinear system may often be approximate by the orbits of an associate linear system. We will show how this associate linear system can be foun in the next section. Once the local solutions near each critical point have been approximate, it may then be possible to blen the local pictures to form a global picture of the solution for the nonlinear system. Suppose X t is an orbit for a system an a a,..., a n is a critical point. Then for any t, X t a x t a 2 x n t a n 2 equals the istance from the point X t on the orbit, to the critical point a. We say that the critical point is stable if every orbit that originates near a remains near a. More precisely, a is stable if for every 0 there is a 0 such that X 0 a implies X t a for all t 0. We say that the critical point a is asymptotically stable if every orbit that originates sufficiently near a is rawn continually closer to a. More precisely, a is asymptotically stable if for some 0 X 0 a implies X t a 0 as t. Any critical point that is asymptotically stable is stable but the converse is false. Any critical point that is not stable is sai to be unstable. At this point, we are going to restrict our consierations to the 2-imensional case, n 2. This will allow us to gain insight into our

2 ynamical systems by representing the system orbits graphically. 4.2 Dynamical Systems in two imensions In orer to iscuss nonlinear ynamical systems, it will be helpful to first consier linear ynamical systems. Linear ynamical systems are systems of linear ODE s like those consiere in the previous chapter but here we are going to focus more on qualitative information about the solutions rather than trying to construct these solutions Linear Dynamical Systems A 2-imensional linear system is of the form U t AU t 2. i.e., t ut vt a c b ut vt Such a system has one critical point at U 0. We assume that the coefficient matrix A is nonsingular hence U 0 is the only point U u, v where the ynamical system is stationary. For a linear ynamical system there are six ifferent types of critical points, which may be characterize accoring to the nature of the eigenvalues of A. We will list the six types an provie a sketch of the orbits for each of the cases. If A has Complex Eigenvalues: i 0, 0 is a CENTER figure if 0 0, 0 is a SPIRAL SINK figure 2a if 0 0, 0 is a SPIRAL SOURCE figure 2b if 0 If A has Real Eigenvalues:, 2 0, 0 is a SOURCE figure 3 if 0 2 0, 0 is a SINK figure 3 if 2 0 0, 0 is a SADDLE POINT figure 4 if 0 2 We will consier some examples to show the structure of the orbits near each of the six types of critical point ) Center: Let A 0 c 2 0 : Then u t vt v t c 2 ut In this case, the eigenvalues of A are imaginary, ic. Note that the equations of the system imply, c 2 u u t v v t c 2 uv c 2 uv 0. 2

3 But c 2 u u t v v t 2 t c2 u 2 v 2 hence an t c2 u 2 v 2 0 c 2 ut 2 vt 2 Const. Eviently, the trajectories or orbits of this system are the curves C ut, vt : t in the u-v plane such that ut an vt solve the ynamical system, then c 2 ut 2 vt 2 Const implies that the plots of u versus v for this system are ellipses. The behavior of the system can be unerstoo from stuying the plots of the system orbits in the u-v plane. We refer to these plots as phase plane portraits of the system. V U Fig Re 0 Here we have sketche one orbit for each of the cases, c 2 2,,. Note that since 2 u t vt, u is increasing when v is positive an ecreasing when v is negative. This implies that the orbits are traverse from left to right in the upper half plane an right to left in the lower half plane. In other wors, the curves are trace in the clockwise irection as t increases. Since the orbits of this system are close curves, the system assumes the same values for u an v over an over again. This means that the solutions to the system must be perioic functions of time. That is, close curve orbits correspon to perioic solutions for the ynamical system. 2) Spiral Sink/Source: Let A 0 c 2 u t vt 2b v t c 2 ut 2bvt, with b 2 c 2. Then In this case the eigenvalues of A are b b 2 c 2, a complex conjugate pair with negative real parts. Note that c u u t v v t cuv cuv 2bv 2 2bv 2 0 hence 2 t c u2 v 2 bv 2 0, an it follows that R 2 t c ut 2 vt 2 is a ecreasing function of time. Since R 2 t is 3

4 closely relate to the square of the istance from the point ut, vt to the origin, a point on the orbit raws closer to the origin as t increases so this orbit must procee inwars. Plotting orbits in this case leas to v u Fig 2 Re 0 Since the curve is trace in the clockwise irection, this is a spiral moving in towar the origin an the origin is classifie as a spiral sink. In the case that b is negative, but b 2 c 2, the orbit is again a clockwise spiral but in this case, it is moving outwar from the origin. This correspons to the conjugate pair of eigenvalues having a positive real part an in this case the origin is classifie as a spiral source. Clearly, the spiral sink is an asymptotically stable critical point while the spiral source is unstable. 3) Sale Point- Let A 0 0. In this case the eigenvalues are with eigenvectors E, associate with an E, associate with the negative eigenvalue. In particular, since the two real eigenvalues have opposite signs, this causes the orbits to have the following appearance. v u Fig In this picture, the orbits all approach the origin along the irection of the eigenvector X, associate with the negative eigenvalue. These are the orbits that have negative slope an we refer to this as the irection of approach. The orbits recee from the origin in the irection of the eigenvector X, associate with the positive eigenvalue. These are the orbits with positive slopes an we refer to this as the escape irection. Since not all trajectories which start near the origin, remain there, a sale point is an unstable critical point. 4

5 4) Source/Sink: Let A 0, , eigenvectors: In this case the two eigenvalues an 2 2 are both negative so ut an vt both ten to zero as t tens to infinity. Then all orbits approach the origin an the trajectories look like v u Fig In the case that A has two positive eigenvalues, ut an vt both grow without boun as t tens to infinity with the result that the trajectories appear as in figure 4 but moving away from the origin. The critical point in this case is calle a stable noe or sink when the eigenvalues are negative an an unstable noe or source if they are positive. Summarizing, every linear ynamical system has a single critical point locate at 0, 0 an that critical point must be one of these six types. We are assuming here that A is nonsingular Nonlinear Dynamical Systems A nonlinear ynamical system is a system of the form u t fu, v v t gu, v 2. 2 an instea of trying to construct solutions for these systems, we are going to try to sketch families of curves, calle orbits or trajectories, ut, vt : t, where u an v solve the equations. This is calle sketching a phase plane portrait for the system. Here we are assuming that f an g o not epen explicitly on t, an in this case, we say the ynamical system is autonomous. Systems which contain explicit epenence on t are calle nonautonomous an they are more ifficult to analyze. The reason that autonomous systems are simpler is that the orbits are coherent. This means that istinct orbits cannot cross each other an the reason they cannot cross is easy to explain. Suppose we have two orbits for the system above that start from istinct points but later cross at a point, u, v. If they cross each other then the tangent T to the first curve oes not have the same slope at u, v as T 2, the tangent to the secon curve. Then if we enote by, mt k, the slope of T k, k, 2, at the point u, v where the curves 5

6 intersect, then mt v u C v t u t C fu, v gu, v v t u t C 2 v u C 2 mt 2. But this says that since both curves are orbits, their slopes are etermine solely by the values of fu, v, an gu, v. But then the slopes at u, v cannot be ifferent, which is to say, the curves cannot cross. We have efine a critical point for the system to be any point u, v, where fu, v gu, v 0. Then a critical point is a stationary point for the system since u t v t 0. Now the process of sketching the phase plane portrait for a ynamical system will consist of sketching the local picture for the system in a neighborhoo of each of its critical points an then using the fact that the orbits are coherent to blen all the local pictures into a global portrait. We will now escribe how we are going to iscover the local picture in the neighborhoo of a critical point. Local Analysis Suppose u, v is a critical point for the ynamical system 2. 2 an write the first few terms of the Taylor series expansions for f an g about the point u, v, fu, v fu, v u fu, v u u v fu, v v v gu, v gu, v u gu, v u u v gu, v v v Taking into account that u, v is a critical point, fu, v gu, v 0, an ropping all the higher orer (nonlinear in u an v) terms, we have fu, v u fu, v u u v fu, v v v gu, v u gu, v u u v gu, v v v Then the equations of the original ynamical system imply that u t u fu, v u u v fu, v v v v t u gu, v u u v gu, v v v, or, since u, v are constants, ut u t u fu, v u u v fu, v v v vt v t u gu, v u u v gu, v v v. But this is now a linear system of the form t ut u vt v u fu, v v fu, v u gu, v v gu, v ut u vt v where the coefficient matrix 6

7 Ju, v u fu, v v fu, v u gu, v v gu, v is constant. The only critical point of this linear system is the point u, v, where ut u vt v 0. A famous theorem known as the Hartman-Grobman theorem asserts that the orbit structure for the linear system near its critical point is very close to the orbit structure for the nonlinear system at the point u, v. We will illustrate with examples. Examples. Unampe Penulum The ynamical system of this example is a mathematical moel for a frictionless penulum consisting of a mass, m, on a rigi weightless ro of length L. If the angular eflection from the rest position is enote by, then the component of the gravity force in the irection normal to the ro is equal to mg sin. Then conservation of angular momentum asserts that ml t mg sint. Letting ut t, vt t leas to u t vt, v t 2 sinut where 2 g/l represents the frequency associate with the oscillating penulum. For simplicity we will choose in our example. Then the system, becomes, u t vt v t sinut Then fu, v v, gu, v sinu an the critical points for the system are all points where v 0 an sinu 0; i.e., there are infinitely many critical points, namely all the points k, 0 : k any integer. Then Ju, v 0 cosu 0 0 cosk 0 We will sketch the local pictures near the critical points, 0,0, 0 an, 0 : At, 0 an, 0, cos, so J, 0 The eigenvalues in this case are, real an of opposite signs, so the critical point is a sale point. The eigenvectors are : X,, for, an X 2, for 2. Recall that X is the irection of escape for orbits near this critical point since this is the eigenvector associate with the positive eigenvalue. The other eigenvector is associate with the negative eigenvalue an is therefore the approach irection for this critical point. At 0, 0, cos0, so J0, ,, an in this case the eigenvalues are 7

8 i. This critical point is a center. The orbits near a center look like the close orbits seen in figure while the orbits near a sale resemble the picture in figure 3. We then try to blen the three local pictures into one global picture using the fact that the orbits are coherent. The result looks like this. y x Figure 5 Unampe Penulum We can see that there are three kins of orbits here. There is an orbit in the upper half plane joining the critical point at, 0 to the critical point at, 0 an another orbit in the lower half plane which goes from, 0 back to, 0. These are not two parts of one orbit but are, in fact two ifferent orbits since an orbit that starts near one critical point an goes to another must stop when it reaches the estination critical point (a critical point is a stationary point). However, the curve compose of these two orbits taken together separates the phase plane into two parts an, for this reason, this curve is calle a separatrix for this system. All the orbits insie the separatrix are close curves an therefore correspon to perioic solutions of the ynamical system. All the curves lying outsie the separatrix are curves that are not close an in fact assume all values for u as t goes from to. Along the nonclose curves in the upper half plane, the variable vt varies between two positive values, while along the nonclose curves in the lower half plane, the variable vt varies between two negative values. Notice that the approach an escape irections for the two sales o, in fact, lie along the irections of the two corresponing eigenvectors an the orbits are traverse in the clockwise irection in conformity with the equations of the system. Observe that the equations of the system imply that sinu u t sinu v hence Here we have chosen v v t sinu u t t v v t sinu v 2 vt2 cosut 0. Eu, v 2 vt2 cosut as the antierivative for v v t sinu u t instea of Gu, v 2 vt2 cosut, since Eu, v is zero when u v 0 an G is not zero. This quantity Eu, v is an expression which can be interprete as the total energy in the physical ynamical system an with this choice of E we have calibrate the energy to be zero when the system is at rest. The equation asserts that the energy is constant along any orbit an along the separtrix in particular, we have E, 0 2. Then any orbit for which Eu, v 2 is a close orbit corresponing to a 8

9 perioic solution an any orbit for which Eu, v 2 is an orbit outsie the separatrix corresponing to an unboune solution. Now we can see that the three types of orbits in our example correspon to three ifferent kins of motions of the penulum. The close orbits insie the separatrix correspon to back an forth motions of the penulum. The small almost circular orbits close to the origin correspon to oscillations of small amplitue an for these orbits, the approximation sinu u is vali. That is, u t 2 ut on these orbits an this linear equation can be easily solve. The solution is referre to as the small amplitue approximation for the penulum. Close orbits of larger amplitue are much less circular an for these orbits the small amplitue approximation is not vali an the solution is not expressible in terms of sines an cosines. The separatrix correspons to a motion in which the penulum starts straight up in the position corresponing to. The penulum is then given a small push so that it swings own an then back up again so as to once again come to rest in the straight up position. The orbits outsie the separatrix correspon to motions in which the penulum is set in motion with so much energy that the penulum loops the loop an goes over the top. Since there is no mechanism for energy issipation in the moel (i.e., no friction) the penulum will continue looping forever. 2. The Dampe Penulum To see what happens when friction is ae, moify the momentum equation for the penulum as follows, ml t mg sint c t, c 0. The term c t represents a force which is in opposition to the momentum an is proportional to the spee of the penulum, e.g., a force create by friction with the air. To see that this term oes in fact issipate energy, let ut t, vt t an write u t vt, v t 2 sinut C vt, C 0. Then v v t 2 sinu u t t 2 vt2 2 cosut Cv 2 0. i.e., t Eut, vt Cv2 0, so energy ecreases along orbits of this system. To see what the phase plane portrait looks like we note that the new system has the same critical points as the frictionless system. Then (letting, for simplicity) we compute Ju, v 0 cosu C 0 cosk C. At, 0 an, 0, cos, so 9

10 J, 0 0 C. Assuming C 2 4, the eigenvalues are once again real an of opposite signs so this critical point is a sale point. The eigenvalues can be foun to be The corresponing eigenvectors are : 2 C 2 C2 4 0 an 2 2 C 2 C X C C , for 0 an X 2 C C for 2 0. At 0, 0, cos0, so J0, 0 0 C, an the eigenvalues are 2 C 2 C2 4, an 2 2 C 2 C2 4, i.e., the eigenvalues are a complex conjugate pair with negative real part so the critical point is a stable focus with orbits similar to those in figure 2. Then the plot analogous to the frictionless phase plane portrait shows that the close orbits aroun the origin have become, in the presence of friction, spirals tening towar the origin (i.e., the rest position). We interpret this to mean that as the penulum loses energy to friction, the amplitue of the motion ecreases to zero. The unboune orbits in the upper phase plane have become spirals that ten towar the next critical point locate at x, while the some of the unboune orbits in the lower phase plane are attracte to the critical point at the origin. Figure 6 Dampe Penulum Note that some of the orbits in this picture win up at the critical point at 0, 0, while other orbits eventually go to the critical point at 2, 0. There is also one orbit that comes from 0

11 above the horizontal axis an terminates at the sale point at, 0. This orbit separates those orbits going to the origin from those orbits which go to the next critical point. The region to the left of this orbit is referre to as the "basin of attraction" for the critical point at 0, 0, while the region to the right of this orbit is basin of attraction for the critical point at 2, 0. Of course there are also orbits that are eventually attracte to other critical points so these basins of attraction have some limits to their size. It is also necessary to note that the orbits shown in Figures 5 an 6 are orbits of the linear approximations to the nonlinear systems. This means that the true orbits may vary slightly from the picture orbits. Of course a slight perturbation of a spiral is still a spiral, an a slight variation of the vaguely hyperbolic curves near a sale points are still more or less hyperbolic, but a slight variation of the close orbits seen in Figure 5 may be no longer close. For this reason, when a nonlinear system has a critical point which is a center, it is not automatically possible to conclue that the orbits near the center are close curves even though this is the case for the orbits of the linear approximation to the nonlinear system. Instea, we must fin some inepenent evience that the orbits are inee close curves. In the case of the unampe penulum, this evience is provie by the energy expression that asserts that the energy expression, Eu, v 2 vt2 cosut is constant on orbits of the nonlinear system, an when this constant value is less than 2, this is the equation of a close curve. In general, it is not always easy to verify that the orbits of a nonlinear system aroun a center are in fact close. We say that the source, sink an sale type of critical points are "structurally stable" while the center is a "structurally unstable" critical point. This just refers to the fact that the orbit pictures aroun source, sink an sale points are not significantly altere by slight changes in the system in contrast to the picture near a center where a small change can turn close orbits into spirals or some other form of non-close curve. To illustrate this point, consier the system u t vt ut 3 v t ut Clearly the only critical point is at 0, 0 an Ju, v 3u 2 0. Then the eigenvalue equation for J0, 0 is 2 0, which implies that the critical point is a center. However, utu t vtv t 2 t u2 v 2 u 4. When is positive, this implies that a point on an orbit moves away from the origin with increasing t so that the critical point must be a spiral source. When is negative, the point is a spiral sink. In either case, the origin is a center for the ynamical system but the orbits near the origin are not close curves. 3. A Preator-Prey Moel We consier two populations, a preator population, whose number at any t is enote by Pt, an a prey population where pt enotes the size of the population at time t. If these populations occupie the same space but i not interact, then the equations governing P an p woul be p t bpt pt P t BPt DPt where b, enote the birth an eath rates for pt an B an D play the same role for Pt. If the populations interact with the preator population feeing off the prey, then we coul

12 assume an B epen on P an p respectively. We coul assume, for simplicity that kp, to reflect the fact that the presence of preators increases the eath rate of the prey in an amount proportional to the population size of the preators. Similarly, we coul suppose the B Kp, to reflect the fact that the presence of plenty of foo (in the form of prey) causes the birth rate of the preators to increase proportional to the size of the prey population. Then the system becomes p t bpt kptpt P t KPtpt DPt For purposes of this example, let us choose simple values for the constants in the system so that we have p t 6pt 3Ptpt P t 2Ptpt 4Pt. Then there are critical points at 0, 0 an 2, 2. We compute the Jacobian matrix, Jx, y 6 3P 3p 2P 2p 4 an fin J0, 0 J2, ,. Then the origin is a sale point with eigenvectors E 6, 0, the escape irection, an E 4 0,, the irection of approach. The point 2, 2 is a center an we will show that it is surroune by close curve orbits. From the first equation in the system, we see that p 0 when P 2, an p 0 when P 2. This tells us that the orbits aroun the center are traverse in the counter-clockwise irection. In this example we will be able to show that the orbits aroun 2, 2 are close curves. We rewrite the system equations in the following separate form p p6 3P P P2p 4 2p 4 or p p 6 P 3P P The equations are separable an we can integrate to fin 2p 3P 4 ln p 6 ln P C. This solution is in implicit form an is not reaily recognizable as some well known curve, but it can be plotte an seen to be a close curve. The implication of this solution is that the two populations will perioically increase an ecrease in time, but as long as the close curve orbit stays away from the origin, neither population will ie out. If the value of either population falls below some threshol value require for sustaining reprouction, then both populations ie out. The critical point a is calle a hyperbolic critical point for the system if Re 0 for all eigenvalues of Ja. Then every hyperbolic critical point is asymptotically stable if it is a sink 2

13 an it is unstable if it is a source or a sale. A center is not a hyperbolic critical point. Exercises In each system, fin all the critical points an classify them. Sketch the orbits in the neighborhoo of each critical point. Try to blen the local portraits into a global phase plane portrait. Ientify any perioic solutions or other significant features of the solutions. If a critical point is a center, try to fin an implicit solution for close curve orbits u t vt v t ut vt 3 u t vt v t 4ut utvt 2 u t vt vt 2 v t ut 2 ut p t 64 ptpt 3Ptpt P t 2Ptpt 4Pt. 5. A linear ynamical system has a single critical point at the origin. Why? How is that critical point classifie? 6. A nonlinear ynamical system may have many (even infinitely many) critical points. What is the purpose of forming the Jacobian matrix an evaluating it at the critical points? 7. How o you etermine the irections of escape an approach at a sale point? 8. At a critical point that is a center are the orbits of the nonlinear system necessarily close curves? Explain. 9. Which types of critical points are asymptotically stable? Which are stable but not asymptotically stable? Which are unstable? 0. What is a separatrix? What is a basin of attraction? 4.3 Aitional Topics First, let us summarize what we have alreay foun. For linear systems, the origin is the only critical point. In 2-imensions, the origin is a sink if it is a stable noe or stable focus an in this case, every orbit of the system is attracte to the origin asymptotically. The origin is unstable an repels all orbits if 0, 0 is an unstable noe or unstable focus or if it is a sale. If the origin is a center, then it is stable (but not asymptotically stable) an all orbits are close curves aroun the origin. The six possible types of critical point for a linear 2-imensional system are summarize as follows: 3

14 , 2 i stable noe 2 0 sink stable focus 0 unstable noe 2 0 source unstable focus 0 sale 0 2 center 0 For nonlinear systems in the plane there can be multiple critical points, each of which can exhibit any of the above behaviors. If the critical points are all foun an classifie (i.e., ientifie as one of the 6 types liste above) then the sketch of the phase plane portrait that is consistent with the type of each critical point can be mae in the neighborhoo of each critical point. Finally, using the fact that the orbits are coherent, it shoul be possible to blen the local pictures into a global phase plane portrait for the nonlinear system. Limit Cycles Although an approach base on these few points is successful on many examples, there are other examples where this is not sufficient. The approach fails for example when the system has a critical point that is encircle by a simple close curve calle a limit cycle. To illustrate this situation, consier the system x t y x x 2 y 2 y t x y x 2 y 2. The origin is the only critical point for this system an it turns out to be an unstable focus, a source. By multiplying the first equation by x an the secon by y an aing, we fin xtx t yty t x 2 y 2 x 2 y 2, or 2 t x2 y 2 x 2 y 2 x 2 y 2. Now r 2 x 2 y 2 is the square of the istance from the origin to xt, yt, a point on an orbit an expressing the last equation in terms of r leas to 2 t r2 t r 2 r 2. Then rt is increasing as long as r an it is ecreasing when r. If we let enote the isc x, y : x 2 y 2, then any orbit that originates in can never leave (since in orer to cross the circle r we woul have to have r 2 t 0 at r t which is impossible because of the equation). Similarly, if we let enote the annular region x, y : x 2 y 2 4, then any orbit that originates in must remain there (rt is steaily ecreasing in an yet no orbit can cross the inner bounary of ). Then orbits in this annular region spiral inwar towar a limit cycle. The phase plane picture is the following. 4

15 Limit Cycle In this example we can actually solve explicitly for rt. If we let ut rt 2, u t 2u u an if we separate an integrate we fin then an ut rt 2 ln u ln u 2t C 0 or u u C e 2t C e 2t C as t for any C. This shows that rt tens to for all orbits (the initial point of the orbit etermines C but here the value of the constant has no effect on the limit of rt as t increases). If we moify the example to rea x t y xx 2 y 2 y t x yx 2 y 2. then the critical point at the origin becomes a center but the orbits of the nonlinear system are not close curves about the critical point. The origin is a spiral sink for this example with the following phase plane picture, Spiral Sink Since in this example we can show that 2 t r2 t rt 4, 5

16 this behavior is not unexpecte. A less contrive example of a limit cycle arises from the so calle Van er Pol equation, x t xt 2 x t xt 0 which translates into the following first orer ynamical system, u t vt v t ut ut 2 This system has a single critical point, a spiral source at the origin. However, the equations of the system imply that utu t vtv t ut 2 vt 2 or 2 t u2 v 2 u 2 v 2. It follows from this last equation that u 2 v 2 is increasing when u 2 but increasing for u 2,. suggesting the presence of a limit cycle. The phase plane picture associate with this system is as follows, vt which clearly shows the presence of a limit cycle. Equations that Separate In general, as the previous examples illustrate, it is often ifficult to etermine what the phase plane portrait looks like aroun a center type critical point for a nonlinear ynamical system. There are special cases, however, when the situation becomes clear. Consier the following example: xt 4 2yt yt 2 3xt 2 This system has critical points at 2, 2 an 2, 2. We fin an then: Jx, y 0 2 6x 0 6

17 A2, 2 has eigenvalues 24, an 24 A2, 2 has eigenvalues i 24, an 2 i 24 2, 2 is a sale 2, 2 is a center We are now face with the problem of eciing whether or not there are close orbits aroun the center at 2, 2. If we write then y x y x 2 3x2 4 2y 4 2yy 2 3x 2 x; Integration leas to 4y y 2 2x x 3 C, or y 2 2 x2 x 2 C. This means that for we have Fx, y y 2 2 x2 x 2, t Fxt, yt xf xt y F yt 2 3x 2 4 2y 2y 42 3x 2 0 Then F is constant on orbits of the ynamical system which is the same as saying the level curves of Fx, y are the orbits of the ynamical system. For example, F2, 2 6, hence the curve y 2 2 x2 x 2 6 is the orbit that passes through the sale at 2, 2. Plotting this an several other level curves looks like this Since the first equation of the system implies that, x 0 when y 2, we can see that the close orbits aroun 2, 2 must be traverse in the counter clockwise irection. The orbit, C, passing through the sale point begins an ens at the same critical point. This orbit is calle a homoclinic orbit an is furthermore a separatrix for the system, since it separates the close orbits (insie C) from the orbits that are not close curves (these are outsie C). The region insie of C is calle a trapping region for the system since any system orbit that begins insie this region never leaves the region. The close curve orbits in the trapping region correspon to perioic solutions for the ynamical system since the state variables xt, yt continue to pass through the same set of points over an over as t goes from to. This example shows that when the system is sufficiently simple to allow us to solve, even implicitly, for the orbits of the system, then the question of how the orbits look aroun a center can be settle by simply plotting the orbits (in this example, the level curves of F ). In general, this simple approach will not be possible. 7

18 Other Simple Systems There are several important physical systems whose mathematical moels lea to ynamical systems of the following general form: x xp ax by y yq cx y Here a, b, c,, p an q enote constant parameters. Moels leaing to systems of this form inclue:. Preator-prey moels- Here the preator population, yt, survives by eating the prey, whose population, xt, has a maximum sustainable level of X. The term xy represents the rate at which the prey population is iminishe by the preators (the prey ecrease rapily when there are a lot of preators present). The term xy reflects the fact that the preators increase more rapily when there is a sufficient foo supply present in the form of prey. x k X x y x y x xy 2. Competition moels- Here there are two populations competing for the same, finite, pool of resources. Each population has a maximum sustainable level an the terms xy, xy represent the interaction between the two populations as they compete for the limite available resources. x r X x x xy y s y y xy Y 3. Combat moels- Here the variables x an y represent troop levels for two armies. The linear terms ax an by represent losses ue to noncombat causes (e.g. accients, illness) while the terms xy, xy represent losses ue to encounters with the enemy. x ax xy y by xy In orer to analyze moels of the general form x xp ax by y yq cx y We make use of null clines; i.e., curves along which x 0 ( x null clines) or on which y 0 ( y null clines). For our system, x 0 if x 0 or p ax by 0 y 0 if y 0 or q cx y 0 Apparently, there are four null clines; the y axis an the line, ax by p are x null clines, 8

19 while the y null clines are the x- axis an the line, cx y q. At a point where two null clines meet, we have x y 0, so intersection points of x null clines with y null clines etermine the critical points for the system. For our example we have critical points at: x 0 y 0 2 x 0 y q/ 3 y 0 x p/a p bq qa cp 4 x y if et a bc 0 a bc a bc The critical point 4) at the point where the two lines meet will turn out to be either a sale point or a stable noe. If it is a sale, the critical points 2) an 3) then turn out to be stable noes. In this case, all orbits originating in the first quarant ten to one of the critical points 2) or 3) where one of the populations is extinct. If the critical point 4) is a stable noe, then all orbits originating in the first quarant ten to this critical point, an the populations stabilize at this cooperative equilibrium point where neither population is extinct. We consier two examples of this type: x x x y Here, we have critical points at: We fin Jx, y y y3 2x 4y x 0 y 0 2 x 0 y 3/4 3 y 0 x 4 x /2 y /2 2x y x 2y 3 2x 8y an J0, has eigenvalues, 3 Unstable Noe J0, 3/4 /4 0 3/2 3 has eigenvalues 3, /4 Sale J, 0 0 has eigenvalues, Sale J 2, has eigenvalues Stable Noe 9

20 The phase plane picture looks like this, from which it is evient that every orbit originating in the first quarant is attracte to the critical point at. 5,. 5. On the other han consier the example x x x y y y2 3x y In this case, we have critical points at: x 0 y 0 Here, Jx, y 2 x 0 y 2 3 y 0 x 4 x /2 y /2 2x y x 3y 2 3x 2y an J0, has eigenvalues, 2 Unstable Noe J0, has eigenvalues, 2 Stable Noe J, 0 0 has eigenvalues, Stable Noe J 2, , has eigenvalues 2 3 2, Sale Now the phase plane picture looks like this, 20

21 In this case all orbits that originate in the first quarant are attracte to one of the two stable noes 0, 2 or, 0 where one or the other of the populations is extinct. This means that this system has no cooperative equilibrium point where the two populations can coexist. Note that in the first example the eterminant a bc for the system of equations to fin the point of intersection of the two lines ha the value 2 while in the secon example a bc 2. It is possible to show in general that when the eterminant is positive, the intersection point of the lines is a stable noe an when the eterminant is negative, the intersection point is a sale. Liapunov functions Another tool for examining the stability of nonhyperbolic critical points is the so calle Liapunov function. For example, the previously consiere energy function, Eu, v, for the nonlinear penulum is constant along orbits for the unampe penulum while Eu, v ecreases along orbits for the ampe penulum. In the ampe case the orbits cross the level curves for energy (i.e., the curves along which energy is constant) in the irection of ecreasing energy. The energy function for the nonlinear penulum is a special case of a Liapunov function. Liapunov functions can be use not only in the case of 2-imensional systems but can be extene to higher imensions as well. We will present the iscussion in the two imensional setting but we will also give some examples in three imensions. Suppose that the system x t Fxt, yt y t Gxt, yt has a nonhyperbolic critical point at a x 0, y 0. Suppose also that we have a function E Ex, y which is a smooth function on R 2 such that Ex 0, y 0 0, an Ex, y 0 if x, y x 0, y 0. Then i If Ext, yt 0 on all orbits of the system t then a x 0, y 0 is asymptotically stable 2

22 ii If Ext, yt 0 on all orbits of the system t then a x 0, y 0 is stable iii If Ext, yt 0 on all orbits of the system t then a x 0, y 0 is unstable Of course the success of this metho epens on being able to fin a function E, which is calle a Liapunov function for the system. There are no universal recipes for fining Liapunov functions but by stuying the examples, we can at least get some iea of how to fin an E. Note that on orbits xt, yt of the system, Ext, yt E/x t x t E/y y t F x E G y E an if the expression on the right is zero then the erivative is zero. This means that E is constant along all orbits of the system; i.e., orbits of the system are level curves of the function E. If this erivative is not zero but instea is negative for all orbits of the system, this means that the orbits are not level curves of E but as a point moving along an orbit cuts a level curve, it goes from the exterior of the level curve into the interior. Finally, if this erivative is positive for all orbits of the system, the orbits are not level curves of E but as a point moving along an orbit cuts a level curve, it goes from the interior of the level curve into the exterior. To illustrate, consier the system x t y 3 y t x 3 The only critical point for this system is 0, 0 an the Jacobian has only the eigenvalue zero at this point. Then this is a nonhyperbolic critical point. Now consier the function Ex, y x 4 y 4. For this E we have E0, 0 0 an Ex, y 0 for x, y 0, 0 an Ext, yt t 4x3 x t 4y 3 y t 0. This means that level curves of E are orbits of the system an the origin is stable but not asymptotically stable. Now consier the 3-imensional example xt 2y yz yt x xz zt xy. Then 0, 0, 0 is the only critical point an we have 22

23 J0, 0, , with eigenvalues: 0, i 2,i 2 Then the origin is a nonhyperbolic critical point. If we try an E of the form then Ex, y, z ax 2 by 2 cz 2, Ext, yt 2ax2y yz 2byx xz 2czxy t 2b 4axy 2a b cxyz an if we choose the constants such that b 2a, c a 0 then E0, 0, 0 0, E 0 for x, y, z 0, 0, 0, an Ext, yt 0 on all orbits of the system; t i.e., all orbits are level curves for the ellipsoi x 2 2y 2 z 2 C 2. This is a stable critical point but it is not asymptotically stable. If we change the system slightly to xt 2y yzx 3 yt x xzy 3 zt xy z 3 then the critical point is the same an J has the same eigenvalues at the critical point. The same Liapunov function, E x 2 2y 2 z 2, now satisfies E0, 0, 0 0, E 0 for x, y, z 0, 0, 0, an Ext, yt t 2x4 2y 4 z 4 0 for x, y, z 0, 0, 0. Then the origin is an asymptotically stable critical point for this system even though all of the eigenvalues have zero real part. Finally consier the secon orer equation x"t qx 0, where qx enotes a smooth function such that q0 0 an xqx 0 for x 0. For example, the penulum equation with qx sinx satisfies this for x. If we let x t yt then x t yt y t qx an there is a critical point at the origin. If we let Ex, y 2 y2 0 x qs s 23

24 then Ext, yt y t y qxx yqx qxy 0. Then the orbits of the system are level curves of E (which is just the total energy function) an the origin is therefore a stable critical point. 24