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# 130 Class Note for MATH 251 at PSU

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Date Created: 02/06/15

Differential Equations LECTURE 37 Applications of Nonlinear Systems In today s lecture we ll look at some particular nonlinear systems generally inspired from physics or biology We ll then use the linearization method we ve been discussing for the last few lectures to try to understand the behavior of these systems 1 Competing Species One of the classical examples of a nonlinear system is the Lotka Volterra model of competition between two species let s take them to be rabbits and kangaroos Both species are competing for the same food supply the grass and there s a limited amount of this resource We ll assume these populations exist in an environment without predators What are the main considerations we ll need to keep in mind to write down this system 1 When we discussed the logistic equation as an example of an autonomous rst order equation we mentioned the existence of an environmental carrying capacity that is given a species which consumes resources at a certain rate there is a certain upper limit for the population of this species that the environment can support Here since we re assuming that we ve got a kangaroo population and a rabbit population and no others we can assume that in the absence of the other population each would grow to its carrying capacity Thus we ll incorporate logistic growth into the equations for each species We ll also assign rabbits a higher intrinsic growth rate due to the well known ability of rabbits to reproduce 2 What happens when the two species encounter each other while grazing Sometimes the rabbit might get to eat but the kangaroo being signi cantly larger they will normally push the rabbit aside and start eating We can assume that these encounters will occur at a rate proportional to the size of each population In light of the previous comment we ll also assume that the con icts reduce the growth rate for each species but that the rabbit population is more severely affected In particular if mt is the rabbit population at time t and yt is the kangaroo population at time t the model m z37m7 2y 371 M 24071711 372 incorporates these assumptions Our rst task is to nd the xed points of this system We solve z y 0 and obtain four xed points 0 0 0 2 30 and 11 Next let s use the linearization method to try to classify them The Jacobian of the system 371 is 7 3 7 2m 7 2y 72m A 7 lt 7y 2 7 m 7 2y 39 Let s now consider each xed point to determine the nearby behavior 0 00 HereA g 0 0 is an unstable node as non degenerate or star nodes are preserved in the original This has eigenvalues 1 3 and 2 2 so we can conclude that Differential Equations Lecture 37 Applications of Nonlinear Systems nonlinear system Recall that right at a node7 typical trajectories are tangential to the slow eigensolution7 which in this case is the eigensolution corresponding to 2 2 The eigenvector corresponding to 2 is i lt0 1 so trajectores will leave the node along the y axis o 07 2 Here A 32 the xed point is a stable node A typical trajectory approaches it along the eigensolution This matrix has eigenvalues 1 71 and 2 72 hence corresponding to A1 which is in the direction of the eigenvector i o 30 Here A lt73 76 The eigenvalues are 1 73 and 2 71 so this is also 0 71 a stable node The slow eigensolution is in the direction of the eigenvector of A2 namely 3 77 72 39 71 72 o 11 HereAi lt71 71 We can further note that the m axis is a straight line trajectory7 as z 0 when x O Similarly7 the y axis is a straight line trajectory Lines where z 0 or y 0 are called nullclines7 and in this case these nullclines are also solutions Putting this all together and using some common sense to ll in the rest of the trajectories7 we obtain a phase portrait that looks something like Figure 371 gt and 12 71i This is a saddle point J 111 i KzKK Ti ALlA ff KKKK fv Klt sn KKKK gt 444 I FIGURE 371 Phase portrait for the rabbits vs kangaroo model 371 Notice the xed points and their stability This has an interesting biological interpretation it shows that7 in general7 one species drives the other to extinction If we start above the stable solution of the saddle point7 the kangaroos drive the rabbits to extinction7 but if we start below it7 the rabbits drive the kangaroos to extinction This phenomenon shows up in more complex models of competing species as well7 and it has led to the formulation of the principle of competitive exclusion which states that7 in general7 two species competing for the same resources cannot coexist This is why releasing pets into the wild is a bad idea its species might drive native populations competing for the same resources out Differential Equations Lecture 37 Applications of Nonlinear Systems 2 Nonlinear Pendulum In the absence of any damping or external driving the motion of a pendulum is governed by the equation 120 g i w Zs1n0 0 where 0t is the angle from the downward vertical 9 is acceleration due to gravity and L is the length of the pendulum This is derived from the rotational formulation of Newton s Second Law 739 Ia where 739 7mgLsin0 is the torque I mL2 is the moment of inertia and Oz is the rotational acceleration Let s write we Then our equation is s wg sin0 0 373 For very small angles this equation will be linearized using sin0 z 9 this is generally done in high school for example but using phase plane methods we can study the nonlinear equation even for large angles Writing 373 as a system yields 0 z 37 4 1 flag sin0 where 1 is the angular velocity The xed points are of the form mr 0 where n is any integer The Jacobian of the system is A lt7w030slt0gt 5 Notice that there is no difference between angles that differ by 27139 either physically or formally so we ll focus on the two xed points 00 and 7r0 Near 0 0 the linearized system is 117 011 77w00 The eigenvalues of the coef cient matrix are 12 iwoi which says that the linearized system has a center As we discussed last class however that doesn t mean we have a nonlinear center at the xed point we could have a spiral So what do we do It turns out that the system 374 is an example of a reversible system that is if we reverse time by changing t to it and 1 to 71 since reversing time will reverse the velocity as well the system stays the same Physically this should make sense if we tape a pendulum s motion and play it backwards we won t see any physical absurdities It turns out that for reversible systems if the origin is a linear center it will be a nonlinear center as well The idea is that we take a trajectory close to the origin which swirls around the origin as the origin will have to be a spiral or a center Reversing time and velocity re ects this to a twin trajectory with the same endpoints but with the arrow reversed which closes the orbit This is illustrated in Figure 372 Thus 00 and hence 2k7r0 are all centers What happens near 7r0 The linearized system is 701 illiwooill The coef cient matrix has eigenvalues 12 iwwo Hence it and any xed point of the form 2k 17r0 is a saddle point Differential Equations Lecture 37 Applications of Nonlinear Systems FIGURE 372 Why an origin that is a linear center is also a nonlinear center in a reversible system the trajectory to the left also has its mirror image as a trajectory Now we can draw the phase portrait near the xed points How do we physically interpret the classi cation we just found The centers correspond to a state of neutrally stable equilibrium the pendulum is at rest and is hanging straight down as 9 21m If we move the pendulum slightly away from there and possibly give it a little initial velocity we ll get oscillation back and forth The saddles on the other hand correspond to the cases where the pendulum is at rest but is balanced perfectly up They re unstable as if we move the pendulum slightly from this balance they ll swing back down What happens away from the xed points This corresponds to giving the pendulum a lot of initial velocity We ll actually end up with the pendulum rotating around and around the axis so that the phase portrait looks like Figure 373 EXERCISE Free Damped Pendulum If we add a damper with damping coef cient 39y gt 0 to the nonlinear pendulum modeled by 373 the new equation governing the motion of the pendulum is 0 39y0 I we sin0 0 After writing this equation as a system of rst order equations nd and classify the xed points for all 39y gt 0 and plot the phase portraits for each qualitatively different case FIGURE 373 Phase portrait for the free undamped nonlinear pendulum 374 Differential Equations Lecture 37 Applications of Nonlinear Systems 3 Next Steps We ve only dipped our toes into the theory of nonlinear systems7 but hopefully we ve gotten the sense that there s a lot of interesting phenomena occuring here Let s brie y list some of the directions there are go in from here most of these would get at least a brief discussion in a second course in differential equations 31 Limit Cycles and Periodic Solutions After equilibria7 the next most important fea ture of nonlinear systems is the possibility of periodic solutions These also very strongly affect the overall phase portrait There are several global methods that can be brought to bear on the problem of detecting and understanding periodic solutions There are many questions that can be asked here If we have a closed trajectory7 must it always circle a xed point What kinds of xed points are permitted inside of one ls there a restriction on the number of closed trajectories we can have or where they can be located ln fact7 we ve seen examples where closed trajectories occur in nested families They can also be isolated such a solution is called a limit cycle Nearby solutions7 which aren t closed7 can limit to or away from the limit cycle Stable limit cycles are very important scienti cally7 as they model self sustained oscillations7 such as in the human heart These are7 as we ve seen7 inherently nonlinear phenomena if a linear system has closed solutions7 so are nearby solutions 32 Bifurcat ions We ve only looked at what happens if we have a single system of nonlinear differential equations What if we have a family of them7 differing by some parameter For example7 what if we have a nonlinear pendulum or an electrical system driven by a constant external force7 which we proceed to increase Can the behavior change as we vary the parameter The answer is yes Many different things can happen here the number or stability of xed points or closed orbits can change7 for example Such a change is called a bifurcation This is very important if you think about an externally driven electrical circuit7 for example understanding where and what bifurcations can occur will tell us how much force we can safely apply to the circuit Bifurcations can also occur for onedimensional equations7 as well7 and some very interesting ones occur for discrete systems7 where we take some function and iterate it repeatedly7 looking for stable and unstable xed points 33 Chaos It can be shown that for a two dimensional system7 things behave generally nicely and limit cycles are typical in some sense However7 in three dimensions7 all bets are off In 19637 while modeling convection rolls in the upper atmosphere7 Edward Lorenz wrote down the following system of differential equations m7amby y 7mzrmiy z zyibz where 011 are some constants and r gt This looks like a relatively simple system7 but it turns out to be surprisingly complicated The solutions don t ever settle down to a periodic orbit or xed point7 but they also don t run off to in nity lnstead7 they wrap around two equilibria in a fairly crazy manner The manner in which they wrap around these equilibria depends very strongly on the initial conditions small changes lead to solutions which behave very differently This is known as sensi tivity to initial conditions7 and it s one of the hallmarks of a chaotic system It s become known as the butter y effect a butter y aps its wings in Beijing7 and in New York it rains a few days Differential Equations Lecture 37 Applications of Nonlinear Systems later instead of being sunnyl It s why we can t predict the weather very far in advance since in struments and computers only have a certain number of decimal places they can measurecompute to but down the line the small bits that are lost in the process become very important 1There s another possibility for why this is called the butter y e ect77 There s a classic science ction story and an episode of The Simpsons parodying it about a time traveler who goes back in time to hunt dinosaurs and steps on a butter y When he returns to his time everything has changed in a substantial way Killing that one butter y changed the initial conditions77 of the universe and upon moving forward a signi cant amount of time everything evolved in a di erent way

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