Nonlinear OrdinaryDifferentialEquations

1 Nonlinear Ordinary Differential Equationsby Peter J. OlverUniversity of Minnesota1. notes are concerned with initial value problems for systems of ordinary dif-ferential equations . Here our emphasis will be on nonlinearphenomena and properties,particularly those with physical relevance. Finding a solution to a differential equationmay not be so important if that solution never appears in the physical model representedby the system , or is only realized in exceptional circumstances. Thus, equilibrium solu-tions, which correspond to configurations in which the physical system does not move,only occur in everyday situations if they are stable. An unstable equilibrium will not ap-pear in practice, since slight perturbations in the system or its physical surroundings willimmediately dislodge the system far away from course, very few Nonlinear systems can be solved explicitly, and so one must typ-ically rely on a numerical scheme to accurately approximatethe solution. Basic methodsfor initial value problems, beginning with the simple Eulerscheme, and working up tothe extremely popular Runge Kutta fourth order method, will be the subject of the finalsection of the chapter.

2 However, numerical schemes do not always give accurate results,and we briefly discuss the class of stiff differential equations , which present a more seriouschallenge to numerical some basic theoretical understanding of the natureof solutions, equilibriumpoints, and stability properties, one would not be able to understand when numerical so-lutions (even those provided by standard well-used packages) are to be trusted. Moreover,when testing a numerical scheme, it helps to have already assembled a repertoire of nonlin-ear problems in which one already knows one or more explicit analytic solutions. Furthertests and theoretical results can be based on first integrals(also known as conservationlaws) or, more generally, Lyapunov functions. Although we have only space to touch onthese topics briefly, but, we hope, this will whet the reader s appetite for delving into thissubject in more depth. The references [2,9,13,15,17] can be profitably First Order Systems of Ordinary Differential us begin by introducing the basic object of study in discrete dynamics: the initialvalue problem for a first order system of ordinary differential equations .

3 Many physicalapplications lead to higher order systems of ordinary differential equations , but there is asimple reformulation that will convert them into equivalent first order systems. Thus, wedo not lose any generality by restricting our attention to the first order case , numerical solution schemes for higher order initial value problems are entirelybased on their reformulation as first order 2022 Peter J. OlverScalar Ordinary Differential EquationsAs always, when confronted with a new problem, it is essential to fully understandthe simplest case first. Thus, we begin with a single scalar, first order ordinary differentialequationdudt=F(t, u).( )In many applications, the independent variabletrepresents time, and the unknown func-tionu(t) is some dynamical physical quantity. Throughout this chapter, all quantitiesare assumed to be real. (Results on complex ordinary differential equations can be foundin [14].) Under appropriate conditions on the right hand side (to be formalized in thefollowing section), the solutionu(t) is uniquely specified by its value at a single time,u(t0) =u0.

4 ( )The combination ( 2) is referred to as aninitial value problem, and our goal is to deviseboth analytical and numerical solution differential equation is calledautonomousif the right hand side does not explicitlydepend upon the time variable:dudt=F(u).( )All autonomous scalar equations can be solved by direct integration. We divide both sidesbyF(u), whereby1F(u)dudt= 1,and then integrate with respect tot; the result isZ1F(u)dudtdt=Zdt=t+k,wherekis the constant of integration. The left hand integral can beevaluated by thechange of variables that replacestbyu, wherebydu= (du/dt)dt, and soZ1F(u)dudtdt=ZduF(u)=G(u),whereG(u) indicates a convenient anti-derivative of the function 1/F(u). Thus, thesolution can be written in implicit formG(u) =t+k.( )If we are able to solve the implicit equation ( ), we may thereby obtain the explicitsolutionu(t) =H(t+k)( ) Technically, a second constant of integration should appear here, but this can be absorbedinto the previous constantk, and so proves to be 2022 Peter J.

5 To u= terms of the inverse functionH=G 1. Finally, to satisfy the initial condition ( ), wesett=t0in the implicit solution formula ( ), wherebyG(u0) =t0+k. Therefore, thesolution to our initial value problem isG(u) G(u0) =t t0,or, explicitly,u(t) =H t t0+G(u0) .( )Remark: A more direct version of this solution technique is to rewrite the differentialequation ( ) in the separated form duF(u)=dt,in which all terms involvingu, including its differentialdu, are collected on the left handside of the equation, while all terms involvingtand its differential are placed on the right,and then formally integrate both sides, leading to the same implicit solution formula:G(u) =ZduF(u)=Zdt=t+k.( )Before completing our analysis of this solution method, letus run through a coupleof elementary the autonomous initial value problemdudt=u2,u(t0) =u0.( )To solve the differential equation, we rewrite it in the separated formduu2=dt,and then integrate both sides: 1u=Zduu2=t+ 2022 Peter J.

6 OlverSolving the resulting algebraic equation foru, we deduce the solution formulau= 1t+k.( )To specify the integration constantk, we evaluateuat the initial timet0; this impliesu0= 1t0+k,so thatk= 1u0 , the solution to the initial value problem isu=u01 u0(t t0).( )Figure 1 shows the graphs of some typical the critical valuet =t0+ 1/u0from below, the solution blows up ,meaningu(t) ast t . The blow-up timet depends upon the initial data thelargeru0>0 is, the sooner the solution goes off to infinity. If the initial data is negative,u0<0, the solution is well-defined for allt > t0, but has a singularity in the past, att =t0+ 1/u0< t0. The only solution that exists for all positive and negativetime is theconstant solutionu(t) 0, corresponding to the initial conditionu0= general, the constantequilibrium solutionsto an autonomous ordinary differentialequation, also known as itsfixed points, play a distinguished role. Ifu(t) u is a constantsolution, thendu/dt 0, and hence the differential equation ( ) implies thatF(u ) = , the equilibrium solutions coincide with therootsof the functionF(u).

7 In pointof fact, since we divided byF(u), the derivation of our formula for the solution ( )assumed that we werenotat an equilibrium point. In the preceding example, our finalsolution formula ( ) happens to include the equilibriumsolutionu(t) 0, correspondingtou0= 0, but this is a lucky accident. Indeed, the equilibrium solution doesnotappear inthe general solution formula ( ). One must typically take extra care that equilibriumsolutions do not elude us when utilizing this basic integration a population of people, animals, or bacteria consists ofindividuals, the aggregate behavior can often be effectively modeled by a dynamical systemthat involves continuously varying variables. As first proposed by the English economistThomas Malthus in 1798, the population of a species grows, roughly, in proportion toits size. Thus, the number of individualsN(t) at timetsatisfies a first order differentialequation of the formdNdt= N,( )where the proportionality factor = measures the rate of growth, namely thedifference between the birth rate 0 and the death rate 0.

8 Thus, if births exceeddeaths, >0, and the population increases, whereas if <0, more individuals are dyingand the population the very simplest model, the growth rate is assumed to be independent of thepopulation size, and ( ) reduces to a simple linear ordinary differential equation whose1/7/224c 2022 Peter J. Olversolutions satisfy the Malthusian exponential growth lawN(t) =N0e t, whereN0=N(0)is the initial population size. Thus, if >0, the population grows without limit, while if <0, the population dies out, soN(t) 0 ast , at an exponentially fast rate. TheMalthusian population model provides a reasonably accurate description of the behaviorof an isolated population in an environment with unlimited a more realistic scenario, the growth rate will depend upon the size of the populationas well as external environmental factors. For example, in the presence of limited resources,relatively small populations will increase, whereas an excessively large population will haveinsufficient resources to survive, and so its growth rate willbe negative.

9 In other words,the growth rate (N)>0 whenN < N , while (N)<0 whenN > N , where thecarrying capacityN >0 depends upon the resource availability. The simplest class offunctions that satifies these two inequalities are of the form (N) = (N N), where >0 is a positive constant. This leads us to the Nonlinear population modeldNdt= N(N N).( )In deriving this model, we assumed that the environment is not changing over time; adynamical environment would require a more complicated non-autonomous analyzing the solutions to the Nonlinear populationmodel, let us make a pre-liminary change of variables, and setu(t) =N(t)/N , so thaturepresents the size ofthe population in proportion to thecarrying capacityN . A straightforward computationshows thatu(t) satisfies the so-calledlogistic differential equationdudt= u(1 u),u(0) =u0,( )where =N , and, for simplicity, we assign the initial time to bet0= 0. The logisticdifferential equation can be viewed as the continuous counterpart of the logistic map studiedin my Notes on Nonlinear Systems.

10 However, unlike its discrete namesake, the logisticdifferential equation is quite sedate, and its solutions easily , there are two equilibrium solutions:u(t) 0 andu(t) 1, obtained by settingthe right hand side of the equation equal to zero. The first represents a nonexistentpopulation with no individuals and hence no reproduction. The second equilibrium solutioncorresponds to a static populationN(t) N that is at the ideal size for the environment,so deaths exactly balance births. In all other situations, the population size will vary integrate the logistic differential equation, we proceedas above, first writing it inthe separated formduu(1 u)= both sides, and using partial fractions, t+k=Zduu(1 u)=Z 1u+11 u du= log u1 u ,1/7/225c 2022 Peter J. tou =u(1 u).wherekis a constant of integration. Thereforeu1 u=c e t,wherec= foru, we deduce the solutionu(t) =c e t1 +c e t.( )The constant of integration is fixed by the initial condition. Solving the algebraic equationu0=u(0) =c1 +cyieldsc=u01 the result back into the solution formula ( ) and simplifying, we findu(t) =u0e t1 u0+u0e t.