A BRIEF OVERVIEW OF NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS JOHN THOMAS Abstract. This paper discusses the basic techniques of solving linear ordinary differential equations, as well as some tricks for solving nonlinear systems of ODE's, most notably linearization of nonlinear systems. The paper proceeds to talk more thoroughly about the van der Pol system from Circuit Theory and the FitzHugh-Nagumo system from Neurodynamics, which can be seen as a generalization of the van der Pol system. Contents 1. General Solution to Autonomous Linear Systems of Differential Equations 1 2. Sinks, Sources, Saddles, and Spirals: Equilibria in Linear Systems 4 2.1. Real Eigenvalues 5 2.2. Complex Eigenvalues 5 3. Nonlinear Systems: Linearization 6 4. When Linearization Fails 8 5. The van der Pol Equation and Oscillating Systems 9 6. Hopf Bifurcations 12 7. Example: Neurodynamics 13 7.1. Ignoring I 13 7.2. Acknowledging I 14 7.3. Altering Parameters and Bifurcations 15 Acknowledgments 17 References 18 1. General Solution to Autonomous Linear Systems of Differential Equations Let us begin our foray into systems of differential equations by considering the simple 1-dimensional case (1.1) x0 = ax for some constant a. This equation can be solved by separating variables, yielding at (1.2) x = x0e Date: August 14, 2017. 1 2 JOHN THOMAS where x0 = x(0). Before proceeding to examine higher dimension linear, au- tonomous systems, it seems prudent to define "linear" and "autonomous" in this context. But first, a bit of notation. Notation 1.3. Let 0 x1 = f1(t; x1; x2; :::; xn) 0 x2 = f2(t; x1; x2; ::; xn) . 0 xn = fn(t; x1; x2; :::; xn) be a system of differential equations. I will write this as X0 = F (t; X) where x1 . X = . xn Unless otherwise specified, we will assume here that X 2 Rn and F (t; X): Rn+1 ! Rn. Definition 1.4. An n-dimensional system of differential equations X0 = F (t; X) is autonomous if F (t; X) in fact depends only on X. Thus in discussion of autonomous systems, we write X0 = F (X). Definition 1.5. An n-dimensional system of differential equations X0 = F (t; X) is linear if there exists A 2 Rn×n such that X0 = AX. That is, the system takes 0 0 1 x1 = a11x1 + ::: + a1nx1 B . C the form @ . A. 0 xn = an1x1 + ::: + annxn It is worth noting that any linear system of equations must also be autonomous. Let us now consider the very simple 2-dimensional system x0 = ax (1.6) y0 = by By repeating the 1-dimensional separation of variables and "ignoring" either x or at x0e 0 y, we can see that X(t) = and X(t) = bt are solutions to (1.6). In fact, 0 y0e at x0e we will show that X(t) = bt is also a solution to (1.6). y0e Theorem 1.7. Let X0 = AX be a linear system of differential equations with solutions X(t) and Y (t). Then, (X + Y )(t) is also a solution to the system. Proof. We know that (X + Y )0(t) = X0(t) + Y 0(t) and that X0(t) + Y 0(t) = AX + 0 AY = A(X + Y ) by linearity. Therefore (X + Y ) (t) = A(X + Y ) as required. at x0e Then, we have that bt is indeed a solution to (1.6). This solution is more y0e interesting than it may at first appear. To clarify, let us rewrite (1.6) as a 0 (1.8) X0 = X 0 b A BRIEF OVERVIEW OF NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS 3 and the aforementioned solution as 1 0 (1.9) X(t) = x eat + y ebt 0 0 0 1 Now we can clearly observe that, quite interestingly, each of the two terms is of the form ceλt(V ), where V is an eigenvector of A and λ its corresponding eigenvalue. Fortunately, this property is not unique to this system. Theorem 1.10. X(t) is a solution to the equation X0 = AX if and only if X(t) = V eλt for some eigenvector V of A, where λ is the corresponding eigenvalue to V . Proof. Let X(t) = V eλt. Then, X0 = λV eλt. Since V is an eigenvector of A to λ, X0 = AV eλt. Therefore X0 = AX as required. Conversely, if X(t) is a solution to X0 = AX, X(t) = Beαt for some B and α. Therefore, αBeαt = ABeαt. This implies that α is an eigenvalue of A with eigenvector B. Therefore, for any linear system of differential equations X0 = AX, all solutions will be of the form λ1t λ2t λkt (1.11) X(t) = α1V1e + α2V2e + ::: + αkVke where λi are the eigenvalues of A, Vi eigenvectors to λi, and αi constants. Two other concepts important to define now are the Poincar´emap and nullclines. Definition 1.12. Let X0 = F (t; X) be a system of differential equations. Suppose that for any initial condition X(0), we know X(1). Then we can define a function P such that P (X(0)) = X(1). We call this function a Poincar´emap. While the Poincar´emap may not be particularly useful for linear systems, since we can solve them explicitly, it is a very useful tool for modeling the behavior of messy nonlinear systems. Definition 1.13. Given the system x0 = f(x; y) y0 = g(x; y) the x-nullcline is the set of points such that f(x; y) = 0. The y-nullcline is defined similarly. Note that nullclines are not a construct used only in 2-dimensional systems. In higher dimensional systems, we will also have z-nullclines, etc. Before we continue, we should be sure that our efforts in solving differential equations is not in vain. We want to be sure that each system with given initial condition has a solution and that the solution is unique. Fortunately, there's a theorem for that. Theorem 1.14. Given the initial value problem 0 X = F (X);X(t0) = X0 n n n 1 for X0 2 R . Suppose that F : R ! R is C . Then, there exists a unique solution to this initial value problem. That is, there exists an a > 0 and a solution n X :(t0 − a; t0 + a) ! R of the differential equation such that X(t0) = X0. 4 JOHN THOMAS We will not delve into the totality of the proof of this theorem in this paper. Suffice to say, the proof relies on the technique of Picard iteration. The basic idea of this technique is to construct a sequence of functions which converges to the solution of the differential equation. The sequence of functions pk(t) is defined by p0(t) = x0, our initial condition, and Z t pk+1 = x0 + pk(s)ds 0 This technique is useful not only for proving this theorem, but also for approximat- ing solutions to difficult or impossible to solve equations. Let us also consider an example which highlights the importance of the C1 condition in theorem 1.14. Example 1.15. Consider the differential equation x0 = ln(x) 0 1 Since f = x is not continuous, this equation fails the condition of theorem 1.14. Next, consider the initial condition x(0) = −1. But then we have x0(0) = ln(−1), which is nonsense. Therefore, we have no solution with initial condition x(0) = −1. One last important result from theorem 1.14 is that solution curves to a differ- ential equation which satisfies the conditions of theorem 1.14 do not intersect. Another important concept is the ”flow" of an n-dimensional differential equa- tion. The flow is a function n φ ! R × R such that φ(t; X0) is the solution at time t with φ(0;X0) = X0. Then we have the theorem Theorem 1.16. Consider the system X0 = F (X) where F is C1. Then φ(t; X) is 1 @φ @φ C , i.e. @X and @t exist and are continuous. Again, in the interest of time, we will not delve into the prof of this theorem. @φ Worth noting is that we can calculate @t for any t provided we know the solution through X0. We have @φ (t; X ) = F (φ(t; X )) @t 0 0 We also have @φ (t; X ) = Dφ (X ) @X 0 t 0 where Dφt is the Jacobian of X ! φt(X) and φt(X) is φ(t; X) with constant t. @φ Note that @X requires knowledge not only of the solution through X0, but also through all nearby initial conditions. 2. Sinks, Sources, Saddles, and Spirals: Equilibria in Linear Systems Definition 2.1. An equilibrium point of the n-dimensional autonomous system of differential equations X0 = F (X) is a point Z 2 Rn such that X0 = 0 at X = Z. In particular, 0 is always an equilibrium point of a linear system. Let us now restrict our discussion to 2-dimensional linear systems X0 = AX. Specifically, let us look at the eigenvalues of A. A BRIEF OVERVIEW OF NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS 5 Theorem 2.2. Let X0 = AX be a 2-dimensional linear system. If det(A) 6= 0, then X0 = AX has a unique equilibrium point (0,0). Proof. An equilibrium point X = (x; y) of the system X0 = AX is a point that satisfies AX = 0. We know from linear algebra that this system has a nontrivial solution if and only if det(A) = 0. Therefore if det(A) 6= 0, the only solution to AX = 0 is (0; 0). 2.1. Real Eigenvalues. If we ignore for now the possibility that λi = 0 and that λ1 = λ2, then we are left with three cases: (1) 0 < λ1 < λ2 (2) λ1 < λ2 < 0 (3) λ1 < 0 < λ2 Let us first consider case (1): Example 2.3.