Differential Equations Brannan Copyright © 2010 by John Wiley & Sons, Inc. All rights reserved. Chapter 07: Nonlinear Differential Equations and Stability

Chapter 7 Nonlinear Differential Equations and Stability In this chapter, we take up the investigation of systems of nonlinear equations. Such systems can be solved by analytical methods only in rare instances. Numerical approximation methods provide one means of dealing with nonlinear systems. Another approach, presented in this chapter, is geometrical in character and leads to a qualitative understanding of the behavior of solutions rather than to detailed quantitative information.

Chapter 7 - Nonlinear Differential Equations and Stability 7.1 Autonomous Systems and Stability 7.2 Almost Linear Systems 7.3 Competing Species 7.4 Predator–Prey Equations 7.5 Periodic Solutions and Limit Cycles 7.6 Chaos and Strange Attractors: The Lorenz Equations

7.1 Autonomous Systems and Stability We first introduced two-dimensional systems of the form dx/dt = F(x, y), dy/dt = G(x, y) (1) in Section 3.6. Recall that the system (1) is called autonomous because the functions F and G do not depend on the independent variable t. In Chapter 3, we were mainly concerned with showing how to find the solutions of homogeneous linear systems, and we presented only a few examples of nonlinear systems. Now we want to focus on the analysis of two dimensional nonlinear systems of the form(1). Unfortunately, it is only in exceptional cases that solutions can be found by analytical methods. One alternative is to use numerical methods to approximate solutions. Software packages often include one or more algorithms, such as the Runge–Kutta method discussed in Section 3.7, for this purpose.

Nonlinear autonomous systems of equations We are concerned with systems of two simultaneous differential equations dx/dt = F(x, y), dy/dt = G(x, y) (1) where F and G are continuous and have continuous partial derivatives in some domain D of the xy-plane. If (x0, y0) is a point in this domain, then there exists a unique solution x = φ(t), y = ψ(t) of the system (1) satisfying the initial conditions x(t0) = x0, y(t0) = y0.

Stability and Instability. Consider the autonomous systems of the form x' = f(x). The points, if any, where f(x)=0 are called critical points of the autonomous system. A critical point x0 of the system is said to be stable if, given any ε > 0, there is a δ > 0 such that every solution x = φ(t) of the system, which at t = 0 satisfies ||φ(0) − x0|| < δ, exists for all positive t and satisfies ||φ(t) − x0|| < ε for all t ≥ 0. A critical point that is not stable is said to be unstable.

Asymptotically stable

The Oscillating Pendulum. For the Oscillating Pendulum, the angular momentum about the origin, mL2(dθ/dt), is the product of the mass m, the moment arm L, and the velocity Ldθ/dt. Thus the equation of motion is

The Oscillating Pendulum. By straightforward algebraic operations, we can write this eq. in the standard form d2θ/dt2+ γdθ/dt+ ω2 sin θ = 0, where γ = c/mL and ω2 = g/L. To convert this Eq. to a system of two first order equations, we let x = θ and y = dθ/dt; then dx/dt= y, dy/dt = −ω2 sin x − γ y. Since γ and ω2 are constants, the system is an autonomous system of the form (1). The critical points are found by solving the equations y = 0, −ω2 sin x − γ y = 0. We obtain y = 0 and x = ±nπ, where n is an integer.

The Importance of Critical Points. Critical points correspond to equilibrium solutions, that is, solutions in which x(t) and y(t) are constant. For such a solution, the system described by x and y is not changing; it remains in its initial state forever. It might seem reasonable to conclude that such points are not very interesting. However, recall that in Section 2.4 and later in Chapter 3, we found that the behavior of solutions in the neighborhood of a critical point has important implications for the behavior of solutions farther away.

Examples - Undamped Pendulum

Example

7.2 Almost Linear Systems TABLE 7.2.1

THEOREM 7.2.1 The critical point x=0 of the linear system x' = Ax. is asymptotically stable if the eigenvalues λ1, λ2 are real and negative or are complex with negative real part; stable, but not asymptotically stable, if λ1 and λ2 are pure imaginary; unstable if λ1 and λ2 are real and either is positive, or if they are complex with positive real part.

Effect of Small Perturbations. The eigenvalues λ1, λ2 are the roots of the polynomial equation det(A − λI) = 0. It is possible to show that small perturbations in some or all of the coefficients are reflected in small perturbations in the eigenvalues.

Linear Approximations to Nonlinear Systems. Let us consider what it means for a nonlinear system x‘=f(x) (3) to be “close” to a linear system (1). Accordingly, suppose that x' = Ax + g(x) (4) and that x = 0 is an isolated critical point of the system (4). This means that there is some circle about the origin within which there are no other critical points. In addition, we assume that det A = 0, so x = 0 is also an isolated critical point of the linear system x' = Ax. For the nonlinear system (4) to be close to the linear system x = Ax, we must assume that g(x) is small. More precisely, we assume that the components of g have continuous first partial derivatives and satisfy the limit condition ||g(x)||/||x||→0 as x → 0, that is, ||g|| is small in comparison to ||x|| itself near the origin. Such a system is called an almost linear system in the neighborhood of the critical point x = 0.

Examples 1. 2.

Jacobian matrix Let us now return to the general nonlinear system which we write in the scalar form x' = F(x, y), y' = G(x, y). (10) We assume that (x0, y0) is an isolated critical point of this system. The system (10) is almost linear in the neighborhood of (x0, y0) whenever the functions F and G have continuous partial derivatives up to order 2. To show this, we use Taylor expansions about the point (x0, y0) to write F(x, y) and G(x, y) in the form F(x, y) = F(x0, y0) + Fx (x0, y0)(x − x0) + Fy (x0, y0)( y − y0) + η1(x, y), G(x, y) = G(x0, y0) + Gx (x0, y0)(x − x0) + Gy (x0, y0)( y − y0) + η2(x, y), where {η1(x, y)/[(x − x0)2 + ( y − y0)2]1/2}→0 as (x, y)→(x0, y0), and similarly for η2. Note that F(x0, y0) = G(x0, y0) = 0; also dx/dt = d(x − x0)/dt and dy/dt = d( y − y0)/dt.

Jacobian matrix Then the system reduces to or where u1 = x − x0 and u2 = y − y0.

Jacobian matrix The matrix which appears as the coefficient matrix in above equation is called the Jacobian matrix of the functions F and G with respect to the variables x and y. We need to assume that det(J) is not zero at (x0, y0) so that this point is also an isolated critical point of the linear system, (13)

Example

THEOREM 7.2.2 Let λ1 and λ2 be the eigenvalues of the linear system (1), x' = Ax, corresponding to the almost linear system (4), x' = Ax + g(x). Assume that x = 0 is an isolated critical point of both of these systems. Then the type and stability of x = 0 for the linear system (1) and for the almost linear system (4) are as shown in Table 7.2.2.

Table 7.2.2

Damped Pendulum. Discuss the Damped Pendulum whose characteristic equation is λ2 + γλ + ω2 = 0, 3 cases 1. If γ2 − 4ω2 > 0, then the eigenvalues are real, unequal, and negative. The critical point (0, 0) is an asymptotically stable node of the linear system and of the almost linear system. 2. If γ2 − 4ω2 = 0, then the eigenvalues are real, equal, and negative. The critical point (0, 0) is an asymptotically stable (proper or improper) node of the linear system. It may be either an asymptotically stable node or spiral point of the almost linear system. 3. If γ2 − 4ω2 < 0, then the eigenvalues are complex with a negative real part. Insert images 7.2.3 and 7.2.4

7.3 Competing Species Let x and y be the populations of the two species at time t. Assume that the population of each of the species, in the presence of the other, is governed by a logistic equation. dx/dt= x(ε1 − σ1x− α1 y), dy/dt= y(ε2 − σ2 y− α2x), respectively, where ε1 and ε2 are the growth rates of the two populations, and ε1/σ1 and ε2/σ2 are their saturation levels and where α1 is a measure of the degree to which species y interferes with species x and α2 is a measure to which species x interferes with species y.

Example

Example (Ctd.) - A phase portrait of the system