Lecture 7 3 Ordinary differential equations (ODEs) (continued) 6 Linear equations of second order 7 Systems of differential equations Please note Please note that there is no lecture on Tuesday, 15 November 2011. Additional tutorial The additional tutorial will start on Friday, 11 November 2011, 17.30-19.00, Infotech seminar room. 1/21
Linear equations of second order XIV We consider a damped harmonic oscillator driven by a force F(t). The differential equation is mẍ + dẋ + cx = F(t), where m > 0 is the mass and d > 0 is the damping. We divide the equation by m and set 2α = d m, ω2 0 = c m and f (t) = 1 mf(t). Then ẍ + 2αẋ + ω 2 0x = f (t). We first consider the homogeneous equation with f (t) 0 (no external force). The characteristic polynomial is λ 2 + 2αλ + ω0 2 with zeroes λ 1,2 = α ± α 2 ω0 2. There are three cases depending on the sign of α 2 ω 2 0. 2/21
Linear equations of second order XV 1st case: α 2 > ω0 2. In this case (strong damping) we have two real zeroes λ 1, λ 2, both of which are negative. The general solution is x(t) = c 1 e λ 1t + c 2 e λ 2t. The constants c 1 and c 2 can be determined from initial values for x(t) and ẋ(t). Since both λ 1, λ 2 are negative, we have x(t) 0 for large times t. There is no oscillation. 2nd case: α 2 = ω0 2. In this case (aperiodic limit case) we have one real zero λ 1,2 = α. The general solution of the ODE is x(t) = (c 1 + c 2 t)e αt. Again, there is no oscillation and x(t) 0 for large times t. 3/21
Linear equations of second order XVI 3rd case: α 2 < ω0 2. In this case (weak damping) we have two complex zeroes λ 1,2 = α ± iω 1, where ω 1 denotes the so-called eigenfrequency ω0 2 α2 > 0. The general solution is x(t) = e αt (c 1 cos ω 1 t + c 2 sinω 1 t). Again, we have x(t) 0 for large times t but this time there is an oscillation. We now consider the inhomogeneous equation with a harmonic driving force of frequency ω: ẍ + 2αẋ + ω 2 0x = B cos ωt. Since the solutions of the homogeneous equation all satisfy x h (t) 0 for large times t, in the end there will only remain the particular solution x p (t) of the inhomogeneous equation. 4/21
Linear equations of second order XVII We consider the equation with complexified right hand side ẍ + 2αẋ + ω 2 0x = Be iωt and use the ansatz x(t) = Ce iωt for the particular solution. This leads to C = The real part of x(t) is therefore x p (t) = B ω0 2 ω2 + 2iαω = B(ω2 0 ω2 2iαω) (ω0 2 ω2 ) 2 + 4α 2 ω 2. B (ω 2 0 ω2 ) 2 + 4ω 2 α 2((ω2 0 ω 2 )cos ωt + 2αω sinωt). Using trigonometric identities this can be written as x p (t) = B 2αω cos(ωt φ), with φ = arctan (ω 20 ω2 ) 2 + 4ω 2 α 2 ω0 2 ω2. 5/21
Linear equations of second order XVIII The factor A(ω) = 1 (ω0 2 ω2 ) 2 + 4ω 2 α 2 is called amplification factor. Note that A(ω) 0 for large ω. Hence driving forces with a very high frequency are of no effect. The amplification factor A(ω) has a maximum if the denominator has a minimum. This happens precisely for 4(ω 2 0 ω 2 )ω + 8ωα 2 = 0, hence for the frequency ω = ω r, where ω 2 r = ω 2 0 2α 2 = ω 2 1 α 2 and ω 1 is the eigenfrequency. The frequency ω r is called resonance frequency and exists only for damping constant α < 1 2 ω 0. 6/21
Systems of differential equations I It often happens in applications that several scalar functions x i (t) are coupled in the sense that the first derivative of one function depends on the variable t and on the value of all other functions: ẋ i = v i (t, x 1,...,x n ), 1 i n, where v i are certain functions of n + 1 variables. These n equations form a system of ordinary differential equations. It is useful to write this system in the following form: Let x denote the time dependent column vector (x 1,...,x n ) and v the column vector (v 1,...,v n ). Then we can write x = v(t, x). A solution of this system of ODEs is a map x : I R n, where I R is an interval, such that x(t) = v(t, x(t)). If we demand in addition that x(t 0 ) = x 0 we have an initial value problem. 7/21
Systems of differential equations II If we have a system of ODEs x = v(t, x), we can think of v as a time dependent vector field on a subset of R n. A solution x(t) is a curve in this subset of R n which is at each instance of time t tangential to this vector field. Example Consider the non-linear system of equations ẋ = 3y 3x ẏ = (1 + a 2 )x y xz ż = xy z, where a is a constant. It is a simplified model of atmospheric turbulence. Depending on the value of a this system admits so-called chaotic solutions related to the Lorenz attractor. 8/21
Systems of differential equations III Note that any explicit differential equation of order n for a scalar function x(t) x (n) = f (t, x, ẋ,...,x (n 1) ) can be transformed into an equivalent system of ODEs of first order ẋ 0 = x 1, ẋ 1 = x 2,. ẋ n 2 = x n 1, ẋ n 1 = f (t, x 0, x 1,...,x n 1 ). The scalar function x(t) is a solution of the n-th order explicit ODE if and only if x(t) = (x(t), ẋ(t),...,x (n 1) (t)) T is a solution of the system of ODEs. 9/21
Systems of differential equations IV There are certain fundamental theorems which guarantee the existence and uniqueness of solutions to initial value problems of first order ODEs or systems of ODEs. We will only formulate the general version for first order systems of ODEs. Theorem (Peano) Suppose that the time dependent vector field v : G R n, with G R n+1, is continuous. Then the initial value problem x = v(t, x), x(t 0 ) = x 0, with (t 0, x 0 ) G, has at least one solution. Therefore continuity of v guarantees the existence of a solution. However, it is not sufficient to guarantee the uniqueness. 10/21
Systems of differential equations V Example Consider the initial value problem ẋ = 2 x, x(0) = 0. This has the constant solution x 1 (t) 0. However, it also has the solution { t 2 if t 0 x 2 (t) = t 2 if t < 0. Hence the solution is not unique even though the function 2 x is continuous. 11/21
Systems of differential equations VI Let G R n+1 be some subset and v : G R n a vector field on G. Definition We say that v satisfies a Lipschitz condition on G if there exists a constant L 0 such that for all (t, x) and (t, x ) in G. v(t, x) v(t, x ) L x x, Here the norm of a vector is defined by x = x1 2 +... + x2 n. 12/21
Systems of differential equations VII We can now formulate a uniqueness theorem for initial value problems: Theorem (Picard-Lindelöf) Suppose that the vector field v(t, x) satisfies a Lipschitz condition on [a, b] R n. Then the initial value problem x = v(t, x), x(t 0 ) = x 0, for each x 0 R n has a unique solution defined on the interval I = [a, b]. 13/21
Systems of differential equations VIII We will only consider systems of the following form: x = A x + b(t), where A R n n is a constant matrix. These systems are called linear systems with constant coefficients. If b(t) 0, the system is called homogeneous. Theorem The space of solutions of the homogeneous system with A R n n is an n-dimensional vector space. Let x p (t) denote some particular solution to the inhomogeneous equation. Then any solution of this equation can be written as x(t) = x p (t) + x h (t), where x h (t) is a solution of the homogeneous equation. 14/21
Systems of differential equations IX To describe the general solution of a homogeneous linear system of equations we need the concept of the exponential function of a matrix. First, let p(x) = a 0 + a 1 x +... + a n x n denote some polynomial in the variable x. Then for an n n-matrix we define p(a) = a 0 I n + a 1 A +... + a n A n. Similarly we can define a power series in the square matrix A, where convergence means element-wise convergence. In particular we can define e A = k=0 1 k! Ak = I n + A + 1 2 A2 + 1 3! A3 +... and hence e ta = k=0 t k k! Ak = I n + ta + t2 2 A2 + t3 3! A3 +... 15/21
Systems of differential equations X The exponential function of a square matrix has the following properties: Exponential function of a square matrix Special case: e 0 = I n, where 0 denotes the n n zero-matrix. Functional equation: If AB = BA, then e A e B = e B e A = e A+B. Inverse matrix: The matrix e A is always invertible with inverse (e A ) 1 = e A (this follows from the first two properties). Derivative: d dt eta = Ae ta = e ta A. Note that the second property only holds in general if A and B commute. 16/21
Systems of differential equations XI Theorem Every solution of a homogeneous linear system of ODEs with constant coefficients x = A x is of the form x(t) = e ta c, defined for all t R, where c is any column vector in R n. Proof. Note that x(t) = e ta c is a solution of the differential equation, because x(t) = Ae ta c = A x(t). Let c 1,..., c n denote a basis of R n and x 1,..., x n the corresponding solutions. Then the solutions are linearly independent at each time t, hence the space spanned by these solutions is n-dimensional. Since the space of solutions is also n-dimensional, every solution is of this form for some vector c. 17/21
Systems of differential equations XII In general the exponential e ta is difficult to calculate if A is any given matrix. Note the following: Lemma If u is an eigenvector of A with eigenvalue λ, then e ta u = e tλ u. The exponential on the right hand side is simply the exponential of a number. Proof. We have e ta u = (I n + ta + t2 2 A2 + t3 3! A3 +...) u = u + tλ u + t2 2 λ2 u + t3 3! λ3 u... = e tλ u. 18/21
Systems of differential equations XIII The lemma implies: Theorem Suppose that the matrix A R n n has a basis of (complex) eigenvectors u 1,..., u n with (complex) eigenvalues λ 1,...,λ n. Then the time dependent vectors e tλ 1 u 1,...,e tλn u n form a basis of the solution space of the linear homogeneous system x = A x. Note that a basis of (real) eigenvectors exist in particular if the matrix A is symmetric. Theorem If λ is a non-real complex eigenvalue with complex eigenvector u C n, then both Re(e tλ u) and Im(e tλ u) are linearly independent real solutions of the system x = A x. 19/21