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Formulas Trigonometric Identities sin(x+y) = sin x cos y + cos x sin y cos(x+y) = cos x cos y - sin x sin y sin x sin y = ½ [ cos(x-y) - cos(x+y) ] cos x cos y = ½ [ cos(x-y) + cos(x+y) ] sin x cos y = ½ [ sin(x+y) + sin(x-y) ] sin2x = ½ [ 1 - cos(2x) ] cos2x = ½ [ 1 + cos(2x) ] cos -sin R = sin cos = matrix for a counter-clockwise rotation of the plane through an angle . Electric Circuits Kirchoff's current law: At any junction, the sum of the currents going into the junction is equal to the sum of the currents going out of the junction, i.e. ins = outs. Kirchoff's voltage law: In any loop, the sum of the voltage changes is zero. In other words the sum of the voltage increases is equal to the sum of the voltage decreases, i.e. ups = downs. Resistors: Let VR be the voltage decrease from one end of a resistor to the other going in the direction of the current. We shall assume VR depends only on the current, i.e VR = f(i). A resistor is linear if VR is a linear function of the current, i.e. VR = iR,where R is a constant called the resistance of the resistor, this relationship is called Ohm's law. Inductors: Let VL be the voltage decrease from one end of a inductor to the other going di in the direction of the current. Usually one may assume VL = L dt, where L is a constant called the inductance of the inductor. Capacitors: Let q be the charge on plate of the capacitor that the current is flowing into dq (call this the top plate and the other plate the bottom plate). Then i = dt . Let VC be the voltage decrease from the top plate to the bottom plate. Usually q one may assume VC = C ,where C is a constant called the capacitance of the capacitor. Eigenvalues and Eigenvectors is an eigenvalue of A Au = u for some u 0 det(A - I) = 0. Let be the eigenvalues of A.1, 2, …, p X is an eigenvector of A corresponding to the eigenvalue AX = X (A - I)X = 0. Let X1, X2, …, Xp be eigenvectors of A corresponding to 1, 2, …, p. X is a generalized eigenvector of A corresponding to the eigenvalue (A - I)kX = 0 for some positive integer k. A = TDT -1 T = matrix whose columns are the eigenvectors of A, D = diagonal matrix with the eigenvalues of A on the diagonal. An = TDnT -1 Dn = diagonal matrix with the powers of the eigenvalues on the diagonal. etA = TetDT -1 etD = diagonal matrix whose diagonal entries are ejt. cos( A t) = T cos( D t) T -1 cos( D t) = diagonal matrix whose diagonal entries are cos( j t). Note: cos(i) = cosh() -1/2 -1/2 A sin( A t) = T D sin( D t) T -1 D-1/2 sin( D t) = diagonal matrix whose diagonal entries are sin( j t)/ j. Note: sin(i)/i = sinh() A = rTRT -1 A = 22 matrix with complex eigenvalues = i = r ( cos i sin ), T = matrix whose columns are the imaginary and real parts of an eigenvector corresponding to +, R = matrix for a rotation by an angle . An = rnTRnT -1 etA = etTRtT -1 A = TJT -1 An = TJ nT -1 etA = TetJT -1 A = 22 matrix with repeated eigenvalue and single eigenvector X (up to constant multiples), T = matrix whose columns are X and Y where (A - I)Y = X, 1 J = 0 . n nn-1 J n = 0 n . et tet etJ = . 0 et Difference Equations un+1 = Aun un = Anuo un = c11nX1 + … + cppnXp un = c1rn(cos(n)Y - sin(n)Z) + c2rn(cos(n)Z + sin(n)Y) if A is a 22 matrix with complex eigenvalue r( cos + i sin ) whose eigenvector is Y + iZ, n un = c1 X + c2(nY + nn-1X) if A is a 22 matrix with repeated eigenvalue and single eigenvector X and vector Y satisfying (A - I)Y = X Differential Equations du dt = Au u(t) = etAu(0) u(t) = c1e1tX1 + … + cpeptXp u(t) = c1et(cos(t)Y - sin(t)Z) + c2et(cos(t)Z + sin(t)Y) if A is a 22 matrix with complex eigenvalue i whose eigenvector is Y + iZ, u(t) = c1etX + c2et(Y + tX) if A is a 22 matrix with repeated eigenvalue and single eigenvector X and vector Y satisfying (A - I)Y = X du dt = Au + f(t) u(t) = up(t) + uh(t) du where up(t) is a solution to dt = Au + f(t) and uh(t) is the general du solution to dt = Au -tA up(t) = etA e f(t) dt Special case: If f(t) = etY then try up(t) = etX, plug into equation and solve for X. This works if is not an eigenvalue of A, but needs modification if is an eigenvalue of A. d2u dt2 = -Au du u(t) = cos( A t) u(0) + A-1/2 sin( A t) dt (0) Nonlinear Systems of Differential Equations dx dt = f(x,y) dy dt = g(x,y) Suppose f(x,y) and g(x,y) and their first partial derivatives are continuous in an open set D of the xy-plane. The set D will be referred to below. The phase plane is the plane of the dependent variables. In this case it is the xy-plane. If (x(t), y(t)) is a solution to the system then the corresponding trajectory is the directed curve in the phase plane defined by (x, y) = (x(t), y(t)) as t varies. dx The x-nullclines are the curves in the phase plane defined by dt = 0, i.e. f(x,y) = 0. dy The y-nullclines are the curves in the phase plane defined by dt = 0, i.e. g(x,y) = 0. (x*, y*) is an equilibrium point (x(t), y(t)) = (x*, y*) for all t is a solution to the system. f(x*, y*) = 0 and g(x*, y*) = 0 (x*, y*) is a point of intersection of an x-nullcline and y-nullcline. xf yf Linearization: Let (x*, y*) be an equilibrium point and A = g g where the partial x y derivatives are evaluated at (x*, y*). The phase portrait near (x*, y*) is similar to the phase du portrait of the linear system dt = Au near the origin provided all the eigenvalues of A have non-zero real part. Conservation Laws: A conservation law for the system is a function V(x, y) defined in D that is constant along trajectories, i.e. for every solution (x(t), y(t)) of the system there is a constant C such that (x(t), y(t)) = C for all t. This will occur if d V(x(t), y(t)) dt = 0 for any solution (x(t), y(t)). In this case the trajectories lie on the level curves of of V(x, y), i.e. the curves defined by V(x, y) = C for various values of C. To help draw these trajectories, here are some properties of these level curves. i. Suppose V(x, y) = p(x) + q(y) where q(y) decreases from to 0 as y increases from - to 0 and increases from 0 to as y increases from 0 to . Let y = q-1+(z) and y = q-1- (z) be the inverse functions of z = q(y). Then the level curve of V(x, y) = C can be made by taking the portion of the curve z = C – p(x) that lies above the x axis and applying y = q-1+(z) and y = q-1- (z). In the first case this gives a curve similar to the part of z = C – p(x) lying above the x axis and in the second it gives a curve similar to the reflection of the part of z = C - p(x) lying above the x axis across the x axis. xV2 xV y V V ii. Suppose x = 0 and y = 0 at a point (x*, y*). Let A = 2V 2V be the matrix of 2 2 x y y2 second derivatives evaluated at (x*, y*). If the eigenvalues of A are both strictly positive or both strictly negative, then, near (x*, y*), the level curves of V(x, y) are closed curves about (x*, y*). If one eigenvalue of A is strictly positive and the other is strictly negative, then, near (x*, y*), the level curves of V(x, y) have a saddle point structure. Positively Invariant Sets: A set P in the phase plane is positively invariant if P is contained in D and any trajectory that enters P remains in P thereafter. Liapunov Functions: A Liapunov function for the system is a function L(x, y) defined in D that is non-increasing along trajectories. This will occur if d L(x(t), y(t)) dt 0 for any solution (x(t), y(t)). If L(x, y) is a Liapunov function then the following is true. a. Liapunov’s Theorem (basic version): Suppose the following are true. i. L(x, y) is a Liapunov function for the system. ii. C is any number. iii. P is a connected component of the set { (x, y): (x, y) is in D and L(x, y) C }. iv. P is closed and bounded. v. The only trajectories in P on which L(x, y) is constant are equilibrium points. vi. There are only a finite number of equilibrium points in P. Then any trajectory that enters P approaches an equilibrium point as t . b. Liapunov’s Theorem (extended version): Suppose the following are true. i. L(x, y) is a Liapunov function for the system. ii. P is a posively invariant set. iii. P is closed and bounded. iv. The only trajectories in P on which L(x, y) is constant are equilibrium points. v. There are only a finite number of equilibrium points in P. Then any trajectory that enters P approaches an equilibrium point as t . c. If L(x, y) is a Liapunov function then any connected component of the set { (x, y): (x, y) is in D and L(x, y) C } is positively invariant. Periodic Solutions: A solution (x(t), y(t)) is periodic (with periond T) if (x(t), y(t)) = (x(t+T), y(t+T)) for all t. This occurs if and only if the trajectory corresponding to the solution is a closed curve. Poincare-Bendixson Theorem. Suppose a solution (x(t), y(t)) that remains in D and is bounded as t . Then one of the following are true. i. (x(t), y(t)) is an equilibrium solution. ii. (x(t), y(t)) approaches an equilibrium solution as t . iii. (x(t), y(t)) is a periodic solution. iv. (x(t), y(t)) approaches a periodic solution as t . v. (x(t), y(t)) approaches the boundary of D as t .