Hello everyone, Best, Josh
|
|
- Jasper Davis
- 5 years ago
- Views:
Transcription
1 Hello everyone, As promised, the chart mentioned in class about what kind of critical points you get with different types of eigenvalues are included on the following pages (The pages are an ecerpt from Edwards and Penny s Differential equations tet). The chart summarizes the situation for the case where (0, 0) is a critical point. For all of our cases, the system of differential equations is linear, with the eception of the last section. For the non-linear problem, you can linearize and get the equilibrium points and locally (in a small region around the critical point), the phase portrait will resemble the corresponding picture. Also included are some eamples of each kind so you can get an idea of what the homework is asking you to sketch for the phase portrait. On the last page, there are some theorems about stability of the critical points which we will need later, so included that page as well for your reference. hope this will clear up some of the material related to the phase portraits. Best, Josh 1
2 500 Chapter 7 Nonlinear Systems and Phenomena we plot trajectories of the system 2 FGURE Phase portrait for the system in Eq. () FGURE Phase portrait for the system corresponding to Eq. (5). d - = 2-2y, The result is shown in Fig s. dy 2 - = 2y - y. Plot similarly some solution curves for the following differential equations. dy - 5y - = d 2 + y dy - 5y = d 2 - y dy d - 2-5y dy 2y d - _ 2 y2 dy 2 + 2y = d y2 + 2y - y Now construct some eamples of your own. Homogeneous functions like those in Problems 1 through 5-rational functions with numerator and denominator of the same degree in and y-work well. The differential equation dy 25 + y(1-2 - y2)( _ 2 _ - d _ -25y + (1-2 - y2)( - 2 of this form generalizes Eample 6 in this section but would be inconvenient to solve eplicitly. ts phase portrait (Fig ) shows two periodic closed trajectoriesthe circles r = 1 and r = 2. Anyone want to try for three circles? y2) y2) () (5) We now discuss the behavior of solutions of the autonomous system d dy =!(, y), - = g(, y) near an isolated critical point (o, Yo) where!(o, Yo) = g(o, Yo) = O. A critical point is called isolated if some neighborhood of it contains no other critical point. We assume throughout that the functions! and g are continuously differentiable in a neighborhood of (o, Yo). We can assume without loss of generality that Xo = Yo = O. Otherwise, we make the substitutions u = - Xo, v = y - Yo. Then d/ = du/ and dy/ = dv/, so (1) is equivalent to the system du - =!(u + o, v + Yo) =!\ (u, v), dv - = g(u + o, v + Yo) = g\ (u, v) that has (0, 0) as an isolated critical point. (1) (2)
3 7. Linear and Almost Linear Systems 501 Ea m ple 1 The system d 2 - = - - y = ( - - y), dy = y + y 2 _ y = y(1 - + y) has (1, 2) as one of its critical points. We substitute u = -, v = y - 2; that is, = u +, y = v + 2. Then and - - y = - (u + 1) - (v + 2) = -u - v y = 1 - (u + 1) + (v + 2) = -u + v, so the system in () takes the form () (1, 2) for the system ' = - X2 y' - y, = y + y2 y _ FGURE The saddle point of Eample 1. du 2 - = (u + 1)(-u - v) = -u - v - u - uv, dv = (v + 2) ( -u + v) = -6u + 2v + v2 - uv and has (0, 0) as a critical point. f we can determine the trajectories of the system in () near (0, 0), then their translations under the rigid motion that carries (0, 0) to (1, 2) will be the trajectories near (1, 2) of the original system in (). This equivalence is illustrated by Fig (which shows computer-plotted trajectories of the system in () near the critical point (1, 2) in the y-plane) and Fig (which shows computer-plotted trajectories of the system in () near the critical point (0, 0) in the uv-plane). () Figures and 7..2 illustrate the fact that the solution curves of the ysystem in (1) are simply the images under the translation (u, v) (u + o, v + Yo) of the solution curves of the uv-system in (2). Near the two corresponding critical points-(o, Yo) in the y-plane and (0, 0) in the uv-plane-the two phase portraits therefore look precisely the same. FGURE The saddle point (0, 0) for the equivalent system u' = -u - v - u2 - uv, v' = -6u + 2v + v2 - uv. u Linearization Near a Critical Point Taylor's formula for functions of two variables implies that-if the function f (, y) is continuously differentiable near the fied point (o, Yo)-then f(o + u, Yo + v) = f(o, Yo) + f Co, Yo)u + fy (o, Yo)v + r(u, v) where the "remainder term" r(u, v) satisfies the condition r(u, v) lim (U, v)--+ (O,O) + v2 = o. (Note that this condition would not be satisfied if r(u, v) were a sum containing either constants or terms linear in u or v. n this sense, r(u, v) consists of the "nonlinear part" of the function f (o + u, Yo + v) of u and v.)
4 502 Chapter 7 Nonlinear Systems and Phenomena f we apply Taylor's formula to both f and g in (2) and assume that (o, Yo) is an isolated critical point so f(o, Yo) = g(o, Yo) = 0, the result is du = f (o, yo)u + f y (o, yo)v + r(u, v), dv = g (o, yo)u + g y(o, yo)v + s (u, v) where r (u, v) and the analogous remainder term s (u, v) for g satisfy the condition r(u, v) lim (U, v) -+ (O,O) + v2 s(u, v) lim = O. (u, v)-+ (O,O) + v2 Then, when the values u and v are small, the remainder terms r(u, v) and s(u, v) are very small (being small even in comparison with u and v). f we drop the presumably small nonlinear terms r(u, v) and s(u, v) in (5), the result is the linear system du = f Co, yo)u + f y(o, yo)v, (7) dv = g (o, yo)u + g y(o, yo)v whose constant coefficients (of the variables u and v) are the values f (o, Yo), fy(o, Yo) and g Co, Yo), gy (o, Yo) of the functions f and g at the critical point (o, Yo). Because (5) is equivalent to the original (and generally) nonlinear system u' = f(o + u, Yo + v), v' = g(o + u, Yo + v) in (2), the conditions in (6) suggest that the linearized system in (7) closely approimates the given nonlinear system when (u, v) is close to (0, 0). Assuming that (0, 0) is also an isolated critical point of the linear system, and that the remainder terms in (5) satisfy the condition in (6), the original system ' = f (, y), y' = g (, y) is said to be almost linear at the isolated critical r point (o, Yo). n this case, its linearization at (o, Yo) is the linear system in (7). n short, this linearization is the linear system u ' = Ju (where u = [ u v ) whose coefficient matri is the so-called Jacobian matri of the functions f and g, evaluated at the point (o, Yo). (5) (6) (8) Ea mple 1 Continued J(, y) = [ y - ] [ ] -y 1 + 2y - ' so J(1, 2) = -6 2 ' Hence the linearization of the system ' = y, y' = y + y 2 - y at its critical point ( 1, 2) is the linear system U = -u - v, v' = -6u + 2v that we get when we drop the nonlinear (quadratic) terms in ().
5 7. Linear and Almost Linear Systems 50 t turns out that in most (though not all) cases, the phase portrait of an almost linear system near an isolated critical point (o, Yo) strongly resemblesqualitatively-the phase portrait near the origin of its linearization. Consequently, the first step toward understanding general autonomous systems is to characterize the critical points of linear systems. Critical Points of Linear Systems We can use the eigenvalue-eigenvector method of Section 5. to investigate the critical point (0, 0) of a linear system (9) with constant-coefficient matri A. Recall that the eigenvalues A and A2 of A are the solutions of the characteristic equation det(a - A) = a - A b c d _ A = (a - A)(d - A) - bc = O. We assume that (0, 0) is an isolated critical point of the system in (9), so it follows that the coefficient determinant ad - bc of the system a + by = 0, c + dy = 0 is nonzero. This implies that A = 0 is not a solution of (9), and hence that both eigenvalues of the matri A are nonzero. The nature of the isolated critical point (0, 0) then depends on whether the two nonzero eigenvalues A and A2 of A are real and unequal with the same sign; real and unequal with opposite signs; real and equal; comple conjugates with nonzero real part; or pure imaginary numbers. u (u, v) _ - -1 V FGURE 7... The oblique u v-coordinate system determined by the eigenvectors V and V2. These five cases are discussed separately. n each case the critical point (0, 0) resembles one of those we saw in the eamples of Section 7.2-a node (proper or improper), a saddle point, a spiral point, or a center. UNEQUAL REAL EGENVALUES r WTH THE SAME SGN : n this case the matri A has linearly independent eigenvectors V and V2, and the general solution (t) = [ (t) yet) of (9) takes the form This solution is most simply described in the oblique uv-coordinate system indicated in Fig. 7.., in which the u- and v-aes are determined by the eigenvectors V and V2. Then the uv-coordinate functions u (t) and vet) of the moving point (t) are simply its distances from the origin measured in the directions parallel to the vectors V and V2, so it follows from Eq. (10) that a trajectory of the system is described by where Uo = u (O) and Vo = v(o). f Vo = 0, then this trajectory lies on the u ais, whereas if Uo = 0, then it lies on the v-ais. Otherwise-if Uo and Vo are (10) (1 1)
6 50 Chapter 7 Nonlinear Systems and Phenomena both nonzero-the parametric curve in (1 1) takes the eplicit form v = Cu k where k = A2/)"1 > O. These solution curves are tangent at (0, 0) to the u-ais if k > 1, to the v-ais if 0 < k < 1. Thus we have in this case an improper node as in Eample of Section 7.2. f A and A2 are both positive, then we see from (10) and (1 1) that these solution curves "depart from the origin" as t increases, so (0, 0) is a nodal source. But if Al and A2 are both negative, then these solution curves approach the origin as t increases, so (0, 0) is a nodal sink Ea mple 2 FGURE 7... The improper nodal source of Eample 2. 5 (a) The matri A = g [ 7-1 ] has eigenvalues Al = 1 and A2 = 2 with associated eigenvectors V = [ 1 f and V 2 = [ 1 f. Figure 7.. shows a direction field and typical trajectories of the corresponding linear system ' = A. Note that the two eigenvectors point in the directions of the linear trajectories. As is typical of an improper node, all other trajectories are tangent to one of the oblique aes through the origin. n this eample the two unequal real eigenvalues are both positive, so the critical point (0, 0) is an improper nodal source. (b) The matri B = -A = g [ -7 -] -17 has eigenvalues Al = - 1 and A2 = -2 with the same associated eigenvectors V = [ 1 f and V2 = [ 1 f. The new linear system ' = B has the same direction field and trajectories as in Fig. 7.. ecept with the direction field arrows now all reversed, so (0, 0) is now an improper nodal sink FGURE The saddle point of Eample. Ea m ple 5 UNEQUAL REAL EGENVALUES WTH OPPOSTE SGNS : Here the situation is the same as in the previous case, ecept that A2 < 0 < Al in (1 1). The trajectories with Uo = 0 or Vo = 0 lie on the u- and v-aes through the critical point (0, 0). Those with Uo and Vo both nonzero are curves of the eplicit form v = Cu k, where k = A2/A < O. As in the case k < 0 of Eample in Section 7.2, the nonlinear trajectories resemble hyperbolas, and the critical point (0, 0) is therefore an unstable saddle point. _.. The matri -] has eigenvalues Al = 1 and A2 = - 1 with associated eigenvectors V = [ 1 f and V 2 = [ 1 f. Figure 7..5 shows a direction field and typical trajectories of the corresponding linear system ' = A. Note that the two eigenvectors again point in the directions of the linear trajectories. Here k = - 1 and the nonlinear trajectories are (true) hyperbolas in the oblique uv-coordinate system, so we have the saddle point indicated in the figure. Note that the two eigenvectors point in the directions of the asymptotes to these hyperbolas. -5
7 Ea m ple 7. Linear and Almost Linear Systems 505 EQUAL REAL ROOTS : n this case, with A = A = A2 i= 0, the character of the critical point (0, 0) depends on whether or not the coefficient matri A has two linearly independent eigenvectors V and V2. f so, then we have oblique uvcoordinates as in Fig. 7.., and the trajectories are described by u (t) = uoe A t, vet) = voeat (12) as in (1 1). But now k = A2/A =, so the trajectories with Uo i= 0 are all of the form v = Cu and hence lie on straight lines through the origin. Therefore, (0, 0) is a proper node (or star) as illustrated in Fig. 7.2., and is a source if A > 0, a sink if A < O. f the multiple eigenvalue A i= 0 has only a single associated eigenvector V, then (as we saw in Section 5.6) there nevertheless eists a generalized eigenvector V2 such that (A - AJ)V2 = V, and the linear system ' = A has the two linearly independent solutions We can still use the two vectors V and V2 to introduce oblique uv-coordinates as in Fig Then it follows from (1) that the coordinate functions u(t) and vet) of the moving point (t) on a trajectory are given by (1) u (t) = (uo + vot)e A t, vet) = voe A t, (1) where Uo = u(o) and Vo = v(o). f Vo = 0 then this trajectory lies on the u-ais. Otherwise we have a nonlinear trajectory with dv dv/ AvOeAt AVO = = du du/ voeat + A(UO + vot)eat Vo + A(UO + vot) We see that dv/du 0 as t ±oo, so it follows that each trajectory is tangent to the u-ais. Therefore, (0, 0) is an improper node. f A < 0, then we see from (1) that this node is a sink, but it is a source if A > O _._-. _- _. _. _- --_.. _- - -._----_._.._.._. _. ---_. _ _......_.._ - _ _.. -_. The matri 9J A _.! [ - l '" FGURE The improper nodal sink of Eample. has the multiple eigenvalue A = - 1 with the single associated eigenvector V = [ 1 r. t happens that V2 = [ 1 r is a generalized eigenvector based on VJ, but only the actual eigenvector shows up in a phase portrait for the linear system ' = A. As indicated in Fig. 7..6, the eigenvector V determines the u-ais through the improper nodal sink (0, 0), this ais being tangent to each of the nonlinear trajectories. COMPLEX CONJUGATE EGENVALUES : Suppose that the matri A has eigenvalues A = P + qi and = p - qi (with p and q both nonzero) having associated comple conjugate eigenvectors V = a + bi and v = a - bi. Then we saw in Section 5.-see Eq. (22) there-that the linear system ' = A has the two independent real-valued solutions (t) = ept (a cos qt - b sin qt) and 2(t) = ept (b cos qt + a sin qt). (15)
8 506 Chapter 7 Nonlinear Systems and Phenomena Ea m ple 5 Thus the components (t) and yet) of any solution (t) = CX (t) +C2X2(t) oscillate between positive and negative values as t increases, so the critical point (0, 0) is a spiral point as in Eample 5 of Section 7.2. f the real part p of the eigenvalues is negative, then it is clear from (15) that (t) 0 as t +00, so the origin is a spiral sink. But if p is positive, then the critical point is a spiral source. The matri 1 [ - 10 A = - 15 ] has the comple conjugate eigenvalues A = - ± i with negative real part, so (0, 0) is a spiral sink. Figure 7..7 shows a direction field and a typical spiral trajectory approaching the origin as t +00. PURE MAGNARY EGENVALUES : f the matri A has conjugate imaginary eigenvalues A = qi and = -qi with associated comple conjugate eigenvectors v = a + bi and v = a - bi, then (15) with p = 0 gives the independent solutions Xl (t) = a cos qt - b sin qt and X2(t) = b cos qt + a sin qt (16) FGURE The spiral sink of Eample Ea mple 6 of the linear system ' = A. Just as in Eample of Section 7.2, it follows that any solution (t) = CX (t) + C2X2(t) describes an ellipse centered at the origin in the y-plane. Hence (0, 0) is a stable center in this case. The matri 1 [ -9 A = - 15 has the pure imaginary conjugate eigenvalues A = ±i, and therefore (0, 0) is a stable center. Figure 7..8 shows a direction field and typical elliptical trajectories enclosing the critical point. For the two-dimensional linear system ' = A with det A ==- 0, the table in Fig lists the type of critical point at (0, 0) found in the five cases discussed here, according to the nature of the eigenvalues A 1 and A2 of the coefficient matri A. Our discussion of the various cases shows that the stability of the critical point (0, 0) is determined by the signs of the real parts of these eigenvalues, as summarized in Theorem. Note that if Al and A2 are real, then they are themselves their real parts. ] FGURE The stable center of Eample 6. Real, unequal, same sign Real, unequal, opposite sign Real and equal Comple conjugate Pure imaginary mproper node Saddle point Proper or improper node Spiral point Center FGURE Classification of the critical point (0, 0) of the two-dimensional system ' = A.
9 7. Linear and Almost Linear Systems 507 TH EOREM 1 Stability of Linear Systems Let A 1 and A2 be the eigenvalues of the coefficient matri A of the twodimensional linear system with ad - d = a + by, dy = e + dy be : O. Then the critical point (0, 0) is 1. Asymptotically stable if the real parts of Al and A2 are both negative; 2. Stable but not asymptotically stable if the real parts of Al and A2 are both zero (so that A, A2 = ±qi);. Unstable if either A 1 or A2 has a positive real part. (17) 1-11 = r + si FGURE The effects of perturbation of pure imaginary roots. FGURE The effects of perturbation of real equal roots. t is worthwhile to consider the effect of small perturbations in the coefficients a, b, e, and d of the linear system in (17), which result in small perturbations of the eigenvalues A 1 and A2. f these perturbations are sufficiently small, then positive real parts (of A 1 and A2) remain positive and negative real parts remain negative. Hence an asymptotically stable critical point remains asymptotically stable and an unstable critical point remains unstable. Part 2 of Theorem is therefore the only case in which arbitrarily small perturbations can affect the stability of the critical point (0, 0). n this case pure imaginary roots At. A2 = ±qi of the characteristic equation can be changed to nearby comple roots Lt. L2 = r ± si, with r either positive or negative (see Fig ). Consequently, a small perturbation of the coefficients of the linear system in (7) can change a stable center to a spiral point that is either unstable or asymptotically stable. There is one other eceptional case in which the type, though not the stability, of the critical point (0, 0) can be altered by a small perturbation of its coefficients. This is the case with Al = A2, equal roots that (under a small perturbation of the coefficients) can split into two roots L 1 and L2, which are either comple conjugates or unequal real roots (see Fig ). n either case, the sign of the real parts of the roots is preserved, so the stability of the critical point is unaltered. ts nature may change, however; the table in Fig shows that a node with Al = A2 can either remain a node (if Ll and L2 are real) or change to a spiral point (if Ll and L2 are comple conjugates). Suppose that the linear system in (17) is used to model a physical situation. t is unlikely that the coefficients in (17) can be measured with total accuracy, so let the unknown precise linear model be d - = a * + b * y, dy - = e * + d * (17 * ) y. f the coefficients in (17) are sufficiently close to those in (17*), it then follows from the discussion in the preceding paragraph that the origin (0, 0) is an asymptotically stable critical point for (17) if it is an asymptotically stable critical point for (17*), and is an unstable critical point for (17) if it is an unstable critical point for (17*). Thus in this case the approimate model in (17) and the precise model in (17 * )
Chapter 6 Nonlinear Systems and Phenomena. Friday, November 2, 12
Chapter 6 Nonlinear Systems and Phenomena 6.1 Stability and the Phase Plane We now move to nonlinear systems Begin with the first-order system for x(t) d dt x = f(x,t), x(0) = x 0 In particular, consider
More informationDef. (a, b) is a critical point of the autonomous system. 1 Proper node (stable or unstable) 2 Improper node (stable or unstable)
Types of critical points Def. (a, b) is a critical point of the autonomous system Math 216 Differential Equations Kenneth Harris kaharri@umich.edu Department of Mathematics University of Michigan November
More informationDepartment of Mathematics IIT Guwahati
Stability of Linear Systems in R 2 Department of Mathematics IIT Guwahati A system of first order differential equations is called autonomous if the system can be written in the form dx 1 dt = g 1(x 1,
More informationAppendix: A Computer-Generated Portrait Gallery
Appendi: A Computer-Generated Portrait Galler There are a number of public-domain computer programs which produce phase portraits for 2 2 autonomous sstems. One has the option of displaing the trajectories
More informationStability of Dynamical systems
Stability of Dynamical systems Stability Isolated equilibria Classification of Isolated Equilibria Attractor and Repeller Almost linear systems Jacobian Matrix Stability Consider an autonomous system u
More informationMath 266: Phase Plane Portrait
Math 266: Phase Plane Portrait Long Jin Purdue, Spring 2018 Review: Phase line for an autonomous equation For a single autonomous equation y = f (y) we used a phase line to illustrate the equilibrium solutions
More informationNonlinear Autonomous Systems of Differential
Chapter 4 Nonlinear Autonomous Systems of Differential Equations 4.0 The Phase Plane: Linear Systems 4.0.1 Introduction Consider a system of the form x = A(x), (4.0.1) where A is independent of t. Such
More informationSection 9.3 Phase Plane Portraits (for Planar Systems)
Section 9.3 Phase Plane Portraits (for Planar Systems) Key Terms: Equilibrium point of planer system yꞌ = Ay o Equilibrium solution Exponential solutions o Half-line solutions Unstable solution Stable
More informationMath 312 Lecture Notes Linear Two-dimensional Systems of Differential Equations
Math 2 Lecture Notes Linear Two-dimensional Systems of Differential Equations Warren Weckesser Department of Mathematics Colgate University February 2005 In these notes, we consider the linear system of
More informationCopyright (c) 2006 Warren Weckesser
2.2. PLANAR LINEAR SYSTEMS 3 2.2. Planar Linear Systems We consider the linear system of two first order differential equations or equivalently, = ax + by (2.7) dy = cx + dy [ d x x = A x, where x =, and
More informationMath 3301 Homework Set Points ( ) ( ) I ll leave it to you to verify that the eigenvalues and eigenvectors for this matrix are, ( ) ( ) ( ) ( )
#7. ( pts) I ll leave it to you to verify that the eigenvalues and eigenvectors for this matrix are, λ 5 λ 7 t t ce The general solution is then : 5 7 c c c x( 0) c c 9 9 c+ c c t 5t 7 e + e A sketch of
More informationPhase portraits in two dimensions
Phase portraits in two dimensions 8.3, Spring, 999 It [ is convenient to represent the solutions to an autonomous system x = f( x) (where x x = ) by means of a phase portrait. The x, y plane is called
More informationMath 1270 Honors ODE I Fall, 2008 Class notes # 14. x 0 = F (x; y) y 0 = G (x; y) u 0 = au + bv = cu + dv
Math 1270 Honors ODE I Fall, 2008 Class notes # 1 We have learned how to study nonlinear systems x 0 = F (x; y) y 0 = G (x; y) (1) by linearizing around equilibrium points. If (x 0 ; y 0 ) is an equilibrium
More informationTHE SEPARATRIX FOR A SECOND ORDER ORDINARY DIFFERENTIAL EQUATION OR A 2 2 SYSTEM OF FIRST ORDER ODE WHICH ALLOWS A PHASE PLANE QUANTITATIVE ANALYSIS
THE SEPARATRIX FOR A SECOND ORDER ORDINARY DIFFERENTIAL EQUATION OR A SYSTEM OF FIRST ORDER ODE WHICH ALLOWS A PHASE PLANE QUANTITATIVE ANALYSIS Maria P. Skhosana and Stephan V. Joubert, Tshwane University
More informationCalculus and Differential Equations II
MATH 250 B Second order autonomous linear systems We are mostly interested with 2 2 first order autonomous systems of the form { x = a x + b y y = c x + d y where x and y are functions of t and a, b, c,
More informationHomogeneous Constant Matrix Systems, Part II
4 Homogeneous Constant Matri Systems, Part II Let us now epand our discussions begun in the previous chapter, and consider homogeneous constant matri systems whose matrices either have comple eigenvalues
More information7 Planar systems of linear ODE
7 Planar systems of linear ODE Here I restrict my attention to a very special class of autonomous ODE: linear ODE with constant coefficients This is arguably the only class of ODE for which explicit solution
More informationMath 312 Lecture Notes Linearization
Math 3 Lecture Notes Linearization Warren Weckesser Department of Mathematics Colgate University 3 March 005 These notes discuss linearization, in which a linear system is used to approximate the behavior
More informationLecture 38. Almost Linear Systems
Math 245 - Mathematics of Physics and Engineering I Lecture 38. Almost Linear Systems April 20, 2012 Konstantin Zuev (USC) Math 245, Lecture 38 April 20, 2012 1 / 11 Agenda Stability Properties of Linear
More informationMA 527 first midterm review problems Hopefully final version as of October 2nd
MA 57 first midterm review problems Hopefully final version as of October nd The first midterm will be on Wednesday, October 4th, from 8 to 9 pm, in MTHW 0. It will cover all the material from the classes
More informationMath 331 Homework Assignment Chapter 7 Page 1 of 9
Math Homework Assignment Chapter 7 Page of 9 Instructions: Please make sure to demonstrate every step in your calculations. Return your answers including this homework sheet back to the instructor as a
More information2.10 Saddles, Nodes, Foci and Centers
2.10 Saddles, Nodes, Foci and Centers In Section 1.5, a linear system (1 where x R 2 was said to have a saddle, node, focus or center at the origin if its phase portrait was linearly equivalent to one
More informationLinear Planar Systems Math 246, Spring 2009, Professor David Levermore We now consider linear systems of the form
Linear Planar Systems Math 246, Spring 2009, Professor David Levermore We now consider linear systems of the form d x x 1 = A, where A = dt y y a11 a 12 a 21 a 22 Here the entries of the coefficient matrix
More informationMath 308 Final Exam Practice Problems
Math 308 Final Exam Practice Problems This review should not be used as your sole source for preparation for the exam You should also re-work all examples given in lecture and all suggested homework problems
More informationMOL 410/510: Introduction to Biological Dynamics Fall 2012 Problem Set #4, Nonlinear Dynamical Systems (due 10/19/2012) 6 MUST DO Questions, 1
MOL 410/510: Introduction to Biological Dynamics Fall 2012 Problem Set #4, Nonlinear Dynamical Systems (due 10/19/2012) 6 MUST DO Questions, 1 OPTIONAL question 1. Below, several phase portraits are shown.
More informationMath Notes on sections 7.8,9.1, and 9.3. Derivation of a solution in the repeated roots case: 3 4 A = 1 1. x =e t : + e t w 2.
Math 7 Notes on sections 7.8,9., and 9.3. Derivation of a solution in the repeated roots case We consider the eample = A where 3 4 A = The onl eigenvalue is = ; and there is onl one linearl independent
More information154 Chapter 9 Hints, Answers, and Solutions The particular trajectories are highlighted in the phase portraits below.
54 Chapter 9 Hints, Answers, and Solutions 9. The Phase Plane 9.. 4. The particular trajectories are highlighted in the phase portraits below... 3. 4. 9..5. Shown below is one possibility with x(t) and
More informationODE, part 2. Dynamical systems, differential equations
ODE, part 2 Anna-Karin Tornberg Mathematical Models, Analysis and Simulation Fall semester, 2011 Dynamical systems, differential equations Consider a system of n first order equations du dt = f(u, t),
More informationUnderstand the existence and uniqueness theorems and what they tell you about solutions to initial value problems.
Review Outline To review for the final, look over the following outline and look at problems from the book and on the old exam s and exam reviews to find problems about each of the following topics.. Basics
More informationEven-Numbered Homework Solutions
-6 Even-Numbered Homework Solutions Suppose that the matric B has λ = + 5i as an eigenvalue with eigenvector Y 0 = solution to dy = BY Using Euler s formula, we can write the complex-valued solution Y
More informationHomogeneous Constant Matrix Systems, Part II
4 Homogeneous Constant Matrix Systems, Part II Let us now expand our discussions begun in the previous chapter, and consider homogeneous constant matrix systems whose matrices either have complex eigenvalues
More informationSTABILITY. Phase portraits and local stability
MAS271 Methods for differential equations Dr. R. Jain STABILITY Phase portraits and local stability We are interested in system of ordinary differential equations of the form ẋ = f(x, y), ẏ = g(x, y),
More informationENGI Linear Approximation (2) Page Linear Approximation to a System of Non-Linear ODEs (2)
ENGI 940 4.06 - Linear Approximation () Page 4. 4.06 Linear Approximation to a System of Non-Linear ODEs () From sections 4.0 and 4.0, the non-linear system dx dy = x = P( x, y), = y = Q( x, y) () with
More informationLocal Phase Portrait of Nonlinear Systems Near Equilibria
Local Phase Portrait of Nonlinear Sstems Near Equilibria [1] Consider 1 = 6 1 1 3 1, = 3 1. ( ) (a) Find all equilibrium solutions of the sstem ( ). (b) For each equilibrium point, give the linear approimating
More informationPart II Problems and Solutions
Problem 1: [Complex and repeated eigenvalues] (a) The population of long-tailed weasels and meadow voles on Nantucket Island has been studied by biologists They measure the populations relative to a baseline,
More informationFind the general solution of the system y = Ay, where
Math Homework # March, 9..3. Find the general solution of the system y = Ay, where 5 Answer: The matrix A has characteristic polynomial p(λ = λ + 7λ + = λ + 3(λ +. Hence the eigenvalues are λ = 3and λ
More informationProblem set 7 Math 207A, Fall 2011 Solutions
Problem set 7 Math 207A, Fall 2011 s 1. Classify the equilibrium (x, y) = (0, 0) of the system x t = x, y t = y + x 2. Is the equilibrium hyperbolic? Find an equation for the trajectories in (x, y)- phase
More informationMATH 215/255 Solutions to Additional Practice Problems April dy dt
. For the nonlinear system MATH 5/55 Solutions to Additional Practice Problems April 08 dx dt = x( x y, dy dt = y(.5 y x, x 0, y 0, (a Show that if x(0 > 0 and y(0 = 0, then the solution (x(t, y(t of the
More information520 Chapter 9. Nonlinear Differential Equations and Stability. dt =
5 Chapter 9. Nonlinear Differential Equations and Stabilit dt L dθ. g cos θ cos α Wh was the negative square root chosen in the last equation? (b) If T is the natural period of oscillation, derive the
More information20D - Homework Assignment 5
Brian Bowers TA for Hui Sun MATH D Homework Assignment 5 November 8, 3 D - Homework Assignment 5 First, I present the list of all matrix row operations. We use combinations of these steps to row reduce
More informationIntroduction to the Phase Plane
Introduction to the Phase Plane June, 06 The Phase Line A single first order differential equation of the form = f(y) () makes no mention of t in the function f. Such a differential equation is called
More informationYou may use a calculator, but you must show all your work in order to receive credit.
Math 2410-010/015 Exam II April 7 th, 2017 Name: Instructions: Key Answer each question to the best of your ability. All answers must be written clearly. Be sure to erase or cross out any work that you
More informationENGI 9420 Lecture Notes 4 - Stability Analysis Page Stability Analysis for Non-linear Ordinary Differential Equations
ENGI 940 Lecture Notes 4 - Stability Analysis Page 4.01 4. Stability Analysis for Non-linear Ordinary Differential Equations A pair of simultaneous first order homogeneous linear ordinary differential
More informationPROBLEMS In each of Problems 1 through 12:
6.5 Impulse Functions 33 which is the formal solution of the given problem. It is also possible to write y in the form 0, t < 5, y = 5 e (t 5/ sin 5 (t 5, t 5. ( The graph of Eq. ( is shown in Figure 6.5.3.
More informationAutonomous Systems and Stability
LECTURE 8 Autonomous Systems and Stability An autonomous system is a system of ordinary differential equations of the form 1 1 ( 1 ) 2 2 ( 1 ). ( 1 ) or, in vector notation, x 0 F (x) That is to say, an
More information4 Second-Order Systems
4 Second-Order Systems Second-order autonomous systems occupy an important place in the study of nonlinear systems because solution trajectories can be represented in the plane. This allows for easy visualization
More informationDynamical Systems Solutions to Exercises
Dynamical Systems Part 5-6 Dr G Bowtell Dynamical Systems Solutions to Exercises. Figure : Phase diagrams for i, ii and iii respectively. Only fixed point is at the origin since the equations are linear
More informationSolutions Chapter 9. u. (c) u(t) = 1 e t + c 2 e 3 t! c 1 e t 3c 2 e 3 t. (v) (a) u(t) = c 1 e t cos 3t + c 2 e t sin 3t. (b) du
Solutions hapter 9 dode 9 asic Solution Techniques 9 hoose one or more of the following differential equations, and then: (a) Solve the equation directly (b) Write down its phase plane equivalent, and
More information8.1 Bifurcations of Equilibria
1 81 Bifurcations of Equilibria Bifurcation theory studies qualitative changes in solutions as a parameter varies In general one could study the bifurcation theory of ODEs PDEs integro-differential equations
More information= F ( x; µ) (1) where x is a 2-dimensional vector, µ is a parameter, and F :
1 Bifurcations Richard Bertram Department of Mathematics and Programs in Neuroscience and Molecular Biophysics Florida State University Tallahassee, Florida 32306 A bifurcation is a qualitative change
More informationMATHEMATICS 317 December 2010 Final Exam Solutions
MATHEMATI 317 December 1 Final Eam olutions 1. Let r(t) = ( 3 cos t, 3 sin t, 4t ) be the position vector of a particle as a function of time t. (a) Find the velocity of the particle as a function of time
More informationMathQuest: Differential Equations
MathQuest: Differential Equations Geometry of Systems 1. The differential equation d Y dt = A Y has two straight line solutions corresponding to [ ] [ ] 1 1 eigenvectors v 1 = and v 2 2 = that are shown
More information+ i. cos(t) + 2 sin(t) + c 2.
MATH HOMEWORK #7 PART A SOLUTIONS Problem 7.6.. Consider the system x = 5 x. a Express the general solution of the given system of equations in terms of realvalued functions. b Draw a direction field,
More informationSolutions to Math 53 Math 53 Practice Final
Solutions to Math 5 Math 5 Practice Final 20 points Consider the initial value problem y t 4yt = te t with y 0 = and y0 = 0 a 8 points Find the Laplace transform of the solution of this IVP b 8 points
More information0 as an eigenvalue. degenerate
Math 1 Topics since the third exam Chapter 9: Non-linear Sstems of equations x1: Tpical Phase Portraits The structure of the solutions to a linear, constant coefficient, sstem of differential equations
More informationProblem set 6 Math 207A, Fall 2011 Solutions. 1. A two-dimensional gradient system has the form
Problem set 6 Math 207A, Fall 2011 s 1 A two-dimensional gradient sstem has the form x t = W (x,, x t = W (x, where W (x, is a given function (a If W is a quadratic function W (x, = 1 2 ax2 + bx + 1 2
More informationFIRST-ORDER SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS III: Autonomous Planar Systems David Levermore Department of Mathematics University of Maryland
FIRST-ORDER SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS III: Autonomous Planar Systems David Levermore Department of Mathematics University of Maryland 4 May 2012 Because the presentation of this material
More informationClassification of Phase Portraits at Equilibria for u (t) = f( u(t))
Classification of Phase Portraits at Equilibria for u t = f ut Transfer of Local Linearized Phase Portrait Transfer of Local Linearized Stability How to Classify Linear Equilibria Justification of the
More information10 Back to planar nonlinear systems
10 Back to planar nonlinear sstems 10.1 Near the equilibria Recall that I started talking about the Lotka Volterra model as a motivation to stud sstems of two first order autonomous equations of the form
More informationSection 5.4 (Systems of Linear Differential Equation); 9.5 Eigenvalues and Eigenvectors, cont d
Section 5.4 (Systems of Linear Differential Equation); 9.5 Eigenvalues and Eigenvectors, cont d July 6, 2009 Today s Session Today s Session A Summary of This Session: Today s Session A Summary of This
More informationChapter 4. Systems of ODEs. Phase Plane. Qualitative Methods
Chapter 4 Systems of ODEs. Phase Plane. Qualitative Methods Contents 4.0 Basics of Matrices and Vectors 4.1 Systems of ODEs as Models 4.2 Basic Theory of Systems of ODEs 4.3 Constant-Coefficient Systems.
More informationPhysics: spring-mass system, planet motion, pendulum. Biology: ecology problem, neural conduction, epidemics
Applications of nonlinear ODE systems: Physics: spring-mass system, planet motion, pendulum Chemistry: mixing problems, chemical reactions Biology: ecology problem, neural conduction, epidemics Economy:
More information1 The pendulum equation
Math 270 Honors ODE I Fall, 2008 Class notes # 5 A longer than usual homework assignment is at the end. The pendulum equation We now come to a particularly important example, the equation for an oscillating
More informationSolutions to Dynamical Systems 2010 exam. Each question is worth 25 marks.
Solutions to Dynamical Systems exam Each question is worth marks [Unseen] Consider the following st order differential equation: dy dt Xy yy 4 a Find and classify all the fixed points of Hence draw the
More informationNonlinear Autonomous Dynamical systems of two dimensions. Part A
Nonlinear Autonomous Dynamical systems of two dimensions Part A Nonlinear Autonomous Dynamical systems of two dimensions x f ( x, y), x(0) x vector field y g( xy, ), y(0) y F ( f, g) 0 0 f, g are continuous
More informationFirst Midterm Exam Name: Practice Problems September 19, x = ax + sin x.
Math 54 Treibergs First Midterm Exam Name: Practice Problems September 9, 24 Consider the family of differential equations for the parameter a: (a Sketch the phase line when a x ax + sin x (b Use the graphs
More informationSolutions for B8b (Nonlinear Systems) Fake Past Exam (TT 10)
Solutions for B8b (Nonlinear Systems) Fake Past Exam (TT 10) Mason A. Porter 15/05/2010 1 Question 1 i. (6 points) Define a saddle-node bifurcation and show that the first order system dx dt = r x e x
More informationDifferential Equations and Modeling
Differential Equations and Modeling Preliminary Lecture Notes Adolfo J. Rumbos c Draft date: March 22, 2018 March 22, 2018 2 Contents 1 Preface 5 2 Introduction to Modeling 7 2.1 Constructing Models.........................
More informationF and G have continuous second-order derivatives. Assume Equation (1.1) possesses an equilibrium point (x*,y*) so that
3.1.6. Characterizing solutions (continued) C4. Stability analysis for nonlinear dynamical systems In many economic problems, the equations describing the evolution of a system are non-linear. The behaviour
More information1 The relation between a second order linear ode and a system of two rst order linear odes
Math 1280 Spring, 2010 1 The relation between a second order linear ode and a system of two rst order linear odes In Chapter 3 of the text you learn to solve some second order linear ode's, such as x 00
More informationChapter 8 Equilibria in Nonlinear Systems
Chapter 8 Equilibria in Nonlinear Sstems Recall linearization for Nonlinear dnamical sstems in R n : X 0 = F (X) : if X 0 is an equilibrium, i.e., F (X 0 ) = 0; then its linearization is U 0 = AU; A =
More informationWe have two possible solutions (intersections of null-clines. dt = bv + muv = g(u, v). du = au nuv = f (u, v),
Let us apply the approach presented above to the analysis of population dynamics models. 9. Lotka-Volterra predator-prey model: phase plane analysis. Earlier we introduced the system of equations for prey
More informationIn these chapter 2A notes write vectors in boldface to reduce the ambiguity of the notation.
1 2 Linear Systems In these chapter 2A notes write vectors in boldface to reduce the ambiguity of the notation 21 Matrix ODEs Let and is a scalar A linear function satisfies Linear superposition ) Linear
More informationANSWERS Final Exam Math 250b, Section 2 (Professor J. M. Cushing), 15 May 2008 PART 1
ANSWERS Final Exam Math 50b, Section (Professor J. M. Cushing), 5 May 008 PART. (0 points) A bacterial population x grows exponentially according to the equation x 0 = rx, where r>0is the per unit rate
More informationExamples include: (a) the Lorenz system for climate and weather modeling (b) the Hodgkin-Huxley system for neuron modeling
1 Introduction Many natural processes can be viewed as dynamical systems, where the system is represented by a set of state variables and its evolution governed by a set of differential equations. Examples
More informationMath 20D: Form B Final Exam Dec.11 (3:00pm-5:50pm), Show all of your work. No credit will be given for unsupported answers.
Turn off and put away your cell phone. No electronic devices during the exam. No books or other assistance during the exam. Show all of your work. No credit will be given for unsupported answers. Write
More informationu t v t v t c a u t b a v t u t v t b a
Nonlinear Dynamical Systems In orer to iscuss nonlinear ynamical systems, we must first consier linear ynamical systems. Linear ynamical systems are just systems of linear equations like we have been stuying
More informationMath 232, Final Test, 20 March 2007
Math 232, Final Test, 20 March 2007 Name: Instructions. Do any five of the first six questions, and any five of the last six questions. Please do your best, and show all appropriate details in your solutions.
More informationMath 5490 November 5, 2014
Math 549 November 5, 214 Topics in Applied Mathematics: Introduction to the Mathematics of Climate Mondays and Wednesdays 2:3 3:45 http://www.math.umn.edu/~mcgehee/teaching/math549-214-2fall/ Streaming
More informationMCE693/793: Analysis and Control of Nonlinear Systems
MCE693/793: Analysis and Control of Nonlinear Systems Systems of Differential Equations Phase Plane Analysis Hanz Richter Mechanical Engineering Department Cleveland State University Systems of Nonlinear
More informationMidterm 2. MAT17C, Spring 2014, Walcott
Mierm 2 MAT7C, Spring 24, Walcott Note: There are SEVEN questions. Points for each question are shown in a table below and also in parentheses just after each question (and part). Be sure to budget our
More informationSample Solutions of Assignment 9 for MAT3270B
Sample Solutions of Assignment 9 for MAT370B. For the following ODEs, find the eigenvalues and eigenvectors, and classify the critical point 0,0 type and determine whether it is stable, asymptotically
More informationLesson #33 Solving Incomplete Quadratics
Lesson # Solving Incomplete Quadratics A.A.4 Know and apply the technique of completing the square ~ 1 ~ We can also set up any quadratic to solve it in this way by completing the square, the technique
More informationPolynomial and Synthetic Division
Chapter Polynomial and Rational Functions y. f. f Common function: y Horizontal shift of three units to the left, vertical shrink Transformation: Vertical each y-value is multiplied stretch each y-value
More informationLinearization of Differential Equation Models
Linearization of Differential Equation Models 1 Motivation We cannot solve most nonlinear models, so we often instead try to get an overall feel for the way the model behaves: we sometimes talk about looking
More informationENGI Duffing s Equation Page 4.65
ENGI 940 4. - Duffing s Equation Page 4.65 4. Duffing s Equation Among the simplest models of damped non-linear forced oscillations of a mechanical or electrical system with a cubic stiffness term is Duffing
More informationAutonomous systems. Ordinary differential equations which do not contain the independent variable explicitly are said to be autonomous.
Autonomous equations Autonomous systems Ordinary differential equations which do not contain the independent variable explicitly are said to be autonomous. i f i(x 1, x 2,..., x n ) for i 1,..., n As you
More informationNonlinear dynamics & chaos BECS
Nonlinear dynamics & chaos BECS-114.7151 Phase portraits Focus: nonlinear systems in two dimensions General form of a vector field on the phase plane: Vector notation: Phase portraits Solution x(t) describes
More informationLecture Notes for Math 251: ODE and PDE. Lecture 27: 7.8 Repeated Eigenvalues
Lecture Notes for Math 25: ODE and PDE. Lecture 27: 7.8 Repeated Eigenvalues Shawn D. Ryan Spring 22 Repeated Eigenvalues Last Time: We studied phase portraits and systems of differential equations with
More informationGiven the vectors u, v, w and real numbers α, β, γ. Calculate vector a, which is equal to the linear combination α u + β v + γ w.
Selected problems from the tetbook J. Neustupa, S. Kračmar: Sbírka příkladů z Matematiky I Problems in Mathematics I I. LINEAR ALGEBRA I.. Vectors, vector spaces Given the vectors u, v, w and real numbers
More informationThird In-Class Exam Solutions Math 246, Professor David Levermore Thursday, 3 December 2009 (1) [6] Given that 2 is an eigenvalue of the matrix
Third In-Class Exam Solutions Math 26, Professor David Levermore Thursday, December 2009 ) [6] Given that 2 is an eigenvalue of the matrix A 2, 0 find all the eigenvectors of A associated with 2. Solution.
More information3.3.1 Linear functions yet again and dot product In 2D, a homogenous linear scalar function takes the general form:
3.3 Gradient Vector and Jacobian Matri 3 3.3 Gradient Vector and Jacobian Matri Overview: Differentiable functions have a local linear approimation. Near a given point, local changes are determined by
More informationAPPM 2360: Final Exam 10:30am 1:00pm, May 6, 2015.
APPM 23: Final Exam :3am :pm, May, 25. ON THE FRONT OF YOUR BLUEBOOK write: ) your name, 2) your student ID number, 3) lecture section, 4) your instructor s name, and 5) a grading table for eight questions.
More informationDIFFERENTIATION. 3.1 Approximate Value and Error (page 151)
CHAPTER APPLICATIONS OF DIFFERENTIATION.1 Approimate Value and Error (page 151) f '( lim 0 f ( f ( f ( f ( f '( or f ( f ( f '( f ( f ( f '( (.) f ( f '( (.) where f ( f ( f ( Eample.1 (page 15): Find
More informationKinematics of fluid motion
Chapter 4 Kinematics of fluid motion 4.1 Elementary flow patterns Recall the discussion of flow patterns in Chapter 1. The equations for particle paths in a three-dimensional, steady fluid flow are dx
More informationENGI 9420 Lecture Notes 4 - Stability Analysis Page Stability Analysis for Non-linear Ordinary Differential Equations
ENGI 940 Lecture Notes 4 - Stability Analysis Page 4.0 4. Stability Analysis for Non-linear Ordinary Differential Equations A pair of simultaneous first order homogeneous linear ordinary differential equations
More informationMath 215/255 Final Exam (Dec 2005)
Exam (Dec 2005) Last Student #: First name: Signature: Circle your section #: Burggraf=0, Peterson=02, Khadra=03, Burghelea=04, Li=05 I have read and understood the instructions below: Please sign: Instructions:.
More informationIdentifying second degree equations
Chapter 7 Identifing second degree equations 71 The eigenvalue method In this section we appl eigenvalue methods to determine the geometrical nature of the second degree equation a 2 + 2h + b 2 + 2g +
More informationSystems of Linear ODEs
P a g e 1 Systems of Linear ODEs Systems of ordinary differential equations can be solved in much the same way as discrete dynamical systems if the differential equations are linear. We will focus here
More informationA plane autonomous system is a pair of simultaneous first-order differential equations,
Chapter 11 Phase-Plane Techniques 11.1 Plane Autonomous Systems A plane autonomous system is a pair of simultaneous first-order differential equations, ẋ = f(x, y), ẏ = g(x, y). This system has an equilibrium
More information