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  • Introduction to Ordinary Differential Equations

    Todd Kapitula

    Department of Mathematics and StatisticsUniversity of New Mexico

    September 28, 2006

    E-mail: kapitula@math.unm.edu

  • 1 Todd Kapitula

    Contents

    0. Introduction 20.1. Notation and introductory definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2. Solving linear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40.3. The phase plane for linear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

    0.3.1. Real eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50.3.2. Complex eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70.3.3. Classification of the critical point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

    0.4. The phase plane for conservative nonlinear systems . . . . . . . . . . . . . . . . . . . . . . . . 9

    1. Existence and uniqueness 111.1. Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

    1.1.1. Proof by successive approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.1.2. Proof by polygonal approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

    1.2. Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.3. Continuity with respect to initial data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.4. Extensibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.5. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

    2. Linear systems 192.1. General results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2. Equations with constant coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

    2.2.1. The fundamental matrix solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2.2. The Jordan canonical form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.2.3. Estimates on solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2.4. Linear perturbations: stable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.2.5. Linear perturbations: unstable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.2.6. Nonlinear perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

    2.3. Equations with periodic coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.3.1. Example: periodic forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.3.2. Example: the forced linear Schrodinger equation . . . . . . . . . . . . . . . . . . . . . 372.3.3. Example: linear Hamiltonian systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.3.4. Example: Hills equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

    3. The manifold theorems 42

    4. Stability analysis: the direct method 454.1. The -limit set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.2. Lyapunov functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

    4.2.1. Example: Hamiltonian systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

    5. Periodic Solutions 525.1. Nonexistence: Bendixsons criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.2. Existence: Poincare-Bendixson Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

    5.2.1. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.3. Index theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.4. Periodic vector fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

  • ODE Class Notes 2

    6. Applications of center manifold theory 636.1. Reduction to scalar systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

    6.1.1. Example: singular perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666.1.2. Example: hyperbolic conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

    6.2. Reduction to planar systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696.2.1. The Hopf bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696.2.2. The Takens-Bogdanov bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

    References 70

    0. Introduction

    Some physically and/or biologically interesting mathematical models are:

    (a) The Gross-Pitaevski equation,

    d2dt2

    + 3 = V (t), lim|t|

    |(t)| = 0,

    is a model used in the study of Bose-Einstein condensates (see [9, 10, 13] and the referencestherein). Here () represents the wavefunction of the condensate, and V () represents the appliedexternal potential. The boundary condition guarantees that the condensate is localized, i.e.,experimentally realizable.

    (b) Lotka-Volterra competition model:

    dN1dt

    = r1N1(1N1/K1) b1N1N2,dN2dt

    = r2N2(1N2/K2) b2N1N2.

    Here Ki represents the carrying capacity of the environment for species Ni in the absence ofcompetition, and bi reflects the competition between the two species.

    (c) Fireflys flashing rhythm:ddt

    = +A sin( ), ddt

    = .

    Here represents the phase of the fireflys rhythm, A is the fireflys ability to modify its frequency,and is the periodic stimulus.

    Different questions can be asked for each model. For example, when considering the Lotka-Volterra model,one can ask:

    (a) does one species wipe out the other?

    (b) if not, in which manner do the two species coexist - relatively constant populations, or populationswhich periodically fluctuate?

    The purpose of this course is to acquire and develop the mathematical tools that will allow you to begin toanalyze models such as the above. In the remainder of this section we will quickly review the material thatyou (should) have learned in your undergraduate course in Ordinary Differential Equations (e.g., see [1]), aswell as your introductory course in real analysis (e.g., see [2]).

    0.1. Notation and introductory definitions

    Definition 0.1. A norm | | : Rn 7 R satisfies

    (a) |x + y | |x |+ |y |

  • 3 Todd Kapitula

    (b) |cx | = |c| |x | for all c R

    (c) |x | 0, and equality occurs only if x = 0

    Definition 0.2. Let x = (x1, . . . , xn)T Rn. A norm is given by

    |x | :=ni=1

    |xi|.

    Let A Rnn. The norm of A is given by

    |A| := sup{|Ax | : |x | = 1} =n

    i,j=1

    |aij |.

    Remark 0.3. One has that:

    (a) More generally, a norm can be defined by

    |x |p := (ni=1

    |xi|p)1/p.

    It can be shown that each of these norms are equivalent, i.e., given a 1 p, q , one has thatthere are positive constants c1 and c2 such that

    c1|x |q |x |p c2|x |q.

    For example,|x |2 |x |1

    n|x |2.

    Thus, the choice of the norm is not important.

    (b) |Ax | |A| |x |.

    Definition 0.4. Given an x 0 Rn and > 0, define

    B(x 0, ) := {x Rn : |x x 0| < }, B(x 0, ) := {x Rn : |x x 0| }.

    Regarding calculus for vectors, we write:

    (a)

    x (t) dt = (x1(t) dt, . . . ,

    xn(t) dt)T

    (b) dx/dt = (dx1/dt, . . . ,dxn/dt)T

    Definition 0.5. Let G R Rn be open, and let f : G 7 Rn be continuously differentiable. The matrixDf := f /x Rnn satisfies

    (Df )ij =fixj

    .

    Definition 0.6. Let f : G 7 Rn be continuous. An ordinary differential equation (ODE) is of the form

    x = f (t,x ), :=ddt.

    The function x = (t) solves the ODE on an open interval I R if : I 7 Rn is continuously differentiablewith = f (t, ).

  • ODE Class Notes 4

    Remark 0.7. Consider the second-order scalar equation

    y + y y2 = sin t.

    Upon setting x1 := y, x2 := y, one gets the first-order system

    x1 = x2, x2 = x1 + x21 + sin t,

    i.e.,x = f (t,x ),

    where

    x :=(x1x2

    ), f (t,x ) :=

    (x2

    x1 + x21 + sin t

    ).

    This trick can be used to transform a scalar equation of order n to a first-order system with n equations.

    0.2. Solving linear systems

    Now let us refresh our memories as to how one can explicitly solve linear ODEs of the form

    x = Ax , (0.1)

    where A Rnn. Substitutingx := etv (0.2)

    into equation (0.1) yields(A 1)v = 0 .

    In the above the vector v is known as the eigenvector, and the corresponding eigenvalue is found by solvingthe characteristic equation

    det(A 1) = 0.

    If R, then the solution with real-valued components is given in equation (0.2). If C, i.e., = a+ ib,then the corresponding eigenvector is given by v = p+iq , where v , q Rn, and the two linearly independentsolutions with real-valued components are given by

    x 1 = eat (cos(bt)p sin(bt)q) , x 2 = eat (sin(bt)p + cos(bt)q) .

    If the eigenvalues are simple, then one can find n linearly independent solutions x 1, . . . ,xn via the mannerproscribed above, and the general solution is then given by

    x = c1x 1 + + cnxn,

    where cj R for j = 1, . . . , n.

    0.3. The phase plane for linear systems

    Now suppose that n = 2. The eigenvalues are zeros of the characteristic equation

    2 trace(A)+ det(A) = 0,

    i.e.,

    = :=12

    (trace(A)

    trace(A)2 4 det(A)

    ).

  • 5 Todd Kapitula

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