Appendix A

Systems of ordinary differential equations

Abstract

This appendix serves the purpose of a brief introduction to ordinary differential equations and systems thereof. A concise and nonexhaustive collection of methods and techniques used to find solutions to several types of differential equations, including first-order scalar equations, second order linear equations, and systems of linear equations, is included. In most cases, the reader is directed to more thorough and complete treatises, while this appendix offers a brief idea of the background material required for better understanding the mathematical models and methodologies employed in the analysis of malware diffusion in computer and communications networks, especially those based on epidemics models.

Derivatives; Ordinary differential equations; Linear systems; Initial value problems; Existence; Uniqueness; Epidemics

A.1. Initial Definitions

A differential equation is an equation that contains one or more derivatives of an unknown function. Differential equations are extensively used in mathematical modeling of many systems. When the independent variable is time, then the differential equation describes the temporal dynamics of a monolithic quantity, e.g. population, capital investment, chemical compound, decayed radioactive material, and so on. Other independent random variables such as the spatial coordinates can give more detailed description of a spatiotemporal quantity with spatiotemporal dynamics, e.g. diffusion of heat and fluids. In the rest of this appendix, we assume that the independent variable is time and denote it by timage. Any other interpretation of timage is valid too.
A differential equation that only contains ordinary derivatives is called an ODE. If, on the other hand, one or more partial derivatives are involved as well, it is referred to as a partial differential equations (PDEs) [81]. Each of these two broad categories is further classified based on the highest order of derivative that appears in them. In this appendix, we only consider ODEs.
Depending on the number of unknown functions that are involved, a classification to single ODE and system of ODEs is obtained. If there is a single function to be determined, then one equation is sufficient. However, if there are two or more unknown functions, then a system of equations is required.
Let ximage in a space Xmimage (for some proper dimension mimage of the space) be a function of a single-dimensional real-valued variable timage. Furthermore, for any positive integer iimage, let x(i)(τ)image represent the iimageth order derivative of ximage with respect to timage evaluated at τimage, i.e. x(i)(τ):=di(x(t))dtiimage evaluated at t=τimage. Then, we have the following.

Definition A.1

A differential equation that can be put in the form of

F(t,x(t),x(1)(t),,x(n1)(t),x(n)(t))=0

image (A.1)

for a given F:D×Xn+1Ximage constitutes an ODE of order nimage .
In the rest of this appendix, whenever not ambiguous, we will suppress the explicit dependence on timage for simplicity of notation. Note that the first and second order derivatives are also often denoted by ximage and ximage, and specially when the independent variable is time, also by ẋimage and ẍimage as well.
ODE (A.1) is said to be linear if Fimage is a linear function of the variables x(t),x(1)(t),,x(n1)(t),x(n)(t)image. A similar definition applies to PDEs. An equation not in linear form is called nonlinear ODE or PDE, respectively.
An important class of first-order equations are those in which the independent variable does not appear explicitly. Such equations are called autonomous and have the form x=f(x)image.

Definition A.2

A solution of ODE(A.1)over an interval tIimage is a function φ(t)imagethat identically satisfies(A.1)over the interval of Iimage . That is  (1)  the first nimage-derivatives of ximage exist for any point in the interval of Iimage ;  (2)  for every tIimage , Fimage is defined, i.e. (t,φ,φ,,φ(n)image is in the domain of Fimage ; and (3) F(t,φ,φ,,φn)=0,tIimage .
This definition describes an exact solution of the ODE, as opposed to an approximate one. When a solution of the ODE needs to satisfy the initial condition of φ(t0)=x0image for a given (t0,x0)Dimage, we refer to the problem as an initial value problem (IVP). Some very important questions about a solution of an ODE (PDE) or an IVP is the existence, uniqueness, and stability of solution or solutions.

A.2. First-order Differential Equations

By Eq. (A.1) and when Fimage is invertible with respect to x(n)(t)image, a first-order ODE may also be expressed in the following form:

x=f(t,x),

image (A.2)

for a function f:D×XXimage. For linear first-order ODEs, the integrating factor solution approach can be used to obtain the desired result. Details on the application of such techniques can be found in [40, Section 2.1].
In cases where ximage is a single-valued function of timage, this is just a single ODE. If, on the other hand, ximage is a multivalued function of timage, i.e. X=X1×X2×Xmimage for a positive integer mimage, then the ODE can be decomposed and expressed in terms of each element of the vector function ximage. For such a case, the fimage function is also multivalued, i.e. f=(f1,,fm)image such that each fi:D×XXiimage. This leads to the following system of mimage differential equations:

xi=fi(t,x1,,xm),i=1,,m

image (A.3)

that need to be simultaneously satisfied over the interval of solution. The converse direction yields another interpretation of the above system of first-order ODEs. According to this if there are mimage unknown functions x1,xmimage that need to simultaneously satisfy mimage first-order differential equations, then by introducing an mimage-dimensional vector-valued function one may obtain a single vectorized differential equation.
The form of first-order ODE presented in (A.2) can also model higher order ODEs through introduction of auxiliary dependent variables. Specifically, consider an nimage-th order ODE of the form x(n)=F0(t,x,x,,x(n1))image. Now consider the auxiliary functions y1,,ynimage such that y1=ximage, y2=ximage, y3=ximage, and so on till yn=x(n1)image. Then, we have y1=x=y2image, y2=x=y3,,yn1=x(n1)=ynimage. Moreover, yn=x(n)image, which must be equal to F0(t,x,x,,x(n1))image, hence yn=F0(t,x,x,,x(n1))=F0(t,y1,y2,,yn)image. Hence, the nimageth order ODE x(n)=F0(t,x,x,,x(n1))image is equivalent to the following system of ODEs:

yi=yi+1(i=1,,(n1)),

image

yn=F0(t,y1,y2,,yn).

image

In (A.3), if fiimage functions are linear functions in ximage, specifically,

xi=j=1maij(t)xj+bi(t)i=1,,m,

image (A.4)

where aijimage, biimage are real-valued continuous functions of timage over the interval Iimage, then we have a first-order system of linear ODEs. Using the technique of introducing auxiliary functions for higher order derivatives of the unknown function ximage, we can convert the nimage-th order linear ODE of x(n)=a1(t)x(n1)+a2(t)x(n2)++an(t)x+b(t)image into a first-order system of linear ODEs too [40]. The analysis and properties of (systems of) linear ODEs are well-developed, and consequently, whenever meaningful, nonlinear ODE systems are analyzed by approximating them through “linearization” (around an operation point) [40].
Some first-order ODEs of the form dxdn=f(x,n)image can be transformed as M(x,n)+N(x,n)dxdn=0image and cast in the form M(n)dn+N(x)dx=0image when Mimage is a function of nimage and Nimage a function of ximage. Such equations are called separable. The solution to such equation can be easily obtained. More on separable ODEs and their solution methodology can be found in [40].

A.3. Existence and Uniqueness of a Solution

Existence of a solution to any ODE of the form x=f(x,t)image with initial condition x(0)=x0image is guaranteed for any continuous finite dimensional function fimage. More formally, we have the following.

Theorem A.1

[Solution Existence[73, 2.4.4]] Let Ximage (the mimage space for the values of the unknown function ximage ) be finite dimensional, let Dimage be an open set in ×Ximage , let (t0,x0)Dimage and let f:DXimage be continuous (for tt0image ). Then there exists at least one solution φimage of the ODE x=f(t,x)image satisfying the initial condition of φ(t0)=x0image and reaching the boundary of Dimage on the right.
In (A.2), we say function f:DXimage satisfies a Kimage-Lipschitz condition in ximage over Dimage, where D×Ximage, if there exists a K>0image, such that

for all (t,x),(t,y)D,f(t,x)f(t,y)Kxy.

image

Note that in the above Lipschitz condition, the right hand side does not have any explicit appearance of timage, and the timage in the two terms of the left hand side are the same. For ODEs x=f(t,x)image, where fimage satisfies the Lipschitz condition, there is a result stronger than Theorem A.1, which guarantees not only existence but also uniqueness of the solution.

Theorem A.2

[Solution Existence and Uniqueness[73, 2.7.4]] Let Iimage be an interval in image and assume (t0,x0)I×Ximage . Consider f:I×XXimage to be a continuous function that is Kimage-Lipschitz in ximage over J×Ximage , where Jimage is any compact subinterval of Iimage (Kimage may depend on Jimage ). Then, the IVP of x=f(t,x)image and x(0)=x0image has one and only one solution over Iimage .
The continuity requirement of this theorem for fimage can be relaxed as long as discontinuities constitute a null-measure set [72]. A measure is a function that assigns a non-negative real number or image to (certain) subsets of a set Ximage[210,226]. It must assign zero to the empty set and be (countably) additive: the measure of a “large” subset that can be decomposed into a finite (or countable) number of “smaller” disjoint subsets is the sum of the measures of the “smaller” subsets. Thus, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size [210,226]. Any set of measure zero is called a null-set (or simply a null-measure set).

A.4. Linear Ordinary Differential Equations

In this section, we review some elements of the theory of systems of linear ODEs. These can be helpful in the understanding of the content of Chapters 3, 6 and 7, where systems of ODEs emerge.
Consider the first-order system of linear ODEs as in (A.4). Using a matrix representation, we can rewrite (A.4) in the form

x=A(t)x+b(t),

image

where ximage and bimage are mimage-dimensional vectors of continuous functions from IXimage, and for finite dimensional Ximage, A(t)image is the m×mimage matrix [aij(t)]image where aijimage are all real-valued and continuous. Consider the initial condition of x(0)=(x10,,xm0)image. The existence and uniqueness of the solution to this IVP follows from Theorem A.2. Since the functions are linear, they automatically satisfy the Lipschitz condition, with Kimage associated with the supremum/maximum of A(t)image’s norm over the compact interval mentioned in the statement of Theorem A.2.
When b(t)image is identically zero, the resulting linear ODE, x=A(t)ximage, is called homogeneous linear ODE . Then, x(t)=0image is the unique solution of the homogeneous linear ODE with the initial state of x0=0image. Clearly, addition of a solution of a linear ODE with any solution of its associated homogeneous linear ODE leads to a new solution of that linear ODE. Moreover, subtraction of two solutions of a linear ODE results in a solution of its associated homogeneous counterpart. The solution space of the homogeneous linear ODE forms a vector space. Let φ1,,φpimage be pimage solutions of the homogeneous linear ODE x=A(t)ximage. Then φ1(t),,φp(t)image are linearly independent at each tIimage if and only if they are linearly independent at some point t0Iimage. The proof of this key result can be found in [73, 2.8.2]. Therefore, if Ximage has finite dimension mimage and φ1,,φmimage are mimage linearly independent solutions of the homogeneous linear ODE, then every other solution of it can be written as a unique linear combination of φ1,,φmimage, which is known as the general solution of the homogeneous linear ODE. Hence, if νimage is a solution of the nonhomogeneous linear ODE (called a particular solution ), then every solution of the nonhomogeneous problem ψimage is the sum of the general solution of the associated homogeneous problem and a particular solution, i.e. ψ=j=1mαjφj+νimage for a unique set of scalars α1,,αmimage.
The solution of the IVP x=A(t)x+b(t)image, x(0)=x0image, can be expressed as the following:

x=Λ(t)(Λ(t0)1x0)+Λ(t)t0tΛ(τ)1b(τ)dτ,

image

where Λ(t)image is a fundamental kernel1 of the linear transformation associated with A(t)image. When Ximage has a finite dimension equal to mimage, Λ(t)image is a matrix whose columns are the coordinates of mimage linearly independent solutions of the homogeneous linear ODE x=A(t)ximage. Additionally, when the coefficients of the linear ODE are constant, i.e. we have an autonomous linear ODE , the basis-forming solutions of the homogeneous ODE x=Aximage are of the form eλltϕlp(t)image for l=1,,Limage and p=1,,Plimage for polynomials ϕlpimage of degree of at most Pl1image, where λ1,,λLimage are the distinct eigenvalues of matrix Aimage, and P1,,PLimage are their respective multiplicity.

A.5. Stability

Consider the autonomous ODE of x=f(x)image (where there is no explicit dependence of fimage on timage) with the solution of φ(x0,t)image for the initial value of x(0)=x0image. Then a point xˆXimage is an equilibrium (also known as a steady state, fixed point, or critical state) if the constant function ϕ(t)=xˆ,timage, is a solution of the ODE with the initial condition x(0)=xˆimage, i.e. the solution remains at xˆimage if it starts there as the initial state. We say an equilibrium xˆimage is stable if any solution of the IVP x=f(x)image, x(0)=x0image with x0image close to xˆimage remains in a small neighborhood of xˆimage for all t0image. It is further called asymptotically stable if it is stable and there is a small neighborhood of xˆimage such that if the initial value x0image is in that neighborhood we have limtφ(x0,t)=xˆimage.
The following result governs the stability of equilibrium points derived from f(x)=0image:

Theorem A.3

[[73]] Let xˆimage be an equilibrium of the ODE x=f(x)image . If for every eigenvalue ξimage of the Jacobian matrix of fimage at xˆimage we have Re(ξ)<0image , then xˆimage is asymptotic stable. If there exists an ξimage with Re(ξ)>0image , then xˆimage is unstable.
It should be noted that a finite dimensional space is assumed. In vector calculus, the Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function. Specifically, suppose f:nmimage is a function which takes as input the vector xnimage and produces as output the vector f(x)mimage. Then, the Jacobian matrix Jimage of fimage is an m×nimage matrix, usually defined and arranged as follows:

J=dfdx=[fx1fxn]=[f1x1f1xnfmx1fmxn].

image (A.5)

The Jacobian matrix is important because if the function fimage is differentiable at a point ximage (this is a slightly stronger condition than merely requiring that all partial derivatives exist there), then the Jacobian matrix defines a linear map nmimage, which is the best linear approximation of the function fimage near the point ximage. This linear map is thus the generalization of the usual notion of derivative and is called the derivative or the differential of fimage at ximage.

1 In linear algebra and functional analysis, the kernel (null-space) of a linear map L:VWimage between two vector spaces Vimage and Wimage, is the set of all elements vimage of Vimage for which L(v)=0image, where 0image denotes the zero vector in Wimage. That is, ker(L)={vVL(v)=0}image.

..................Content has been hidden....................

You can't read the all page of ebook, please click here login for view all page.
Reset
18.191.92.107