MATHEMATICA TUTORIAL, Part 2.3: Non-linear Systems of ODEs


This chapter is concerned with initial value problems for systems of ordinary differential equations. The emphasis will be given to nonlinear phenomena and properties, particularly those with physical relevance. Finding a solution to a differential equation may not be so important if that solution never appears in the physical model represented by the system, or is only realized in exceptional circumstances. Unless stated opposite, all quantities are assumed to be real.

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The EquationTrekker package is a great package for plotting and exploring phase space

Nonlinear Systems of Equations

This chapter is devoted to qualitative methods of nonlinear systems of ordinary differential equations (ODEs for short). Interest in nonlinear ODEs is virtually as old as the subject of differential equations itself, which dates back to Newton, Leibniz and Bernoulli brothers. However, little progress was made until the end of nineteen century when two great mathematicians Alexander Lyapunov and Henry Poincaré took up a systematic study of the subject.

Next step in developing the theory of nonlinear oscillations was made in 1920--1930s by the group of Russian scientists leaded by Alexander Andronov. The progress in qualitative methods was summarized in the 1947 book by N. Minorsky. With that and other books, such as Andronov A. and Chaikin C., Theory of Oscillations (Palala Press, 2018), Stoker, J.J, Nonlinear Vibrations in Mechanical and Electrical Systems (Wiley, 1992), the subject appeared in university curricula by the end of the 1950's. With availability of digital computers, the next break through was made by Edward Lorenz and Stephen Smale with discovering chaotic behavior of solutions to differential equations.

In this chapter, we will consider mostly such systems of ordinary equations when the derivative vector can be isolated. A system of first order ordinary differential equations is said to be in normal form if it can be written as

\begin{equation} \label{EqPart3.1} \begin{cases} \dot{x}_1 \equiv {\text d}x_1 /{\text d}t = f_1 (x_1 , x_2 , \ldots , x_n , t) , \\ \dot{x}_2 \equiv {\text d}x_2 /{\text d}t = f_2 (x_1 , x_2 , \ldots , x_n , t) , \\ \quad \vdots \quad \vdots \\ \dot{x}_n \equiv {\text d}x_n /{\text d}t = f_n (x_1 , x_2 , \ldots , x_n , t) , \end{cases} \end{equation}
where superdot stands for the derivative with respect to time variable t. The above system can be presented in compact vector form
\begin{equation} \label{EqPart3.2} \frac{{\text d} {\bf x}}{{\text d}t} = {\bf f}({\bf x} , t ) \qquad\mbox{or} \qquad \dot{\bf x} = {\bf f}({\bf x} , t ) , \end{equation}
\[ {\bf x} (t) = \begin{bmatrix} x_1 (t) \\ x_2 (t) \\ \vdots \\ x_n (t) \end{bmatrix} , \qquad {\bf f} (x_1 , x_2 , \ldots , x_n , t) = \begin{bmatrix} f_1 (x_1 , x_2 , \ldots , x_n ,t) \\ f_2 (x_1 , x_2 , \ldots , x_2 , t) \\ \vdots \\ f_n (x_1 , x_2 , \ldots , x_n , t) \end{bmatrix} \]
are n-column vectors. In engineering and physics, it is a custom to denote a derivative with respect to time variable t by dot: \( \dot{\bf x} = {\text d}{\bf x} / {\text d} t. \)

If an initial position of the vector \( {\bf x} (t) \) is known, we get an initial value problem:

\begin{equation} \label{EqPart3.3} \frac{{\text d} {\bf x}}{{\text d}t} = {\bf f}({\bf x} , t ) , \qquad {\bf x} (t_0 ) = {\bf x}_0 , \end{equation}
where \( {\bf x}_0 \) is a given column vector that fixes the initial position of the desired solution at point t0. In general, the initial conditions serve to uniquely specify a solution to the differential equations — at least for nearby times. To emphasize the dependence of a solution on the initial value x0, we will denote the corresponding solution by x = ϕ(t, x0). This function ϕ : ℝn × ℝ → ℝ is called the flow associated to the differentialequation \eqref{EqPart3.2}. If we hold the variablex0 fixed, then the function
\[ t \mapsto \phi (t, x_0 ) \]
is just an alternative expression for the solution of the initial value problem \eqref{EqPart3.3}.

An autonomous differential equation is a system of ordinary differential equations which does not explicitly depend on the independent variable ( \( t \) in our case). It is of the form

\begin{equation} \label{EqPart3.4} \dot{\bf x} = {\bf f} ( {\bf x} ) . \end{equation}
Here dot stands for the derivative with respect to time variable t and \( {\bf f} ({\bf x}) \) takes values in n-dimensional Euclidean space and t is usually time.

Example: . In 2001, Edwards and Buckmire published a paper with an autonomous differential equation model of how movies make money. The corresponding model of a motion picture released in North America is represented by an initial value problem (IVP):

\[ \begin{split} \frac{{\text d}G}{{\text d}t} = S\,A , \qquad G(0) =0 , \\ \frac{{\text d}S}{{\text d}t} = A - S , \qquad S(0) = S_0 , \\ \frac{{\text d}A}{{\text d}t} = - \alpha \left( \frac{S}{S+\gamma} + \beta\,G \right) A , \qquad A(0) = A_0 . \end{split} \]
Here G(t) is the total gross of a movie at time t, A(t) is the amount of money earned per screen per week, and S(t) is the number of screens on which the film is being presented. The product S A is just the revenue of the film, so initially, the total gross is zero.   ■


Since Andronov (1932), traditionally three different approaches are used for the study of dynamical systems: qualitative methods, analytical methods, and numerical methods. To define the “strategy” of qualitative methods one has to note that the solutions of equations of non linear dynamic systems are in general non classical transcendental functions of the calculus, which are very complex. This “strategy” is of the same type as the one used for the characterization of a function of the complex variable by its singularities: zeros, poles, essential singularities. Here, the complex transcendental functions are defined by the singularities of continuous (or discrete) dynamic systems such as:

The qualitative methods consider the nature of these singularities in the phase (or state) space, and their evolutions when parameters of the system vary, or in the presence of a continuous structure modification of the system (study of the bifurcation sets in the parameter space, or in a functional space).

In fact at the beginning qualitative methods developed from the fundamental studies of circuits of radio-engineering. Indeed in 1927, Andronov, the most famous Mandelstham’s student, defends his thesis with the topic formulated by Mandelstham “The Poincaré’s limit cycles and the theory of oscillations”. This thesis is a first-rank contribution for the evolution of the theory of non linear oscillations, because it opens a new way of applications for the Poincaré’s qualitative theory of differential equations, with a lot of practical consequences. With this work, Andronov was the first to see that the phenomena of free (or self) oscillations, for example that generated by the Van der Pol oscillator, correspond to limit cycles. It is from the study of oscillators that afterwards, Andronov amplifies his activity with a precise purpose: the development of a theory of non linear oscillations, in order to make use of mathematical tools, common to different scientific disciplines.

Andronov and Pontrjagin formulated in 1937 the necessary and sufficient conditions of structural stability for autonomous two-dimensional systems. These conditions are: the system has only a finite number of equilibrium points and limit cycles, which are not in a critical case in the Lyapunov’s sense; no separatrix joins the same, or two distinct saddle points. In this case it is possible to define, in the parameter space of the system, a set of cells inside of which the same qualitative behavior is preserved.

The knowledge of such cells is of first importance for the analysis, and the synthesis of dynamic systems in physics, or engineering. On the boundary of a cell, the dynamic system is structurally unstable, and for autonomous two-dimensional systems (two-dimensional vector fields), structurally stable systems are dense in the functional space. Until 1966, the conjecture of the extension of this result for higher dimensional systems was generally conjectured to be true.

Sufficient conditions of structural stability were formulated by Stephen Smale in 1963. Such systems are now known as Morse--Smale systems.


Poincaré map

A Poincaré map is the intersection of a periodic orbit in the state space of a continuous dynamical system with a certain lower-dimensional subspace, called the Poincaré section, transversal to the flow of the system. Upon integrating Eq.\eqref{EqPart3.3}, we reduce the given initial value problem to the integral equation
\[ \phi(t, {\bf x}_0) = {\bf x}_0 + \int_0^t {\bf f}\left( \phi(s, {\bf x}_0), s \right) {\text d}s , \]
where x(t) = ϕ(t, x0) is a solution to the initial value problem \eqref{EqPart3.3}. If we differentiate this solution with respect to x0, we obtain, using the chain rule:
\[ \frac{\partial \phi}{\partial {\bf x}_0} = 1 + \int_0^t \frac{\partial {\bf f}}{\partial {\bf x}_0}\left( \phi(s, {\bf x}_0), s \right) \cdot \frac{\partial \phi}{\partial {\bf x}_0}\left( s, {\bf x}_0) \right) {\text d}s , \]
\[ \frac{\partial \phi}{\partial {\bf x}_0} = \left[ \frac{\partial \phi_i}{\partial x_{0.j}} \right]_{i,j=1}^n \]
is the matrix of partial derivatives. In order to simplify exposition, we first consider the one-dimensional case (following Hirsch, Smale, Devaney). Let
\[ z(t) = \frac{\partial \phi}{\partial x_0} \left( x_0 , t \right) \qquad \Longrightarrow \qquad z(0) = 1. \]
Differentiatingz with respect to t, we find
\begin{align*} z' (t) &= \frac{\partial f}{\partial x_0} \left( \phi (x_0 , t) , t \right) \cdot \frac{\partial \phi}{\partial x_0} \left( x_0 , t \right) \\ &= \frac{\partial f}{\partial x_0} \left( \phi (x_0 , t) , t \right) \cdot z(t) . \end{align*} Although we don't know the solution ϕ, we know that z(t) solves the differential equation
\[ z' (t) = \frac{\partial f}{\partial x_0} \left( \phi (x_0 , t) , t \right) \, z(t) , \qquad z(0) = 1. \]
Separating of variables and integration yields
\[ z(t) = \exp \left\{ \int_0^t \frac{\partial f}{\partial x_0} \left( \phi (x_0 , s) , s \right) {\text d} s \right\} . \]
Upon setting
\[ p\left( x_0 \right) = \phi \left( x_0 , 1 \right) , \]
we find the second derivative of the Poincaré map
\[ p'' (x_0 ) = p' (x_0 ) \left( \int_0^1 \frac{\partial^2 f}{\partial x_0 \partial x_0} \left( \phi (x_0 ,s), s \right) \cdot \exp \left\{ \int_0^s \frac{\partial f}{\partial x_0} \left( \phi(x_0 ,u) , u \right) {\text d} u\right\} \right) \]


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