# Preface

Introduction to Linear Algebra with Mathematica

# Periodic Solutions

A function $${\bf x}(t) = [x_1 (t), x_2 (t), \ldots , x_n (t)]$$ in n-dimensional space is called periodic if there exist a positive number T such that $${\bf x}(t) = {\bf x}(t+T) .$$ A function $${\bf x}(t) = [x_1 (t), x_2 (t), \ldots , x_n (t)]$$ in n-dimensional space is called quasiperiodic if there exist a periodic function $${\bf \phi}(t) = [t_1 , t_2 , \ldots , t_n ] = {\bf \phi}(t+T)$$ such that $${\bf x}(t) = {\bf \phi}(\omega_1 t , \omega_2 t , \ldots , \omega_n t ) .$$ If all frequencies ω1, ... , ωn are the same, the function is periodic.

In this section, we consider systems of ordinary differential equations

$\dot{\bf x} = {\bf f}\left( {\bf x} , t \right)$
where x represents a column vector in the n-dimensional space, its derivative with respect to time variable t is denoted by $$\dot{\bf x} = {text d}{\bf x}/{text d}t ,$$ and $${\bf f}\,:\, W \subset \mathbb{R}^{n+1} \to \mathbb{R}^{n} is a differentiable vector function. Our main concern is about periodic input (also called forcing or driven) function f, which is assumed to have period T (may be taken as 1 without any loss of generality) in time variable t, that is, f(x, t+ T) = f(x, t). It is well known that such non-autonomous systems may have a wide variety of periodic solutions (often called vibrations or oscillations). The period of these solutions may be T, i.e., the same as the period of the input function, or may be any integer m multiple of T, or any rational or even, in some exceptional cases, irrational number times T. The family of all the periodic solutions is, however, far from being arbitrary. Determination and analysis of periodic solutions for non-autonomous systems of equations is very difficult and interesting topic that is the area of active research. In this section, we mostly demonstrate ability of computational tools to help understand the behavior of periodic solutions, if any. The literature on the subject of periodic solutions is enormous, and we will restrict ourselves to describing and presenting a sequence of famous examples. Let \( \dot{\bf x} = {\bf f}\left( {\bf x} , t \right) = {\bf f}\left( {\bf x} , t+T \right)$$ be a vector differential equation with periodic forcing function f. Any its periodic solution with period T is called harmonic vibration or oscillation. Its solution is called subharmonic of order 1/m if it admits the fundamental period $$m\,T .$$ In particular, a constant solution is considered as harmonic.
Subharmonic vibration refers to the response of a dynamic system to excitation at a whole-number multiple (m) of its natural frequency by vibrating asynchronously at its natural frequency, that is, at (1/m) of the excitation.

Theorem: Suppose that a one dimensional differential equation $$\dot{x} = f(x,t)$$ has a bounded solution for any t greater than a certain finite value t0. Then there exists a harmonic vibration.

Theorem: Suppose that all solutions to a two dimensional differential equation $$\dot{\bf x} = {\bf f}({\bf x},t)$$ exist for any t greater than a certain finite value t0. If one of them is bounded, then there exists a harmonic vibration.

When the dimension of the system $$\dot{\bf x} = {\bf f}({\bf x},t)$$ exceeds 2, then there are known examples that the system has no periodic solutions or solutions are periodic with the period not compatible with the driven term.

Theorem: Let

$A = \min \left\{ 1 + \frac{4}{c}, 1+ \frac{4}{c^2} , \beta^{-1/3} + \frac{4}{c^2} \right\} .$
If c² > 48 Aβ, then there exists a unique periodic solution of the differential equation
$\ddot{x} + c\,\dot{x} + x + \beta\,x^3 = f(t) ,$
where f(t) is a continuous odd periodic function, and all other solutions tend to this periodic solution as t → ∞.

Example 1: The following system has periodic solutions:

$\begin{split} \dot{x} &= x - y - x \left( 3 x^2 + y^2 \right) , \\ \dot{y} &= x + y - x \left( 3 x^2 + y^2 \right) , \\ \dot{z} &= z + xz - z^3 . \end{split}$
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Poincaré maps

The Poincaré mapping is a very useful tool to analyze periodic orbits. In essence,it consists in considering local transverse sections of the flow, rather than the flow as a whole, i.e., the intersections of integral curves with some local hypersurfaces that are not tangent to them. For example, if the flow lies in the two-dimensional space ℝ², we let it go through local line segments that are chosen in such a way that they do not contain any integral curve or parts thereof. One then studies the set of points where the integral curves cross these line segments and tries to analyze the structureof the flow by means of the pattern that one obtains in this way.

A flow in three dimensions is cut locally by planes or other two-dimensional smooth surfaces that are chosen such that they do not contain any integral curves.

Consider solutions to the vector differential equation in ℝ³. A local transverse section S ̃at the point x ∈ ℝ³ is an open neighborhood on a hypersurface of dimension 2 that contains x ̃and is chosen in such a way that, at none of the points sS does the trajectory through s lie in the tangent space to S.
The last condition makes sure that all flow lines going through points s of S do indeed intersect with S and that none of them lies in S. Consider a periodic orbit Γ with period T, and let S be a local transverse section at a point x0 on Γ. Without loss of generality, we may take x0 = 0 (at t = 0). Clearly, we have x0(nT) = 0, for all integers n.
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