\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 175, pp. 1--15.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/175\hfil Oscillatory solutions]
{Oscillatory solutions of the Cauchy problem for linear differential equations}

\author[G. Hovhannisyan, O. Ruff \hfil EJDE-2015/??\hfilneg]
{Gro Hovhannisyan, Oliver Ruff}

\address{Gro Hovhannisyan \newline
Kent State University at Stark\\
6000 Frank Ave. NW\\
Canton, OH 44720-7599, USA}
\email{ghovhann@kent.edu}

\address{Oliver Ruff \newline
Kent State University at Stark\\
6000 Frank Ave. NW\\
Canton, OH 44720-7599, USA}
\email{oruff@kent.edu}

\thanks{Submitted May 6, 2015. Published June 24, 2015.}
\subjclass[2010]{34C10}
\keywords{Linear ordinary differential equation; oscillation; 
\hfill\break\indent initial value problem;
 characteristic polynomial; characteristic roots}

\begin{abstract}
 We consider the Cauchy problem for second and third order linear differential
 equations with constant complex coefficients. We describe necessary and
 sufficient conditions on the data for the existence of oscillatory solutions.
 It is known that in the case of real coefficients the oscillatory behavior
 of solutions does not depend on initial values, but we show that this is no
 longer true in the complex case: hence in practice it is possible to control
 oscillatory behavior by varying the initial conditions.
 Our Proofs are based on asymptotic analysis of the zeros of solutions,
 represented as linear combinations of exponential functions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks

\section{Introduction}

A solution to a differential equation is said to be \emph{oscillatory} 
if it has an unbounded infinite sequence of zeros within some interval 
$(t_0,\infty)$, and \emph{nonoscillatory} otherwise. Since the choice of 
$t_0$ does not affect the determination of whether or not a solution is 
oscillatory, we suppress it in Definition~\ref{oscillatory} below.

In the case where the equation has real  coefficients, the theory of 
oscillatory solutions is well-developed \cite{b1,s1,e1}, and mostly based 
on Sturm's famous comparison theorems. However, the case in which the 
coefficients are complex has not been studied very much, both because 
there are not many immediate physical applications of such equations 
(except the Dirac equation) and because Sturm's comparison theorems no 
longer apply.

In the complex coefficient case the oscillatory behavior of solutions depends 
not only on the roots of the characteristic equation but also on initial values. 
So, in applications it is possible to control the appearance of oscillatory 
behavior by setting appropriate initial conditions.

In this article we study which initial values lead to oscillatory solutions and
 which do not. The main result of the manuscript is the description of the 
initial values and the roots of characteristic equations that produce oscillatory 
solutions in the complex case.

We think that the analysis of the complex case, even in the simplest situation 
where coefficients are constant, will give us a better understanding of real 
coefficients case as well. For example, appearance of oscillatory solutions
is connected with a new algebraic condition (see \eqref{2.17}) which does 
not appear anywhere obvious in the study of the real case.

Proofs are based on analysis of the zeros of the linear combinations of 
various exponential functions. Note that asymptotic behavior of zeros of the 
sums of exponential functions have been studied in the classical paper of 
Langer and others \cite{l1, b2, s2}.

Some oscillation theorems for linear differential equations with complex 
variable coefficients are proved in \cite{h1,h2}
by using the asymptotic theory.

We are grateful to the referee for several helpful observations, 
including pointing out Lemma~\ref{langerlemma} to us.

\section{Differential equations with complex 
constant coefficients}


\subsection{Notation and a preliminary results}

For $z \in \mathbb{C}$, write $\Re[z]$ and $\Im[z]$ for (respectively) 
the real and imaginary parts of $z$.

\begin{definition}\label{oscillatory} \rm
A solution to a differential equation is said to be \emph{oscillatory} 
if it possesses an infinite sequence of real zeros whose limit is $\infty$.
\end{definition}

Since this is a fact we will use frequently, we emphasize that 
Definition~\ref{oscillatory} requires that an oscillatory solution 
must possess real zeros of arbitrarily large magnitude. 
Given a particular oscillatory solution $u$, we denote by 
$\{ t_k : k \geq 1 \}$ an unbounded increasing sequence of its zeros, 
so $u(t_i) = 0$ and $t_i < t_j$ for all $1 \leq i < j$.

For a given linear differential equation of order $n$, write 
$\lambda_1,\dots,\lambda_n$ for the roots of its characteristic polynomial, 
and write
\[
x_i = \Re[\lambda_i], \quad y_i = \Im[\lambda_i], \quad
 \lambda_{ij} = \lambda_i - \lambda_j, \quad 
 x_{ij} = \Re[\lambda_{ij}], \quad y_{ij} = \Im[\lambda_{ij}] 
\]
for all $1 \leq i, j, \leq n$.


We are going to use an important general result connected to the zeros of the 
solutions of linear ordinary differential equation with constant complex 
coefficients.

\begin{lemma}\label{langerlemma} 
A linear $n$-th order ordinary differential equation with constant complex 
coefficients has a nontrivial solution with infinitely many zeros if and only 
if it has two distinct characteristic roots with equal real parts.
\end{lemma}

This result follows from standard facts about the asymptotic zero 
distribution of exponential sums (see \cite{l1,s2}).

\subsection{Second order equations}


\begin{theorem}\label{secondorderthm} 
A nontrivial solution of the initial-value problem
\begin{equation}\label{2.1}
\begin{gathered}
u''(t)+au'(t)+bu(t)=0,\\
u(0)=d_0,\quad u'(0)=d_1,\quad d_0, d_1 \in \mathbb{R},\quad
a,b \in \mathbb{C}
\end{gathered}
\end{equation}
is oscillatory  on $(0,\infty)$ if and only if either
\begin{equation}\label{2.2}
x_{12}=0 ,\quad
d_0\Im[\lambda_1+\lambda_2]=0,
\end{equation}
or the discriminant
\begin{equation}\label{2.3}
D_2=a^2-4b = \lambda_{12}^2
\end{equation}
is real and negative and
\begin{equation}\label{2.4}
d_0\Im[a]=0.
\end{equation}
\end{theorem}

If the coefficients $a$ and $b$ are real, then
\eqref{2.4} is automatically satisfied and the oscillatory behavior 
of the solutions does not depend on the initial conditions. 
However, the following example shows that this is not true for general 
complex coefficients.

\begin{example} \rm
The solution of
\begin{equation}\label{2.5}
u''(t)+(1+2i)u'(t)+iu(t)=0,\quad u(0)=0
,\quad u'(0)=1
\end{equation}
is oscillatory but the solution of
\begin{equation}\label{2.6}
u''(t)+(1+2i)u'(t)+iu(t)=0,\quad u(0)=1
,\quad u'(0)=0
\end{equation}
is nonoscillatory.
Specifically,
\[
 u_1(t)=\frac{2}{\sqrt3}e^{-\frac{t}{2}-it}\sin(t\frac{\sqrt3}{2})
\]
is the oscillatory solution of \eqref{2.5} and
\[
u_2(t)=\frac1{\sqrt3}e^{-\frac{t}{2}-it}
\Big(\sqrt3\cos(t\frac{\sqrt3}{2})+(2i+1)\sin(t\frac{\sqrt3}{2}) \Big)
\]
is the nonoscillatory solution of \eqref{2.6}.
\end{example}

\subsection{Third order equations}

Now we consider the initial-value problem
\begin{equation}\label{2.7}
\begin{gathered}
 u'''(t)+3I_1u'(t)+2I_2u(t)=0, \\ 
u(0)=d_0, \quad u'(0)=d_1, \quad u''(0)=d_2,
\end{gathered}
 \end{equation}
where $d_0,d_1,d_2 \in \mathbb{R}$, and $I_1,I_2\in \mathbb{C}$.

Since the characteristic polynomial associated to \eqref{2.7} is reduced, 
it will always be the case that
\begin{equation}\label{2.8}
 \sum_{i=1}^n \lambda_i = 0 .
\end{equation}
Write
\begin{equation}\label{2.9}
D_3=-I_1^3-I_2^2=\lambda_{12}^2\lambda_{13}^2
\lambda_{23}^2
\end{equation}
for the associated discriminant, and order the roots 
$\lambda_1,\lambda_2,\lambda_3$ in some way so that
\begin{equation}\label{2.10}
x_1 \leq x_2 \leq x_3
\end{equation}

\begin{theorem}\label{mainthirdorderthm} 
A nontrivial solution to  \eqref{2.7} is oscillatory on $(0,\infty)$ 
if and only if one of the following conditions is satisfied:
\begin{equation}\label{2.11}
x_1 = x_2 < 0 < x_3, \quad d_1=d_0x_1,\quad d_2=(x_1^2-y_1^2)d_0,\quad 
\lambda_2=\overline{\lambda}_1,
\end{equation}
or
\begin{equation}\label{2.12}
x_1 < x_2 = x_3, \quad |\lambda_{13}k_2|=|\lambda_{12}k_3|,
\end{equation}
or
\[ 
x_1 = x_2 = x_3 = 0,
\]
there exists a sequence of distinct natural numbers $\{m_k\}_{k=1}^{\infty}$ 
such that
\begin{equation}\label{2.13}
\frac{y_{13}}{y_{23}}\in \mathbb{Q} ,\quad
\frac{y_{13}}{y_{23}} (m_k+\varphi\mp\varphi_0)
-\psi\pm\psi_0\in Z,
\end{equation}
and
\begin{equation}\label{2.14}
|k_3y_{12}|+|k_2y_{13}|\ge |k_1y_{23}|,\quad
- |k_1y_{23}|\le|k_3y_{12}|-|k_2y_{13}|
\le  |k_1y_{23}|,
\end{equation}
where
\begin{gather*}
k_1=d_2-d_1(\lambda_2+\lambda_3) +d_0\lambda_2 \lambda_3,\quad 
k_2=d_2-d_1(\lambda_1+\lambda_3) +d_0\lambda_1 \lambda_3, \\
k_3=d_2-d_1(\lambda_1+\lambda_2) +d_0\lambda_1 \lambda_2, \\
\varphi=\frac1{2\pi}\cos^{-1}
\Big(\pm\frac{|k_1y_{23}|^2-|k_3y_{12}|^2-|k_2y_{13}|^2}{2|k_2k_3y_{12}y_{13}|}\Big),\\
\psi=\frac1{2\pi}\cos^{-1}
\Big(\pm\frac{|k_2y_{13}|^2-|k_1y_{23}|^2-|k_3y_{12}|^2}{2|k_1k_3y_{12}y_{23}|}\Big),
\end{gather*}
\begin{equation}\label{2.15}
\varphi_0=-\frac1{2\pi}\sin^{-1}
\Big(\Im\big[\frac{k_3|k_2|}{k_2|k_3|}\big] \Big),\quad
\psi_0=-\frac1{2\pi}\sin^{-1}
\Big(\Im\big[\frac{k_3|k_1|}{k_1|k_3|}\big]\Big).
\end{equation}
\end{theorem}

Note that condition \eqref{2.14} describes the
region of initial data that produce the oscillatory solutions.
Condition \eqref{2.13} is similar to the condition that the roots 
$\lambda_j$ are \emph{commensurable} (see \cite{l1}), that is 
$\lambda_j=\alpha p_j$, for some
$\alpha\in \mathbb{C}, p_j\in \mathbb{Z}$.
For special initial values the conditions of
Theorem~\ref{mainthirdorderthm} may be simplified.

\begin{theorem}\label{thirdorder01} 
A nontrivial solution to  \eqref{2.7} with $d_0=d_1=0$ is oscillatory if and 
only if one of the following conditions is satisfied:
\begin{equation}\label{2.16}
\lambda_1=x_1<0 <x_2=x_3,
\end{equation}
or
\begin{equation}\label{2.17}
x_1 = x_2 = x_3 =0,\quad \frac{y_{13}}{y_{23}} \in \mathbb{Q}.
\end{equation}
Furthermore, \eqref{2.16},\eqref{2.17} are equivalent, respectively, to
\begin{equation}\label{2.18}
\Im[I_1]=0,\quad \Im[I_1^3+I_2^2]=0,
\quad I_1^3+I_2^2>0,
\end{equation}
\begin{equation}\label{2.19}
\Re[I_2]=\Im[D_3]=0,\quad \Re[D_3]<0,
\quad
\frac{\sqrt3\Im\big[\big(-I_2+\sqrt{I_1^3+I_2^2} \big)^{1/3}\big]}
{\Re\big[\big(-I_2+\sqrt{I_1^3+I_2^2} \big)^{1/3}\big]} \in \mathbb{Q}.
\end{equation}
\end{theorem}

\begin{example} \rm
The solutions to
\begin{equation}\label{2.20}
u'''(t)+2I_2u(t)=0,\quad u(0)=u'(0)=0,\quad
u''(0)=1,\quad I_2\neq 0
\end{equation}
are nonoscillatory since $I_1=0$ and the conditions 
$\Re[I_2]=0$, $ \Re[I_2^2]>0$ are never satisfied.
\end{example}

\begin{example}  \rm
The solutions to
\begin{equation}\label{2.21}
u'''(t)+3(a+ib)u'(t)=0,\quad u(0)=u'(0)=0,\quad
u''(0)=1
\end{equation}
are oscillatory if
\begin{equation}\label{2.22}
\begin{gathered}
\Im[I_1^3+I_2^2]=3a^2b-b^3=0,\quad
\Re[I_1^3+I_2^2]=a^3-3ab^2>0, \\
\frac{\sqrt3 b}{a} \in \mathbb{Q}.
\end{gathered}
\end{equation}
These conditions are satisfied if, for example, $b=0$, $I_1=a>0$
or
\[
a=\sqrt3,\quad b=3,\quad I_1=\sqrt3+3i.
\]
\end{example}

\begin{theorem}\label{thirdorder02} 
A nontrivial solution to  \eqref{2.7} with $d_0=d_2=0$ is oscillatory if and only 
if one of the following conditions is satisfied
\begin{equation}\label{2.23}
\lambda_1=x_1<0<x_2=x_3,\quad I_1, I_2\in \mathbb{R},
\end{equation}
or
\begin{equation}\label{2.24}
0<x_2=x_3,\quad
6x_2^2+ y_{12}^2=3y_2^2,
\end{equation}
or
\begin{equation}\label{2.25}
x_1=x_2= x_3=0,\quad
\frac{y_{13}}{y_{23}} \in \mathbb{Q}.
\end{equation}
\end{theorem}

\begin{theorem}\label{thirdorder12} 
A nontrivial solution to  \eqref{2.7} with $d_1=d_2=0$
is oscillatory if and only if
\begin{equation}\label{2.26}
\lambda_1=x_1<0<x_2=x_3,\quad I_1, I_2\in \mathbb{R},
\end{equation}
or equivalently
\begin{equation}\label{2.27} I_1, I_2 \in \mathbb{R} ,\quad I_2<0,\quad
D_3<0.
\end{equation}
\end{theorem}

\section{Proofs}

\begin{proof}[Proof of Theorem \ref{secondorderthm}]
Note that we may assume that
\begin{equation}\label{3.1}
D_2=a^2-4b = \lambda_{12}^2\neq 0,
\end{equation}
since otherwise there is no distinct roots of
the characteristic polynomial and in view of Lemma~\ref{langerlemma} 
there are no nontrivial oscillatory solutions of \eqref{2.1}.

The solutions to \eqref{2.1} where $D_2\neq 0$ are given by the formula
\begin{equation}\label{3.2}
u(t)=\frac{(d_1-d_0\lambda_2)e^{t\lambda_1}-
(d_1-d_0\lambda_1)e^{t\lambda_2}}{\lambda_{12}}.
\end{equation}
The zeros of \eqref{3.2} satisfy
\[
(d_1-d_0\lambda_2)e^{t\lambda_1}=(d_1-d_0\lambda_1)e^{t\lambda_2};
\]
that is,
\begin{equation}\label{3.3}
 e^{t\lambda_{12}}= \frac{d_1-d_0\lambda_1}{d_1-d_0\lambda_2}.
\end{equation}

Note that we may also assume that
\begin{equation}\label{3.4}
(d_1-d_0\lambda_1)(d_1-d_0\lambda_2)\neq 0,
\end{equation}
otherwise \eqref{3.2} is nonoscillatory.

If $x_{12} >0$ then the left-hand side of \eqref{3.3} becomes unboundedly 
large as $t \to \infty$, so this is impossible in view of
\eqref{3.4}. On the other hand, if $x_{12} < 0$ then the left-hand side 
of \eqref{3.3} approaches $0$ as $t \to \infty$ and this is
impossible as well. Consequently  $x_{12}=0$.

 Solving \eqref{3.3} for $t$, we obtain
\[
t=\frac{\ln\frac{d_1-d_0\lambda_1}{d_1-d_0\lambda_2}}{\lambda_{12}}=
\frac{\ln |\frac{d_1-d_0\lambda_1}{d_1-d_0\lambda_2}|
+i\operatorname{arg}\big(\frac{d_1-d_0\lambda_1}{d_1-d_0\lambda_2}\big)}{i y_{12}}
\]
In order for \eqref{3.2} to be oscillatory, we need an infinite number 
of these t to lie in $\mathbb{R}$, which happens if and only if
\begin{equation}\label{3.5}
x_{12}=0 ,\quad
|d_1-d_0\lambda_1|=|d_1-d_0\lambda_2|.
\end{equation}

So we have the following infinite sequence of zeros:
\[
t_k=\frac{2ki \pi}{\lambda_{12}}
=\frac{2k\pi}{\Im[\lambda_{12}]},\quad k \in \mathbb{Z}.
\]
Simplifying \eqref{3.5} we have
\[
x_{12}=0 ,\quad
d_0^2(y_1^2-y_2^2)=0.
\]
and hence (since $y_{12}=x_{12}=0$ would imply $D_2 = 0$, which is not the case)
\begin{equation}\label{3.6}
x_{12}=0 ,\quad d_0(y_1 + y_2)=0.
\end{equation}
Theorem \ref{secondorderthm} will now follow from the next two lemmas.
\end{proof}

\begin{lemma}\label{L3.1}
\[
\Re\left[\sqrt{m+in}\right]=0,\quad m,n\in \mathbb{R}
\]
if and only if
\[
n=0,\quad m\le 0.
\]
\end{lemma}

\begin{proof}
From the well known formula
\begin{equation}\label{3.7}
\sqrt{2(m+in)}=\sqrt{\sqrt{m^2+n^2}
+m}+i\operatorname{sign}(n)\sqrt{\sqrt{m^2+n^2}-m}
\end{equation}
we obtain
\[
\Re\left[\sqrt{m+in}\right]=\frac1{\sqrt 2}
\sqrt{\sqrt{m^2+n^2}
+m}
\]
which equals $0$ if and only if
$n=0$, $m\le 0$.
\end{proof}

\begin{lemma}
$\Re[\lambda_{12}]=0$ if and only if
\begin{equation}\label{3.8}
\Im[D_2]=0,\quad D_2=a^2-4b\le 0.
\end{equation}
\end{lemma}

\begin{proof}
 As they are the roots of $\lambda^2+a\lambda +b=0$, 
$\lambda_1$ and $\lambda_2$ are given by the quadratic formula
\[
\lambda_1,\lambda_2=\frac{-a\pm\sqrt{m+i n}}2,\quad
\lambda_{12}=\sqrt{m+in},\quad m=\Re[D_2],\quad n=\Im[D_2].
\]
Applying Lemma~\ref{L3.1} we obtain
\[
x_{12}=
\Re\big[\sqrt{m+in}\big]=0
\]
if and only if $n=\Im[D_2]=0, \quad m=\Re[D_2]\le 0$.
\end{proof}


\begin{proof}[Proof of Theorem \ref{mainthirdorderthm}]
First consider the case $D_3=0$. There are two possibilities: $ I_1=0$  
and $I_1\neq 0.$
Since in the case
\begin{equation}\label{3.9}
D_3=-I_1^3-I_2^2=0,\quad I_1= 0
\end{equation}
the equation  $u'''(t)=0$ has nonoscillatory nontrivial solutions 
$u=C_1+C_2t+C_3t^2$,  it is sufficient to consider  the case
\begin{equation}\label{3.10}
D_3=\lambda_{12}^2\lambda_{13}^2
\lambda_{23}^2=0, \quad I_1\neq 0.
\end{equation}
In this case there is one repeated root 
-- for convenience, we assume $\lambda_2 = \lambda_3$. 
(In principle this involves a loss of generality as the $\lambda_i$ are ordered, 
but we will not use the ordering in what follows.) So
$\lambda_2=\lambda_3\neq 0$, $ 3I_1=\lambda_2(2\lambda_1+\lambda_2)\neq 0$, 
and the solutions of
$ u'''(t)+3I_1u'(t)+2I_2u(t)=0 $ are given by
$$
u(t)=C_1e^{t\lambda_1}+C_2e^{t\lambda_2}+C_3te^{t\lambda_2}.
$$
From the initial conditions
\[
C_1+C_2=d_0,\quad
C_1\lambda_1+C_2\lambda_2+C_3=d_1,\quad
C_1\lambda_1^2+C_2\lambda_2^2+
2C_3\lambda_2=d_2,
\]
we obtain
\begin{gather*}
C_1=\frac{d_2+d_0\lambda_2^2-2d_1\lambda_2}{\lambda_{12}^2}, \quad 
C_2=\frac{d_0\lambda_1^2+2d_1\lambda_2 -d_2-2d_0\lambda_1\lambda_2}{\lambda_{12}^2},\\
C_3=-\frac{k_3}{\lambda_{12}},\quad k_3=d_2-d_1(\lambda_1+\lambda_2)
+d_0\lambda_1\lambda_2,
\end{gather*}
and hence the solution is
\begin{equation}\label{3.11}
u(t)= \frac{(d_2+d_0\lambda_2^2-2d_1\lambda_2)
e^{t\lambda_1}+
(d_0\lambda_1(\lambda_1-2\lambda_2)
+2d_1\lambda_2-d_2) e^{t\lambda_2}}{\lambda_{12}^2}-
\frac{k_3te^{t\lambda_2}}{\lambda_{12}}.
\end{equation}

The zeros of \eqref{3.11} satisfy
\[
\frac{(d_2-2d_1\lambda_2+d_0\lambda_2^2)}{t\lambda_{12}^2}e^{t\lambda_{12}}
+\frac{d_0\lambda_1(\lambda_1-2\lambda_2)
+2d_1\lambda_2-d_2}{t\lambda_{12}^2}=\frac{k_3}{\lambda_{12}};
\]
that is,
\begin{equation}\label{3.12}
\frac{d_2-2d_1\lambda_2+d_0\lambda_2^2}{t\lambda_{12}}
e^{t\lambda_{12}}
=k_3+\frac{d_2-2d_1\lambda_2-d_0\lambda_1(\lambda_1-2\lambda_2)}{t\lambda_{12}}
\end{equation}

From Lemma~\ref{langerlemma} it follows that if  \eqref{2.7} is oscillatory 
then the distinct roots $\lambda_1,\lambda_2$ have equal real parts, that is
 $x_{12}=0$.

On the other hand, if $x_{12}= 0$ then the left-hand side of \eqref{3.12} 
approaches $0$ as $t \to \infty$, so  for \eqref{3.11} to be oscillatory 
we must have $k_3=d_2-d_1(\lambda_1+\lambda_2) +d_0\lambda_1\lambda_2=0$.

Further from \eqref{3.12}
\[
e^{t\lambda_{12}}=
\frac{d_2-2d_1\lambda_2-d_0\lambda_1(\lambda_1-2\lambda_2)} {d_2-2d_1\lambda_2
+d_0\lambda_2^2}= \frac{d_1\lambda_{12}-d_0\lambda_1 \lambda_{12}}{d_1\lambda_{12}
-d_0\lambda_2\lambda_{12}}=
\frac{d_1-d_0\lambda_1}{d_1-d_0\lambda_2}
\]
which is impossible as $t\to\infty$ unless $x_{12}=0$, that is
$x_1=x_2=x_3=0.$

Since $d_0,d_1,d_2 \in \mathbb{R}$, from $k_3=0,\Im[k_3]=0$
we obtain
\[
d_1(y_1]+y_2)= d_0x_1(y_1+y_2)
\]
and since $\lambda_1+\lambda_2+\lambda_3=0,\lambda_2=\lambda_3$, and
$x_{12}=0$ it follows that $y_1+y_2\neq 0$.
Then
\[
d_1=d_0x_1,\quad
|e^{t\lambda_{12}}|=|\frac{y_1}{y_2}|=1
\]
yielding $y_1=\pm y_2$,
which is a contradiction (since we know $y_1+y_2 \neq 0$ and $\lambda_{12}\neq 0$).
So: there are no oscillatory solutions in the case  $D_3=0,I_1\neq 0$ .

In the case $D_3=\lambda_{12}^2\lambda_{13}^2
\lambda_{23}^2\neq 0$ ($\lambda_1$, $\lambda_2$, 
$\lambda_3$ are distinct) the solutions of \eqref{2.7} are given by
$u(t)=C_1e^{t\lambda_1}+ C_2e^{t\lambda_2}+C_3e^{t\lambda_3}$,
and in view of the initial conditions we have
\begin{equation}\label{3.13}
u(t)=\frac{k_1\lambda_{23}
e^{t\lambda_1}-k_2\lambda_{13}
e^{t\lambda_2}+k_3\lambda_{12}
e^{t\lambda_3}}{\lambda_{12}\lambda_{13}\lambda_{23}}
\end{equation}
where, as in \eqref{2.15},
\begin{gather*}
k_1=d_2-d_1(\lambda_2+\lambda_3) +d_0\lambda_2 \lambda_3,\quad 
k_2=d_2-d_1(\lambda_1+\lambda_3) +d_0\lambda_1 \lambda_3, \\
k_3=d_2-d_1(\lambda_1+\lambda_2) +d_0\lambda_1 \lambda_2
\end{gather*}
The zeros of \eqref{3.13} satisfy
\[
k_1\lambda_{23}e^{t\lambda_1}
-k_2\lambda_{13}e^{t\lambda_2}
+k_3\lambda_{12}e^{t\lambda_3}=0;
\]
that is,
\begin{equation}\label{3.14}
k_1\lambda_{23}e^{t\lambda_{13}}
-k_2\lambda_{13}e^{t\lambda_{23}}
+k_3\lambda_{12}=0.
\end{equation}
In view of assumption \eqref{2.10} it is sufficient to consider the following 
three cases:
\begin{gather}\label{3.15}
x_1 \leq x_2 <x_3, \\
\label{3.16}
x_1<x_2 =x_3, \\
\label{3.17}
x_1=x_2=x_3.
\end{gather}
From \eqref{2.8} we have
\begin{equation}\label{3.18}
x_1+x_2+x_3=0,\quad y_1+y_2+y_3=0.
\end{equation}

First we consider case \eqref{3.15}.
If \eqref{3.13} is oscillatory (that is, \eqref{3.14} is true for arbitrarily 
large values of $t$) then by letting $t \to \infty$ in \eqref{3.14}  
we obtain $k_3=0$ and
\[
k_1=k_1-k_3=\lambda_{13}(d_1-d_0\lambda_2),\quad
k_2=k_2-k_3=\lambda_{23}(d_1-d_0\lambda_1).
\]
Further, from \eqref{3.14} we have
\[
e^{t\lambda_{12}}=\frac{k_2\lambda_{13}}{k_1\lambda_{23}},
\]
which gives the infinite sequence of zeros
\[
t_n=\frac1{y_{12}}\sin^{-1}
\Big(\Im\big[\frac{k_2\lambda_{13}}{k_1\lambda_{23}}\big]\Big)+\frac{n\pi}{y_{12}}
\]
provided that either
\[
\big|\frac{k_2\lambda_{13}}{k_1\lambda_{23}}\big|= 1,\quad k_3=0,\quad
x_1 = x_2 < x_3
\]
or
\[
\big|\frac{d_1-d_0\lambda_1}{d_1-d_0\lambda_2}\big|=1,\quad x_1=x_2,\quad 
d_2=d_1(\lambda_1+\lambda_2)-d_0\lambda_1\lambda_2,\quad x_2<x_3.
\]
Now, since the $d_j$ are real, from $k_3=0$ we obtain
\begin{gather*}
d_2-d_1(x_1+x_2)+d_0(x_1x_2-y_1y_2)=0,\\
-d_1(y_1+y_2)+d_0(x_1y_2+y_1x_2)=0, \\
\frac{d_1}{d_0}=\frac{x_1y_2+y_1x_2}{y_1+y_2}=x_1,\\
\frac{d_2}{d_0}=y_1y_2-x_1x_2+(x_1+x_2) \frac{d_1}{d_0} 
=y_1y_2+x_1x_2=y_1y_2+x_1^2
\end{gather*}
that is condition \eqref{2.11} of Theorem~\ref{mainthirdorderthm}:
\[
\big|\frac{x_1-\lambda_1}{x_2-\lambda_2}\big|=1\quad\text{or}\quad 
|y_1|=|y_2|,\quad y_2=-y_1,
\]
\begin{equation}\label{3.19}
d_1=x_1d_0=d_0x_1,\quad
d_2=(x_1^2+y_1y_2)d_0,\quad \lambda_2=\overline{\lambda}_1,\quad 
x_1=x_2<0<x_3.
\end{equation}

We move to the next case \eqref{3.16}. Now the left-hand side of 
\eqref{3.14} approaches
\[
-k_2 \lambda_{13}e^{t\lambda_{23}}
+k_3\lambda_{12}=0
\]
as $t \to \infty$, so if \eqref{3.13} is going to be oscillatory we must 
have $k_2\lambda_{13}e^{t\lambda_{23}}=k_3\lambda_{12}$ for certain 
arbitrarily large values of $t$. Since we are in case \eqref{3.16}, 
we know that $x_{23} = 0$, so $\lambda_{23} = i y_{23}$, and also 
$x_{12} = x_{13}$. So in fact
\begin{equation}\label{3.20}
e^{it y_{23}}= \frac{k_3\lambda_{12}}{k_2\lambda_{13}} ,\quad
\big|\frac{k_3\lambda_{12}}{k_2\lambda_{13}}\big|=|e^{it y_{23}}| = 1,
\end{equation}
and we obtain condition~\eqref{2.12} of Theorem~\ref{mainthirdorderthm}:
\[
|k_2\lambda_{13}|=|k_3\lambda_{12}|,\quad
x_1 < x_2 = x_3.
\]

Finally, we turn to case \eqref{3.17}, in which $ x_1 = x_2 = x_3$ 
(so by \eqref{3.18} they are all zero) and so \eqref{3.14} can be written
\begin{equation}\label{3.21}
-k_1y_{23}e^{ity_{13}}+k_2y_{13}e^{ity_{23}} =k_3y_{12}.
\end{equation}
Denoting
\begin{equation}\label{3.22}
b_1=k_3y_{12},\quad b_2=-k_1y_{23},\quad
b_3=k_2y_{13},\quad v=ty_{13},\quad w=ty_{23}.
\end{equation}
we obtain, from \eqref{3.21},
\begin{equation}\label{3.23}
b_2e^{iv}+b_3e^{iw}=b_1,\quad b_1,b_2,b_3\in\mathbb{C}.
\end{equation}

\begin{lemma} \label{lem3.3}
The exponential equation \eqref{3.23} has  solution 
$(v,w)\in\mathbb{R}\times \mathbb{R}$ if and only if
\begin{equation}\label{3.24}
|b_1|+|b_3|\ge|b_2|,\quad
-|b_2|\le |b_1|-|b_3| \le |b_2|.
\end{equation}

These solutions are given by the formulas
\begin{equation}\label{3.25}
\begin{gathered}
v=\cos^{-1}\Big(\pm\frac{|b_1|^2+|b_2|^2-|b_3|^2}{2|b_1b_2|}\Big)
\mp 2\pi\tilde{\psi}_0,\\
w=\cos^{-1}\Big(\pm\frac{|b_1|^2+|b_3|^2-|b_2|^2}{2|b_1b_3|}\Big)
\mp 2\pi\tilde{\varphi}_0,
\end{gathered}
\end{equation}
where
\begin{equation}\label{3.26}
\tilde{\varphi}_0=-\frac1{2\pi}\sin^{-1}
\Big(\frac{\Im[b_1/b_3]}{|b_1/b_3|}\Big),\quad
\tilde{\psi}_0=-\frac1{2\pi}\sin^{-1}
\Big(\frac{\Im[b_1/b_2]}{|b_1/b_2|}\Big).
\end{equation}
\end{lemma}

\begin{remark} \rm
Condition \eqref{3.24}  is invariant with respect to the substitutions
\[
b_j\to b_je^{i\alpha},\quad  \alpha\in\mathbb{R}.
\]
\end{remark}

\begin{remark} \rm
If equation \eqref{3.23} has a solution
$(v_0,w_0)\in\mathbb{R}\times \mathbb{R}$ then it has the infinite 
sequence of solutions
$(v_0+2k\pi,w_0+2m\pi),k,m\in\mathbb{Z}$.
\end{remark}

\begin{proof}[Proof of Lemma \ref{lem3.3}] 
Assuming $b_3\neq 0$ from \eqref{3.23} we have
$\frac{b_1}{b_3}-e^{iw}=\frac{b_2}{b_3}e^{iv}$,
and by taking the absolute value and squaring both sides of this
 equation we obtain
\[
\big|\frac{b_1}{b_3}-e^{iw}\big|^2
=\big|\frac{b_2}{b_3}e^{iv}\big|^2.
\]
Since from the definition of $\tilde{\varphi}_0$,
\[
\frac{b_1}{b_3}=\pm\big|\frac{b_1}{b_3}\big|\cos(2\pi\tilde{\varphi}_0)
-i\big|\frac{b_1}{b_3}\big|\sin(2\pi\tilde{\varphi}_0)
=\pm\big|\frac{b_1}{b_3}\big|  e^{\mp 2i\pi\tilde{\varphi}_0},
\]
we have
\begin{gather*}
\big|\pm |\frac{b_1}{b_3}|e^{\mp 2i\pi\tilde{\varphi}_0}-e^{iw}\big|^2
=|\frac{b_2}{b_3}|^2, \\
\big|\pm|b_1|-|b_3|e^{i(w\pm 2\pi\tilde{\varphi}_0)}\big|^2
= |b_2|^2, \\
\big(\pm|b_1|-|b_3|\cos(w\pm 2\pi\tilde{\varphi}_0)\big)^2+|b_3|^2
\sin^2(w\pm 2\pi\tilde{\varphi}_0)=|b_2|^2,
\end{gather*}
or
\begin{gather*}
\cos(w\pm 2\pi\tilde{\varphi}_0)=\pm
\frac{|b_1|^2+|b_3|^2-|b_2|^2}{2|b_1b_3|}, \\
w=\cos^{-1}\Big(\pm
\frac{|b_1|^2+|b_3|^2-|b_2|^2}{2|b_1b_3|} \Big)\mp 2\pi\tilde{\varphi}_0,
\end{gather*}
where
\[
\tilde{\varphi}_0=-\frac1{2\pi}\sin^{-1}
\Big(\frac{\Im[b_1/b_3]}{|b_1/b_3|}\Big)
=\pm\frac1{2\pi}\cos^{-1}
\Big(\frac{\Re[b_1/b_3]}{|b_1/b_3|}\Big).
\]
Note that in the case $b_3=0$ the real solution
$v$ of \eqref{3.23} exists if and only if
$|b_2|=|b_3|.$ This also follows from
 \eqref{3.24} in that case.

Further the real solution $w$ exists if and only if
\[
\left||b_1|^2+|b_3|^2-|b_2|^2\right|\le 2|b_1b_3|.
\]
To simplify  this inequality one can rewrite it in the form
\[
(|b_1|+|b_3|)^2\ge |b_2|^2,\quad
(|b_1|-|b_3|)^2\le |b_2|^2,
\]
and we obtain condition \eqref{3.24}.


To solve \eqref{3.23} with respect to $v$ we apply
 the substitution
\[
b_2\to b_2e^{iw-iv},\quad
b_3\to b_3e^{iv-iw},
\]
and obtain
\[
b_2e^{iw}+b_3e^{iv}=b_1.
\]
By transposition $b_2\leftrightarrow b_3$ from the formula \eqref{3.25} 
for $w$ we obtain the solution of this equation with respect to $v$
\[
v=\cos^{-1}\Big(\pm\frac{|b_1|^2+|b_2|^2-|b_3|^2}{2|b_1b_2|}\Big)
\mp 2\pi\tilde{\psi}_0,
\]
where
\[
\tilde{\psi}_0=-\frac1{2\pi}\sin^{-1}
\Big(\frac{\Im[b_1/b_2]}{|b_1/b_2|}\Big)
=\pm\frac1{2\pi}\cos^{-1}
\Big(\frac{\Re[b_1/b_2]}{|b_1/b_2|}\Big).
\]
The real solution $v$ exists if and only if
\[
\big||b_1|^2+|b_2|^2-|b_3|^2\big| \le 2|b_1b_2|.
\]
It can be shown that this inequality is equivalent to \eqref{3.24}.
\end{proof}

\begin{remark}\label{higherorderremark} \rm
To study the appearance of oscillatory solutions of the Cauchy problem  for n-th order ordinary differential equations with constant complex coefficients one should study  the existence of real solutions $\{v_j\}_{j=1}^n$ of the exponential equation:
\[
\sum_{j=1}^{n-1}b_je^{iv_j}=b_0, \quad
b_k\in\mathbb{C},\quad k=0,1,2,\dots,n.
\]
\end{remark}

Note that the asymptotic behavior and distribution of zeros of the sums of more general exponential functions have been studied in  \cite{l1, s2, b2}.
and they have a complicated structure.

Continuing the proof of Theorem \ref{mainthirdorderthm} we apply 
to equation \eqref{3.21} the condition  \eqref{3.24} of Lemma \ref{lem3.3},
 and in view of \eqref{3.22} we obtain
\[
|k_3y_{12}|+|k_2y_{13}|\ge |k_1y_{23}|,\quad
- |k_1y_{23}|\le|k_3y_{12}|-|k_2y_{13}| \le  |k_1y_{23}|,
\]
or  condition \eqref{2.14} of Theorem \ref{mainthirdorderthm}.

In view of \eqref{3.22} we have also
\begin{gather*}
\tilde{\varphi}_0=
\varphi_0=-\frac1{2\pi}\sin^{-1}\Big(\frac{\Im(b_1)}{|b_1|}\Big)
= -\frac1{2\pi}\sin^{-1}\Big(\frac{\Im(k_3/k_2)}{|k_3/k_2|}\Big), \\
\tilde{\psi}_0= \psi_0=-\frac1{2\pi}\sin^{-1}\Big(
\frac{\Im(b_1/b_2)}{|b_1/b_2|}\Big)
= -\frac1{2\pi}\sin^{-1}\Big(\frac{\Im(k_3/k_1)}{|k_3/k_1|}\Big).
\end{gather*}
From \eqref{3.25} we have
\begin{gather*}
ty_{13}=\cos^{-1}\Big(\pm
\frac{|k_1y_{23}|^2+|k_3y_{12}|^2-|k_2y_{13}|^2}{2|k_1k_3y_{12}y_{23}|}\Big)
\mp 2\pi\psi_0, \\
ty_{23}=\cos^{-1}\Big(\pm
\frac{|k_3^2|y_{12}^2+|k_2^2|y_{13}^2-|k_1^2|y_{23}^2}{2|k_2k_3y_{12}y_{13}|}\Big)
\mp 2\pi\varphi_0.
\end{gather*}
The  infinite sequence of zeros $\{t_n^*\},\{t_m\}$ is given by
\begin{gather*}
y_{13}t_n^*=2n\pi+2\pi\psi \mp 2\pi\psi_0,\quad n\in\mathbb{Z}, \\
y_{23}t_m=2m\pi+ 2\pi\varphi\mp 2\pi\varphi_0 ,\quad m\in\mathbb{Z},
\end{gather*}
where $\varphi,\psi,\varphi_0,\psi_0$
are as in \eqref{2.15}.

We claim that this occurs precisely when there exists a sequence of distinct
 integers $\{m_k\}_{k=1}^{\infty}$ such that
\[
\frac{y_{13}}{y_{23}}\in \mathbb{Q} ,\quad
\frac{y_{13}}{y_{23}} (m_k+\varphi\mp\varphi_0)
-\psi\pm\psi_0\in Z.
\]
In order for an oscillatory solution to exist, the sequences 
$t_m, t_n^*$ must coincide infinitely many times. That means that there 
must exist sequences  $\{m_k\}_{k=1}^{\infty},\{n_k\}_{k=1}^{\infty}$ 
of distinct integers such that
\[
t_{m_k}=\frac{2\pi(m_k+\varphi\mp\varphi_0)}{y_{23}}
=t_{n_k}^*=\frac{2\pi(n_k +\psi\mp\psi_0)}{y_{13}}
\]
or
\[
n_k=\frac{y_{13}}{y_{23}}
(m_k+\varphi\mp\varphi_0)-\psi\pm\psi_0,\quad
n_k-n_j=\frac{y_{13}}{y_{23}}(m_k-m_j),\quad k,j=1,2,\dots 
\]
and since $\frac{n_k-n_j}{m_k-m_j}\in \mathbb{Q}$, we obtain
\[
\frac{y_{13}}{y_{23}}\in\mathbb{Q},\quad
\frac{y_{13}}{y_{23}}(m_k+\varphi\mp\varphi_0)
-\psi\pm\psi_0\in \mathbb{Z}.
\]
We have now exhausted the cases \eqref{3.15}--\eqref{3.17}, 
which completes the proof.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thirdorder01}]
We deduce Theorem~\ref{thirdorder01} from Theorem~\ref{mainthirdorderthm}.
Note that case \eqref{2.11} is impossible since the condition
 $d_2=(x_1^2+y_1 y_2)d_0$, together with $d_0=d_1=0$, implies $d_2=0$ as well.
Note also that since  $d_0=d_1=0$ we obtain  $k_1=k_2=k_3=d_2$.

From case \eqref{2.12} we have
$|\lambda_{13}| =|\lambda_{12}|$ or
\[
x_{13}^2+y_{13}^2=x_{12}^2+y_{12}^2,
\]
Since in this case $x_{13}=x_{12}$ we obtain $y_{13}^2=y_{12}^2$,  
and in view of \eqref{2.8} we obtain $y_1=0$, that is, case \eqref{2.16}.

Further since $y_{23}^2=(y_{13}-y_{12})^2 =y_{12}^2-2y_{12}y_{13}+y_{13}^2$
the formula (2.15) turns into
\begin{equation}\label{3.27}
\varphi=\psi=\varphi_0=\psi_0=0
\end{equation}
and case \eqref{2.13} turns into case \eqref{2.17}.

It only remains to be shown that \eqref{2.16},\eqref{2.17} are
 equivalent correspondingly to  \eqref{2.18} and \eqref{2.19}.
From Vieta's formulas
\[
\lambda_1+\lambda_2+\lambda_3=0,\quad
3I_1=\lambda_1(\lambda_2+\lambda_3)+
\lambda_2\lambda_3,\quad
2I_2=-\lambda_1\lambda_2\lambda_3,\quad
D_3=\lambda_{12}^2\lambda_{13}^2\lambda_{23}^2
\]
we obtain
\begin{equation}\label{3.28}
I_1,\quad iI_2, \quad D_3 \in \mathbb{R},\quad
D_3\le 0
\end{equation}
Conversely from \eqref{3.29} the cubic equation
with real coefficients
\begin{equation}\label{3.29}
\mu^3-3I_1\mu+2iI_2=0,\quad \tilde{D}_3=-(-I_1)^3-(iI_2)^2=-D_3\ge 0
\end{equation}
has non-negative discriminant, and so it has three real roots; 
that is, $\Im[\mu_j]=0$, and by substitution $\mu=-i\lambda$ equation 
\eqref{3.29} turns to the characteristic equation $\lambda^3+3I_1\lambda+2I_2=0$. 
It follows that the characteristic equation has imaginary roots; 
that is, \eqref{2.18} is true.

 To rewrite the last part of condition~\eqref{2.17}
we will use the cubic formula.
Denote
\begin{equation}\label{3.30}
z_1=-I_2+\sqrt{-D_3},\quad
z_2=-I_2-\sqrt{-D_3}, \quad D_3=-I_1^3-I_2^2.
\end{equation}
 The roots of $\lambda^3+3I_1\lambda+2I_2=0$
are given by the cubic formula:
\begin{equation}\label{3.31}
\begin{gathered}
\lambda_1=z_1^{1/3}+z_2^{1/3}=\xi+\eta,
\quad \xi=(-I_2+\sqrt{D})^{1/3}, \\
 \eta=(-I_2-\sqrt{D})^{1/3}, \quad D=I_1^3+I_2^2,\\
\lambda_2=-\frac{z_1^{1/3}+z_2^{1/3}}2+
\frac{i\sqrt3(z_1^{1/3}-z_2^{1/3})}2=
-\frac{\xi+\eta}2+\frac{i\sqrt3(\xi-\eta)}2, \\
\lambda_3=-\frac{z_1^{1/3}+z_2^{1/3}}2-
\frac{i\sqrt3(z_1^{1/3}-z_2^{1/3})}2=
-\frac{\xi+\eta}2-\frac{i\sqrt3(\xi-\eta)}2,
\end{gathered}
\end{equation}
from which we obtain
\begin{gather*}
\lambda_{23}=i\sqrt3(z_1^{1/3}-z_2^{1/3}) =i\sqrt3(\xi-\eta), \\
\lambda_{13}=\frac32\left(z_1^{1/3}+z_2^{1/3}\right) 
+\frac{i\sqrt{3}}{2}\left(z_1^{1/3}-z_2^{1/3}\right)=
\frac32(\xi+\eta)+\frac{i\sqrt3}2(\xi-\eta).
\end{gather*}
Finally,  we can use the cubic formula to rewrite the last part of 
condition~\eqref{2.18}:
\begin{gather*}
\frac{\Im[\lambda_{13}]}{\Im[\lambda_{23}]}=
\frac{\frac32\Im[\xi+\eta]+\frac{\sqrt{3}}{2}
\Re[\xi-\eta]}{\sqrt{3}\Re[\xi-\eta]}=
\frac{\sqrt{3}\Im[\xi+\eta]}{2\Re[\xi-\eta]}+\frac12
\\
\frac{\Im[\lambda_{13}]}{\Im[\lambda_{23}]}=
\frac{\sqrt3\Im[z_1^{1/3}+z_2^{1/3}] }{2\Re[z_1^{1/3}-z_2^{1/3}]}+\frac12
\end{gather*}
and we have $z_2=-\overline{z_1}$, and so
\[
\frac{\Im[\lambda_{13}]}{\Im[\lambda_{23}]}
=\frac{\sqrt3\Im[z_1^{1/3}-\overline{z_1}^{1/3}]}
{2\Re[z_1^{1/3}+\overline{z_1}^{1/3}]}+\frac12
= \frac{\sqrt3\Im[z_1^{1/3}]}{2\Re[z_1^{1/3}]}+\frac12.
\]
It follows that $\frac{\Im[\lambda_{13}]}{\Im[\lambda_{23}]} \in \mathbb{Q}$ 
if and only if
\[ 
\frac{\sqrt3\Im[(-I_2+\sqrt{-D_3} )^{1/3}]}{\Re[(-I_2+\sqrt{-D_3} )^{1/3}]}
 \in \mathbb{Q}. 
\]
So we have established that \eqref{2.17} is equivalent to \eqref{2.19}.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thirdorder02}]

Again, we deduce Theorem~\ref{thirdorder02} from Theorem~\ref{mainthirdorderthm}.
Note that case \eqref{2.11} leads to the trivial solution since it follows 
from $d_0=d_2=0$ that $d_1=0$.

The case \eqref{2.12}, in view of
\[
k_1=-d_1(\lambda_2+\lambda_3),\quad
k_2=-d_1(\lambda_1+\lambda_3),\quad
k_3=-d_1(\lambda_1+\lambda_2),
\]
turns into
\[
1=\Big|\frac{ \lambda_1^2- \lambda_2^2}{\lambda_1^2- \lambda_3^2}\Big|
=\frac{ (x_{12} + i y_{12} )(x_1 +x_2 + i y_1 + i y_2)}{(x_{13} 
+ i y_{13} )(x_1 + x_3 +i y_1 +i y_3)}.
\]
Further,
\begin{equation}\label{3.32}
0 = \Big| \frac{ (x_{12} + i y_{12} )(x_1 +x_2 + i y_1 + i y_2)}{(x_{13} 
+ i y_{13} )(x_1 + x_3 +i y_1 +i y_3)} \Big|^2 - 1.
 \end{equation}
Noting that we have $x_{12} = x_{13}$, $x_{23} = 0$, and that 
(from \eqref{2.8}) $x_1 = -2x_2$ and $y_3 = -y_1 - y_2$, the right-hand 
side of \eqref{3.32} can now be shown (by some effort) to be equal to
\begin{equation}\label{3.33}
\frac{y_1y_{23}(3y_2^2-6x_2^2-y_{12}^2)}{(9x_2^2+y_{12}^2)(x_2^2+y_3^2)}.
\end{equation}
We may assume that $x_2 \neq 0$, because then $x_2 = x_3$ and \eqref{2.8} 
would imply $x_1 = 0$ and we would not be in case~\eqref{2.12}. 
By assumption, $y_{23} \neq 0$ in this case (otherwise $\lambda_2 = \lambda_3$ 
and so $D_3 = 0$), so this says that in case~\eqref{2.12} the solution is 
oscillatory if and only if either $y_1 = 0$ or $3y_2^2 - 6x_2^2 - y_{12}^2 = 0$.
If $y_1 = 0$, then $\lambda_1 \in \mathbb{R}$ and $y_2 = - y_3$, so $\lambda_2$ 
and $\lambda_3$ are conjugate to one another, so $I_1$ and $I_2$ are real as well. 
This is the first case \eqref{2.23} in the statement of Theorem~\ref{thirdorder02}, 
since $x_1 < x_2$ and $x_1 = -2x_2$ imply that $x_1 < 0 < x_2$.
 On the other hand, if $3y_2^2 - 6x_2^2 - y_{12}^2 = 0$ then we are in the 
second case \eqref{2.24} in the statement of Theorem~\ref{thirdorder02}.

Finally, we deal with case \eqref{2.13}, in which $x_1= x_2= x_3$ 
(and all are zero because of \eqref{2.8}). Now \eqref{2.15} becomes
\[
\varphi=\psi=\varphi_0=\psi_0=0
\]
so \eqref{2.13} turns to \eqref{2.25},
and this completes the proof.
\end{proof}


\begin{proof}[Proof of Theorem~\ref{thirdorder12}]
As in the previous two theorems, case \eqref{2.11} is impossible. 
Indeed, in case \eqref{2.11} from $d_1=d_2=0$ we obtain $d_0x_1=0$, 
which yields $d_0=0$ since here $x_1<0.$

Consider case \eqref{2.12}. In this case from $d_1=d_2=0$ we obtain
\[
k_2=d_0\lambda_1\lambda_3,\quad
k_3=d_0\lambda_1\lambda_2.
\]
so \eqref{2.12} turns into
\[
|\lambda_{13}\lambda_1\lambda_3|=
|\lambda_{12}\lambda_1\lambda_2|,\quad
x_1 < x_2 = x_3.
\]
If $\lambda_1 = 0$, then \eqref{2.8} requires that $x_2+x_3 = 0$,
 hence $x_2 = x_3 = 0$ which contradicts the condition $x_1<x_2$.
If $\lambda_1\neq 0$, then
\[
|\lambda_{13}\lambda_3|=|\lambda_{12}\lambda_2|;
\]
that is,
\[
\Big| \frac{ \lambda_2 \lambda_{12} }{\lambda_3 \lambda_{13}} \Big|^2 = 1 .
\]
This means that
\begin{equation}\label{3.34}
\frac{(x_2^2 + y_2^2)(x_{12}^2 + y_{12}^2)}{(x_3^2 + y_3^2)(x_{13}^2 
+ y_{13}^2)} - 1 = 0,\quad x_1<x_2 = x_3.
\end{equation}
In this case we know that $x_2 = x_3$ and so that $x_1 = -2x_2$ and 
$y_1 + y_2 + y_3 = 0$. Using these facts and performing some calculation, 
\eqref{3.35} becomes
\[ 
- \frac{4 y_1(y_1+2y_2)(3x_2^2+y_1^2+y_1y_2+y_2^2)}{(x_2^2 + y_1^2 
+ 2y_1y_2+y_2^2)(9x_2^2+4y_1^2+4y_1y_2+y_2^2)} = 0. 
\]
If $y_1=0$ then $\lambda_1$ is real (and negative, by \eqref{2.8} 
and \eqref{3.34}), and $y_2 = -y_3$, so $I_1$ and $I_2$ are real and 
we are in the situation given in the statement of the theorem. 
We claim that this is the only possibility. If $y_1+2y_2=0$ 
then \eqref{2.8} gives that $y_2 = y_3$, which in case~\eqref{3.35} means 
that $\lambda_2 = \lambda_3$, contradicting the assumption that $D_3 \neq 0$. 
On the other hand, if
\[ 
3x_2^2+y_1^2+y_1y_2+y_2^2 =0 
\]
then by completing the square we obtain
\[ 
3x_2^2 +\left(y_1 + \frac{y_2}{2}\right)^2 + \frac{3}{4}y_2^2 = 0 .
\]
This requires that $x_2 = y_2 = 0$. But $x_1 < x_2 = x_3 = 0$ violates
 \eqref{2.8}, so this is also impossible.

The final case is  \eqref{2.13}, in which $x_1 = x_2 = x_3 = 0,y_3=-y_1-y_2$, 
from which \eqref{2.15} becomes $\varphi_0=\psi_0=0$ and
\begin{equation}\label{3.35}
\cos(2\pi \varphi)=- \frac{2y_1^4 + 7y_1^3 y_2 + 11 y_1^2 y_2^2 + 8 y_1 y_2^3 
+ 4y_2^4}{2y_1^2 y_2 (y_1 + y_2)}
\end{equation}
which we can put in terms of the variable $\gamma = y_1/ y_2$:
\begin{equation}\label{3.36}
\cos( t y_{23}) = - \frac{ 2\gamma^4 + 7 \gamma^3 + 11 \gamma^2 
+ 8 \gamma + 4}{2\gamma^2 (\gamma+1) } = f(\gamma) .
\end{equation}
Analysis of $f(\gamma)$ reveals that $-1 \leq f(\gamma) \leq 1$ only when 
$\gamma = -2$, and $f(-2) = -1$ -- that is, \eqref{3.35} is only possible 
when $y_1 + 2y_2 = 0$, and that again gives a contradiction as it would 
imply $y_2 = y_3$ and hence $\lambda_2 = \lambda_3$ and $D_3 = 0$. 
Consequently, the case~\eqref{2.13} gives rise to no oscillatory solutions 
and the proof is complete.    
\end{proof}

As noted in Remark~\ref{higherorderremark}, similar results for higher
 order equations will depend on analysis of larger systems of exponential 
equations, which is difficult. Our results have some limited applicability:
 for instance, in degree $4$ if
\[
\sum_{j=1}^4x_j^2\neq 0
\]
then one can reduce to various instances of 
Theorem~\ref{mainthirdorderthm}. However, as with the third order 
results presented here, the most troublesome case is when all the 
$x_i$s are zero.


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\end{document}