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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 140, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/140\hfil Existence of subharmonic solutions]
{Existence of subharmonic solutions \\ to a hysteresis system \\ 
  with sinusoidal  external influence}

\author[A. M. Kamachkin, D. K. Potapov, V. V. Yevstafyeva \hfil EJDE-2017/140\hfilneg]
{Alexander M. Kamachkin, Dmitriy K. Potapov, Victoria V. Yevstafyeva}

\address{Alexander M. Kamachkin \newline
Saint Petersburg State University,
7/9, Universitetskaya nab.,
St. Petersburg, 199034, Russia}
\email{a.kamachkin@spbu.ru}

\address{Dmitriy K. Potapov \newline
Saint Petersburg State University,
7/9, Universitetskaya nab.,
St. Petersburg, 199034, Russia}
\email{d.potapov@spbu.ru}

\address{Victoria V. Yevstafyeva \newline
Saint Petersburg State University,
7/9, Universitetskaya nab.,
St. Petersburg, 199034, Russia}
\email{v.evstafieva@spbu.ru}

\dedicatory{Communicated by Vicentiu Radulescu}

\thanks{Submitted  March 21, 2017. Published May 25, 2017.}
\subjclass[2010]{34C25, 34C55, 93C15, 93C73}
\keywords{Hysteresis nonlinearity; sinusoidal external influence;
\hfill\break\indent subharmonic solutions; switching points}

\begin{abstract}
 We consider a system of ordinary differential equations with nonlinearity
 describing relay hysteresis under sinusoidal external influence.
 Theorems on sufficient conditions for the existence of subharmonic solutions
 to the system being investigated are established.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\allowdisplaybreaks

\section{Introduction and statement of problem}

Dynamics of ordinary differential equation systems with discontinuous right-hand
sides exposed to external influence is of undoubted interest.
The history of such investigations started long ago
(see, for example, \cite{minagava}).
Stable modes in relay systems are examined by iterative methods
in \cite{pokrovskiUst}.
The latest results on the solutions to second-order differential equations with
discontinuous right-hand side are published
in \cite{bonanno9, jacquemard, pot42, pot45, pot48, llibre, nyzhnyk, 
pot38, pot43, samoilenko}; the periodic solutions are considered
in \cite{jacquemard, pot45, pot48, llibre, nyzhnyk, samoilenko}.
Applied problems for these equations are discussed
in \cite{pot18, pot40}.
The existence of periodic solutions to Hamiltonian systems with periodic
influences is proved in \cite{fonda}. Lavrent'ev's problem
on separated flows in the case of non-periodic external influence is analyzed
in \cite{pot40}.
The ordinary differential equation of second-order with superlinear convex
nonlinearity is investigated in \cite{radulescu2005}.
Problems related to control of elliptic type distributed systems
with discontinuous nonlinearity are approached in \cite{pot36}.
The systems of ordinary differential equations with nonlinearity
of non-ideal relay type and external continuous influence
are studied in \cite{pot46, kam, umz, vica, vica2015}.
This work proceeds the researches above.

We consider the automatic control system of the form
\begin{equation}
\dot X=AX+BF(\sigma)+kBf(t),\quad \sigma=(\Gamma,X).
\label{ff1}
\end{equation}
Here $X\in E^d$ ($E^d$ is $d$-dimensional Euclidean space);
$A$ is a real-valued  $(d\times d)$ matrix;
$B$ and $\Gamma$ are real-valued $(d\times 1)$ matrices; $k\in {\mathbb R}$;
$f(t)=\sin(\omega t+\varphi)$, $\omega, \varphi \in {\mathbb R}$;
$(\Gamma,X)$ means the scalar product of vectors $\Gamma$ and $X$.
Ambiguous function $F$ is defined by the relations: $F(\sigma)=m_2$
while $\sigma>l_1$ and $F(\sigma)=m_1$ while $\sigma<l_2$, where
$m_1<m_2$, $l_1<l_2$ ($m_i, l_i\in \mathbb R$, $i=1,2$).

Hence function $F(\sigma)$ describes an asymmetric relay
hysteresis loop being traversed counterclockwise in plane $(\sigma,F(\sigma))$.
Nonlinearities of this kind are often used in applications
(see, e.g., \cite{macki, mayergoyz, visintin}).

Unlike \cite{pokrovskiUst}, in this paper we do not suppose that
system \eqref{ff1} is strong positive and matrix $A$ of the system
is Hurwitz.
In \cite{jacquemard, nyzhnyk}, nonlinearity $F$ corresponds to
the special case when $-m_1=m_2$ and $l_1=l_2=0$.

We pose the problem that is to find out the conditions on the parameters
of the relay hysteresis system under which there exist the periodic modes
similar to the dominant-lock mode or the subharmonic-lock mode \cite{tsypkin}.
The analogy consists only in the locking process, as it is not necessary for
the autonomous system
under considered assumptions to have the self-oscillating mode or even
a periodic solution.


 We shall say that a solution of system \eqref{ff1}
is called \emph{subharmonic} if the period of the forced oscillation be
multiple to the period of the external influence.

Thus in this paper we consider the problem on the existence
of the subharmonic solutions to the hysteresis systems of form
\eqref{ff1} with sinusoidal external influence.

\section{Approach to the problem}

First we present an approach to solving the problem for system \eqref{ff1}.
To construct the forced oscillations of system \eqref{ff1}, we use the general
solution of the system in the Cauchy form
\begin{equation}
X(t)=e^{At}X(0)+\int_0^te^{A(t-\tau)}(BF(\sigma)+kBf(\tau))d\tau.
\label{ff2}
\end{equation}

Moreover, we assume that there is $t=T_B$ such that $X(0)=X(T_B)$.
Then it follows from the solution of \eqref{ff2} that initial vector
$X_0=X(0)$ can be defined by the following expression:
\begin{equation}
X_0=\big(E-e^{AT_B}\big)^{-1}\int_0^{T_B}e^{A(T_B-\tau)}B(F(\sigma)+kf(\tau))d\tau.
\label{ff3}
\end{equation}
Therefore, using \eqref{ff2} and \eqref{ff3},  we can
formally define $T_B$-periodic solution of \eqref{ff1} as follows:
\begin{equation}
\begin{aligned}
X(t)&=e^{At}\big(E-e^{AT_B}\big)^{-1}\int_0^{T_B}e^{A(T_B-\tau)}B(F(\sigma(\tau))+
kf(\tau))d\tau \\
&\quad +\int_0^te^{A(t-\tau)}B(F(\sigma(\tau))+kf(\tau))d\tau.
\end{aligned} \label{ff4}
\end{equation}
Notice that in this case we need to know the properties
of functions $\sigma(t)$ and $f(t)$.

We use \eqref{ff4} to construct the transcendental equations
with respect to the parameters of the periodic solution, which describes
the forced oscillations of the system with the relay hysteresis
given by function $F(\sigma)$.
 	
Let points $X_1$ and $X_2$ belong to the periodic trajectory and
$(\Gamma,X_1)=l_1$, $(\Gamma, X_2)=l_2$. In time $T_B$
the image point returns to the initial position. Then we have
\begin{equation}
\begin{gathered}
X_1=e^{A\tau_1}X_2+\int_0^{\tau_1}e^{A(\tau_1-\tau)}B(m_2+kf(\tau))d\tau, \\
X_2=e^{A(T_B-\tau_1)}X_1+\int_{\tau_1}^{T_B}e^{A(T_B-\tau)}B(m_1+kf(\tau))d\tau,
\end{gathered} \label{ff5}
\end{equation}
where $\tau_1$ is the time it takes the image point to transit
from $X_2$ to $X_1$, $\tau_2$ is the time for return transition from
$X_1$ to $X_2$. Note that $\tau_2=T_B-\tau_1$.

From \eqref{ff5}, we have
\begin{equation}
\begin{aligned}
X_1&=\big(E-e^{AT_B}\big)^{-1}\Big(e^{A\tau_1}
\int_{\tau_1}^{T_B}e^{A(T_B-\tau)}B(m_1+kf(\tau))d\tau\\
&\quad +\int_0^{\tau_1}e^{A(\tau_1-\tau)}B(m_2+kf(\tau))d\tau\Big)\\
&=\big(E-e^{AT_B}\big)^{-1}Q_1
\end{aligned} \label{ff6}
\end{equation}
and similarly
\begin{equation}
\begin{aligned}
X_2&=\big(E-e^{AT_B}\big)^{-1}\Big(e^{A(T_B-\tau_1)}
\int_0^{\tau_1}e^{A(\tau_1-\tau)}B(m_2+kf(\tau))d\tau\\
&\quad +\int_{\tau_1}^{T_B}e^{A(T_B-\tau)}B(m_1+kf(\tau))d\tau\Big) \\
&=\big(E-e^{AT_B}\big)^{-1}Q_2.
\end{aligned} \label{ff7}
\end{equation}

Using the switching conditions and equalities \eqref{ff6}, \eqref{ff7},
we construct the transcendental equations for seeking $\tau_1$ and $\tau_2$,
namely,
\begin{equation}
\begin{gathered}
l_1=(\Gamma,\big(E-e^{AT_B}\big)^{-1}Q_1), \\
l_2=(\Gamma,\big(E-e^{AT_B}\big)^{-1}Q_2).
\end{gathered} \label{ff8}
\end{equation}

Let there exist parameters of system \eqref{ff1} such that
equations \eqref{ff8} is solvable for $\tau_1>0 $, $\tau_2>0$, where
$\tau_1+\tau_2=T_B $. Also, let the solutions of \eqref{ff8} satisfy
system \eqref{ff5}, where $X_1$ and $X_2$ are defined
by \eqref{ff6} and \eqref{ff7} respectively.
Then it is possible to state that the problem at issue is solved.

Let us remark that the solutions of system \eqref{ff8} can be a countable
set. Whence system \eqref{ff1}, generally speaking, can have
a lot of subharmonic solutions.

\section{Real nonzero distinct roots for $d=2$}

Let us write down equations \eqref{ff8} for the case when $d=2$ and
characteristic equation $|A-\lambda E|=0$ has two real nonzero distinct roots
$\lambda_1$ and $\lambda_2$. We perform the nonsingular linear transformation
of system \eqref{ff1} with the matrix of the special
form \cite{kam, umz, vica, vica2015}.
In this case, we have
$B=\begin{pmatrix} 1 \\ 1 \end{pmatrix}$,
$$
e^{At}=\begin{pmatrix} e^{\lambda_1 t} & 0\\ 0
& e^{\lambda_2 t}\end{pmatrix}, \quad
\big(E-e^{AT_B}\big)^{-1}=
\begin{pmatrix} (1-e^{\lambda_1 T_B})^{-1}
& 0\\ 0 & (1-e^{\lambda_2 T_B})^{-1}\end{pmatrix}.
$$
After the transformation, here we return to the original notations for the matrices.
Let $Q_i=\begin{pmatrix} q_1^i\\ q_2^i\end{pmatrix}$, where $i=1,2$.
Component $q_1^1$ is defined by the  equation
\begin{align*}
q_1^1&=\frac{m_1}{\lambda_1}e^{\lambda_1\tau_1}
\left(-1+e^{\lambda_1 (T_B-\tau_1)}\right)
+ke^{\lambda_1 (T_B+\tau_1)} \Big(\frac{-\lambda_1}{\lambda_1^2+\omega^2}
 e^{-\lambda_1 T_B} \sin(\omega T_B+\varphi) \\
&\quad -\frac{\omega}{\lambda_1^2+\omega^2}e^{-\lambda_1 T_B}
\cos (\omega T_B+\varphi)-
\frac{-\lambda_1}{\lambda_1^2+\omega^2}e^{-\lambda_1\tau_1}
\sin(\omega\tau_1+\varphi) \\
&\quad +\frac{\omega}{\lambda_1^2+\omega^2}e^{-\lambda_1\tau_1}
\cos (\omega\tau_1+\varphi)\Big) 
 -\frac{m_2}{\lambda_1}\left(1-e^{\lambda_1\tau_1}\right) \\
&\quad + ke^{\lambda_1\tau_1}\Big(\frac{-\lambda_1}{\lambda_1^2
 +\omega^2}e^{-\lambda_1\tau_1}\sin(\omega\tau_1+\varphi)
 -\frac{\omega}{\lambda_1^2+\omega^2}e^{-\lambda_1\tau_1}
\cos (\omega\tau_1+\varphi) \\
&\quad -\frac{-\lambda_1}{\lambda_1^2+\omega^2}\sin\varphi
+\frac{\omega}{\lambda_1^2+\omega^2}\cos \varphi\Big).
\end{align*}
Component $q_2^1$ is defined by the similar equality
\begin{align*}
q_2^1&=\frac{m_1}{\lambda_2}e^{\lambda_2\tau_1}
\left(-1+e^{\lambda_2 (T_B-\tau_1)}\right)
+ke^{\lambda_2 (T_B+\tau_1)}\Big(\frac{-\lambda_2}{\lambda_2^2
 +\omega^2}e^{-\lambda_2 T_B}
\sin(\omega T_B+\varphi)\\
&\quad -\frac{\omega}{\lambda_2^2+\omega^2}e^{-\lambda_2 T_B}
\cos (\omega T_B+\varphi)-
\frac{-\lambda_2}{\lambda_2^2+\omega^2}e^{-\lambda_2\tau_1}
\sin(\omega\tau_1+\varphi) \\
&\quad +\frac{\omega}{\lambda_2^2+\omega^2}e^{-\lambda_2\tau_1}
\cos (\omega\tau_1+\varphi)\Big)
-\frac{m_2}{\lambda_2}\left(1-e^{\lambda_2\tau_1}\right) \\
&\quad +ke^{\lambda_2\tau_1}
\Big(\frac{-\lambda_2}{\lambda_2^2+\omega^2}e^{-\lambda_2\tau_1}
\sin(\omega\tau_1+\varphi)
-\frac{\omega}{\lambda_2^2+\omega^2}e^{-\lambda_2\tau_1}
\cos (\omega\tau_1+\varphi) \\
&\quad -\frac{-\lambda_2}{\lambda_2^2+\omega^2}\sin\varphi+
\frac{\omega}{\lambda_2^2+\omega^2}\cos \varphi\Big).
\end{align*}
Now, using the coefficients of the original system, we write down the first
transcendental equation of \eqref{ff8} for $\gamma_1=0$.
We can afford these additional assumptions
owing to the choice of the linear transformation for the original system.
Further, we are looking for the subharmonic solutions.

Here and elsewhere $\gamma_i$ ($i=1,2$) are the components of vector
$\Gamma=\begin{pmatrix} \gamma_1 \\ \gamma_2 \end{pmatrix}$.
We emphasize especially that vector $\Gamma$ is obtained as a consequence of
applying this transformation.
From here we obtain
\begin{equation}
\begin{aligned}
\frac{l_1}{\gamma_2}\left(1-e^{\lambda_2 T_B}\right)
&=\frac{1}{\lambda_2}(m_2-m_1)
e^{\lambda_2\tau_1}+\frac{m_1}{\lambda_2}e^{\lambda_2 T_B}
 -\frac{m_2}{\lambda_2} +k\left(e^{\lambda_2 T_B}-1\right) \\
&\quad\times \Big(\frac{\lambda_2}{\lambda_2^2+\omega^2}
\sin(\omega\tau_1+\varphi)+\frac{\omega}{\lambda_2^2+\omega^2}
\cos (\omega\tau_1+\varphi)\Big).
\end{aligned} \label{ff11}
\end{equation}

The second equation for $\tau_2$ can be obtained by similar way.
Values $\tau_1$ and $\tau_2$ are related by $\tau_2=T_B-\tau_1$, where
$T_B$ is the period of forced oscillations that, in particular, 
may be equal to the period of function $f(t)$.
First we write out components $q_1^2$ and $q_2^2$ of vector $Q_2$,
\begin{align*}
q_1^2&=-\frac{m_2}{\lambda_1}e^{\lambda_1(T_B-\tau_1)}
\left(1-e^{\lambda_1\tau_1}\right)+ke^{\lambda_1 T_B}
\Big(\frac{-\lambda_1}{\lambda_1^2+\omega^2}e^{-\lambda_1\tau_1}
\sin(\omega\tau_1+\varphi) \\
&\quad -\frac{\omega}{\lambda_1^2+\omega^2}e^{-\lambda_1\tau_1}
\cos (\omega\tau_1+\varphi)
-\frac{-\lambda_1}{\lambda_1^2+\omega^2}\sin\varphi+
\frac{\omega}{\lambda_1^2+\omega^2}\cos \varphi\Big) \\
&\quad +\frac{m_1}{\lambda_1}\left(-1+e^{\lambda_1 (T_B-\tau_1)}\right)+
ke^{\lambda_1\tau_1}
\Big(\frac{-\lambda_1}{\lambda_1^2+\omega^2}e^{-\lambda_1 T_B}
\sin(\omega T_B+\varphi) \\
&\quad -\frac{\omega}{\lambda_1^2+\omega^2}e^{-\lambda_1 T_B}
\cos (\omega T_B+\varphi)-
\frac{-\lambda_1}{\lambda_1^2+\omega^2}e^{-\lambda_1\tau_1}
\sin(\omega\tau_1+\varphi) \\
&\quad + \frac{\omega}{\lambda_1^2+\omega^2}e^{-\lambda_1\tau_1}
\cos (\omega\tau_1+\varphi)\Big)
\end{align*}
and
\begin{align*}
q_2^2&=-\frac{m_2}{\lambda_2}e^{\lambda_2(T_B-\tau_1)}
\left(1-e^{\lambda_2\tau_1}\right)+ke^{\lambda_2 T_B}
\Big(\frac{-\lambda_2}{\lambda_2^2+\omega^2}e^{-\lambda_2\tau_1}
\sin(\omega\tau_1+\varphi) \\
&\quad -\frac{\omega}{\lambda_2^2+\omega^2}e^{-\lambda_2\tau_1}
\cos (\omega\tau_1+\varphi)-\frac{-\lambda_2}{\lambda_2^2+\omega^2}
\sin\varphi+\frac{\omega}{\lambda_2^2+\omega^2}\cos \varphi\Big) \\
&\quad+\frac{m_1}{\lambda_2}\left(-1+e^{\lambda_2 (T_B-\tau_1)}\right)+
ke^{\lambda_2\tau_1}
\Big(\frac{-\lambda_2}{\lambda_2^2+\omega^2}e^{-\lambda_2 T_B}
\sin(\omega T_B+\varphi) \\
&\quad -\frac{\omega}{\lambda_2^2+\omega^2}e^{-\lambda_2 T_B}
\cos (\omega T_B+\varphi)-
\frac{-\lambda_2}{\lambda_2^2+\omega^2}e^{-\lambda_2\tau_1}
\sin(\omega\tau_1+\varphi)\\
&\quad +\frac{\omega}{\lambda_2^2+\omega^2}e^{-\lambda_2\tau_1}
\cos (\omega\tau_1+\varphi)\Big).
\end{align*}
Then the second transcendental equation of \eqref{ff8} takes the form
\begin{equation}
\begin{aligned}
&\frac{l_2}{\gamma_2}\left(1-e^{\lambda_2 T_B}\right) \\
&=\frac{1}{\lambda_2}(m_1-m_2)e^{\lambda_2 (T_B-\tau_1)}
 +\frac{m_2}{\lambda_2}e^{\lambda_2 T_B}-\frac{m_1}{\lambda_2} \\
&\quad +k\left(1-e^{\lambda_2 (T_B-\tau_1)}\right)
 \Big(\frac{\lambda_2}{\lambda_2^2+\omega^2} \sin(\omega\tau_1+\varphi)
 +\frac{\omega}{\lambda_2^2+\omega^2} \cos (\omega\tau_1+\varphi)\Big) \\
&\quad + k\left(e^{\lambda_2 T_B}-e^{\lambda_2 (\tau_1-T_B)}\right)
\Big(\frac{\lambda_2}{\lambda_2^2+\omega^2}\sin\varphi+
\frac{\omega}{\lambda_2^2+\omega^2}\cos \varphi\Big).
\end{aligned} \label{ff14}
\end{equation}
Next we solve equation \eqref{ff11} with respect to the expression in its right
side in brackets
\begin{equation}
\begin{aligned}
&k\left(\frac{\lambda_2}{\lambda_2^2+\omega^2}\sin(\omega\tau_1+\varphi)+
\frac{\omega}{\lambda_2^2+\omega^2}\cos (\omega\tau_1+\varphi)\right) \\
&=\left(e^{\lambda_2 T_B}-1\right)^{-1} \Big(\frac{l_1}{\gamma_2}
\left(1-e^{\lambda_2 T_B}\right)-\frac{1}{\lambda_2}(m_2-m_1)e^{\lambda_2\tau_1}\\
&\quad -\frac{m_1}{\lambda_2}e^{\lambda_2 T_B}+\frac{m_2}{\lambda_2}\Big).
\end{aligned}\label{ff15}
\end{equation}
Exactly the same expression is in the fourth term of equation \eqref{ff14}.
We substitute expression \eqref{ff15} in equation \eqref{ff14}, then denote
$y=e^{\lambda_2\tau_1}$ (assuming that $\tau_1>0$) and group the coefficients
at $y^2, y^1$ and $y^0$ respectively. We have
\begin{equation}
ay^2+by+c=0, \label{ff16}
\end{equation}
where coefficients $a$, $b$ and $c$ are determined by the following equations:
\begin{equation}
\begin{gathered}
a=\frac{m_2-m_1}{\lambda_2\left(e^{\lambda_2 T_B}-1\right)}-
\frac{k}{e^{\lambda_2 T_B}\sqrt{\lambda_2^2+\omega^2}}\sin(\varphi+\delta), \\
\begin{aligned}
b&=\frac{ke^{\lambda_2 T_B}}{\sqrt{\lambda_2^2+\omega^2}}\sin(\varphi+\delta)+
\frac{l_2}{\gamma_2}\left(1-e^{\lambda_2 T_B}\right)-
\Big(\frac{m_2}{\lambda_2}e^{\lambda_2 T_B}-\frac{m_1}{\lambda_2}\Big)+
\frac{l_1}{\gamma_2} \\
&\quad + \frac{1}{\lambda_2\left(e^{\lambda_2 T_B}-1\right)}
\left(m_1e^{\lambda_2 T_B}-m_2\right)-
\frac{e^{\lambda_2 T_B}(m_2-m_1)}{\left(e^{\lambda_2 T_B}-1\right)\lambda_2},
\end{aligned}  \\
c=-\frac{1}{\lambda_2}e^{\lambda_2 T_B}(m_1-m_2)-\frac{l_1}{\gamma_2}e^{\lambda_2 T_B}+
\frac{e^{\lambda_2 T_B}}{e^{\lambda_2 T_B}-1}
\Big(\frac{m_2}{\lambda_2}-\frac{m_1}{\lambda_2}e^{\lambda_2 T_B}\Big),
\end{gathered}\label{ff17}
\end{equation}
where $\delta=\arctan(\omega/\lambda_2)$.
If the period of the external influence defined by function $f(t)$ is known,
then we also consider value $T_B$ in \eqref{ff17} as known.
This follows from the agreement to search the subharmonic solutions.
We suppose that $T_B=nT$, where $n\in\mathbb N$ and $T$ is the period
of function $f(t)$. To determine $y$ as positive solution
of equation \eqref{ff16}, we also assume that $a=a(T_B)$, $b=b(T_B)$ and
$c=c(T_B)$. In particular, solving equation \eqref{ff16}, where $T_B=T$,
it is necessary to keep in mind that we are only interested in solutions $y$
such that $\tau_1=\lambda_2^{-1}{\rm ln}y<T$.
In addition, roots of equation \eqref{ff16}, i.e.
$y_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$, should be real.
So, we have to impose the condition on the coefficients of equation \eqref{ff16}
\begin{equation}
b^2-4ac\geq 0.
\label{ff18}
\end{equation}

Inequality \eqref{ff18} is a condition that determines the existence domains
for solution $\tau_1$ and, consequently, for the periodic solutions of
the original system in its multidimensional parameter space.
Remind that we are interested in the positive solution $\tau_1$ satisfying
the condition $0<\tau_1<T_B$.
Therefore, if $\lambda_2>0$, then at least one of roots $y_1$ or $y_2$ should
be greater than unity. If $\lambda_2<0$, then at least one of the same roots
should be greater than zero and less than unity. In short, when
condition \eqref{ff18} is valid, the following conditions should also hold:
\begin{equation}
\begin{gathered}
\text{if  $\lambda_2>0$, then  at  least one  of  roots satisfies } y_i>1, \\
\text{if  $\lambda_2<0$,  then at  least  one  of roots satisfies }  0<y_i<1.
\end{gathered}\label{ff19}
\end{equation}
If root $\tau_1$ is found, then by given $T_B$, one can find
$\tau_2$. This means that sufficient conditions \eqref{ff18} and
\eqref{ff19} for parameters $a,b$ and $c$ of equation \eqref{ff16} and,
consequently, for the parameters of the original system, guarantee the existence
of a periodic mode (cyclic behavior).
After substituting $T_B=nT$ and $\tau_1$ in \eqref{ff6}
or \eqref{ff7}, we obtain uniquely the switching points of periodic solutions
in the phase plane, namely, point $ X_1 $ that belongs to switching line
$\sigma=l_1$ or respectively point $X_2$ that belongs to $\sigma=l_2$.

After replacing $T_B$ by $nT$, the solution of equation \eqref{ff16}
is associated with searching the periodic modes similar
to the dominant-lock or subharmonic-lock ones.

Next we formulate the results obtained above as a theorem on
the sufficient condition for the existence of periodic solutions
to \eqref{ff1}.

\begin{theorem} \label{thm1} 
 Let by a nonsingular transformation,
the initial automatic control system be reduced to the form of system \eqref{ff1},
where matrix
$A=\begin{pmatrix} \lambda_1  & 0\\ 0
& \lambda_2 \end{pmatrix}$,
vector $B=\begin{pmatrix} 1 \\ 1 \end{pmatrix}$,
$f(t)=\sin(\omega t+\varphi)$,
function $F(\sigma)$ describes relay hysteresis, $\sigma=(\Gamma,X)$,
$\Gamma=\begin{pmatrix} \gamma_1 \\ \gamma_2 \end{pmatrix}$ and
$\gamma_1=0$.
Let conditions \eqref{ff18},\eqref{ff19} hold and equation \eqref{ff16}
be solvable for $\tau_1>0$. Then the initial automatic control
system has at least one $T_B$-periodic solution, where $T_B=nT$,
$n\in\mathbb N$, $T$ is the period of function $f(t)$.
\end{theorem}

If the discriminant of equation \eqref{ff16} equals zero, then there exists
one root $y$ of \eqref{ff16}. If either $y>1$ for $\lambda_2>0$ or $y<1$
for $\lambda_2<0$, then it means that there exists a unique solution $\tau_1$
and, therefore, after substituting $\tau_1$ in \eqref{ff6} and
\eqref{ff7}, we obtain switching points $X_1$ and $X_2$ of the periodic solution.
Thus the following theorem holds.

\begin{theorem} \label{thm2} 
 Let the conditions of Theorem \ref{thm1} be satisfied.
Then the number of roots $y_i$ of equation \eqref{ff16} determines
the number of periodic solutions of \eqref{ff1} if conditions \eqref{ff18}
and \eqref{ff19} hold. System \eqref{ff1} can not have more than two
periodic solutions for $d=2$.
\end{theorem}

We can formulate a statement similar to Theorem \ref{thm1} for the case when
$\gamma_1\neq 0$ and $\gamma_2=0$. Condition $\gamma_1=0$ (or $\gamma_2=0$)
allows one to reduce the system of transcendental algebraic equations for
searching $\tau_1$ and $\tau_2$, where $T_B=\tau_1+\tau_2$ is given, to
the simple quadratic equation that should have the roots satisfying
condition \eqref{ff19}. In the general case, it is impossible to obtain
analytically the solution of \eqref{ff8} even for two-dimensional
system \eqref{ff1}. However, for $\gamma_1=0$ these equations permit one
to set conditions on the existence of the periodic solutions describing
the forced oscillations such that the frequency equals the frequency
of the external influence or is $1/n$ part of this frequency.

\section{Real nonzero multiple roots for $d=2$}

Let us consider the case when the roots of the characteristic equation are
real nonzero multiple. Suppose that the initial automatic system is reduced
to the system with matrix
$A=\begin{pmatrix} \lambda& 0\\ 1
&\lambda\end{pmatrix}$, vector $B=\begin{pmatrix} 1\\ 0\end{pmatrix}$ 
\cite{derusso},
and, as before, $f(t)=\sin(\omega t+\varphi)$.
Let $\gamma_2=0$. Next we write down system \eqref{ff8}.
We get the matrix
$$
\big(E-e^{AT_B}\big)^{-1}=
\begin{pmatrix} \left(1-e^{\lambda T_B}\right)^{-1}
& 0\\ te^{\lambda T_B}\left(1-e^{\lambda T_B}\right)^{-2}
& \left(1-e^{\lambda T_B}\right)^{-1}\end{pmatrix}.
$$
The first component of vector $Q_1$ has the form
\begin{align*}
q_1^1&=e^{\lambda (T_B+\tau_1)}\int_{\tau_1}^{T_B}e^{-\lambda\tau}
(m_1+k\sin(\omega\tau+\varphi))d\tau \\
&\quad + e^{\lambda\tau_1}\int_0^{\tau_1}
e^{-\lambda\tau}(m_2+k\sin(\omega\tau+\varphi))d\tau,
\end{align*}
and its second component is defined as follows:
\begin{align*}
 q_2^2&=e^{\lambda (T_B+\tau_1)}\int_{\tau_1}^{T_B}
 \Big((T_B+\tau_1)e^{-\lambda\tau}(m_1+k\sin(\omega\tau+\varphi)) \\
 &\quad -\tau e^{-\lambda\tau}(m_1+k\sin(\omega\tau+\varphi))\Big)d\tau \\
&\quad + e^{\lambda\tau_1}\int_0^{\tau_1}
 \Big(\tau_1 e^{-\lambda\tau}(m_2+k\sin(\omega\tau+\varphi)) 
 -\tau e^{-\lambda\tau}(m_2+k\sin(\omega\tau+\varphi))\Big)d\tau.
\end{align*}
This means that for $\gamma_2=0$ the first equation of \eqref{ff8} 
takes the form
\begin{equation}
l_1=\gamma_1\left(1-e^{\lambda T_B}\right)^{-1}q_1^1.\label{ff20}
\end{equation}

After the canonical transformation  under the condition $T_B=nT$, $n\in\mathbb N$,
equation \eqref{ff20} can be rewritten as 
\begin{equation}
\begin{aligned}
\frac{l_1}{\gamma_1}\left(1-e^{\lambda T_B}\right)
&=\frac{1}{\lambda}(m_2-m_1)
e^{\lambda\tau_1}+\frac{m_1}{\lambda}e^{\lambda T_B}-\frac{m_2}{\lambda}+
k\left(e^{\lambda T_B}-1\right) \\
&\quad\times \Big(\frac{\lambda}{\lambda^2+\omega^2}
\sin(\omega\tau_1+\varphi)+\frac{\omega}{\lambda^2+\omega^2}
\cos (\omega\tau_1+\varphi)\Big).
\end{aligned}\label{ff21}
\end{equation}

Note that equation \eqref{ff21} differs from equation \eqref{ff11}
only the denotation of the eigenvalue, namely, $\lambda_2$ is replaced by
$\lambda$.
Now let us consider the second equation of \eqref{ff8}.
We have the first component of vector $Q_2$,
\begin{align*}
q_1^2&=e^{\lambda T_B}\int_0^{\tau_1}e^{-\lambda\tau}
(m_2+k\sin(\omega\tau+\varphi))d\tau \\
&\quad +e^{\lambda T_B}\int_{\tau_1}^{T_B}
e^{-\lambda\tau}(m_1+k\sin(\omega\tau+\varphi))d\tau
\end{align*}
and the second component of vector $Q_2$,
\begin{align*}
q_2^2&=e^{\lambda T_B}\int_0^{\tau_1}
\left(T_Be^{-\lambda\tau}(m_2+k\sin(\omega\tau+\varphi))
-\tau e^{-\lambda\tau}(m_2+k\sin(\omega\tau+\varphi))\right)d\tau \\
&\quad +e^{\lambda T_B}\int_{\tau_1}^{T_B}
\left(T_B e^{-\lambda\tau}(m_1+k\sin(\omega\tau+\varphi))
-\tau e^{-\lambda\tau}(m_1+k\sin(\omega\tau+\varphi))\right)d\tau.
\end{align*}

Then taking into account the form of vector $\Gamma$,
we obtain the second equation of \eqref{ff8} as follows:
\begin{equation}
\begin{aligned}
&\frac{l_2}{\gamma_1}\left(1-e^{\lambda T_B}\right)\\
&=\frac{1}{\lambda}(m_1-m_2)e^{\lambda(T_B-\tau_1)}
 +\frac{m_2}{\lambda}e^{\lambda T_B}-\frac{m_1}{\lambda} \\
&\quad +k\left(1-e^{\lambda (T_B-\tau_1)}\right)
 \Big(\frac{\lambda}{\lambda^2+\omega^2}
\sin(\omega\tau_1+\varphi)+\frac{\omega}{\lambda^2+\omega^2}
\cos (\omega\tau_1+\varphi)\Big) \\
&\quad +k\left(e^{\lambda T_B}-e^{\lambda (\tau_1-T_B)}\right)
\Big(\frac{\lambda}{\lambda^2+\omega^2}
\sin\varphi+\frac{\omega}{\lambda^2+\omega^2}\cos \varphi\Big).
\end{aligned}\label{ff22}
\end{equation}

Therefore, equation \eqref{ff22} differs from equation \eqref{ff14}
by replacing $\lambda_2$ to $\lambda$.
Then the equation of form \eqref{ff16} can be obtained
from the system of transcendental equations \eqref{ff21}, \eqref{ff22} if
$\lambda_2$ is replaced by $\lambda$ in formulas \eqref{ff17} for
defining coefficients $a=a(T_B)$, $b=b(T_B)$, and $c=c(T_B)$.
In this case, $\delta$ is replaced by $\arctan(\omega/\lambda)$
in the formulas for defining $a=a(T_B)$ and $b=b(T_B)$. It is also necessary
to replace the root of the characteristic equation $\lambda_2$ by $\lambda$
under \eqref{ff18} and \eqref{ff19}. Then if \eqref{ff18} holds, we require
the following:
\begin{equation}
\begin{gathered}
\text{if $\lambda>0$,  then  at  least  one  of  roots satisfies } y_i>1,\\
\text{if $\lambda<0$,  then  at  least  one  of  roots satisfies }  y_i<1.
\end{gathered}\label{ff23}
\end{equation}

We now formulate an analogue of Theorem \ref{thm1} on the sufficient condition for the
existence of periodic solutions to~\eqref{ff1}.

\begin{theorem} \label{thm3} 
 Let by a nonsingular linear transformation, the initial
automatic control system be reduced to the form
\begin{gather*}
\dot x_1=\lambda x_1+F(\sigma)+k\sin(\omega t+\varphi),\\
\dot x_2=x_1+\lambda x_2.
\end{gather*}
Here function $F(\sigma)$ describes relay hysteresis, $\sigma=(\Gamma,X)$,
where $\Gamma=\begin{pmatrix} \gamma_1 \\ \gamma_2 \end{pmatrix}$ and 
$\gamma_2=0$.
Let conditions \eqref{ff18}, \eqref{ff23} hold and equation \eqref{ff16}
be solvable for $\tau_1>0$. Then the initial automatic control system
has at least one $T_B$-periodic solution, where $T_B=nT$,
$n\in\mathbb N$, $T$ is the period of function $f(t)=\sin(\omega t+\varphi)$.
\end{theorem}

Thus, in the case of the Jordan block, condition $\gamma_2=0$ makes it also
possible to reduce the problem on existence of periodic solutions to the
problem on resolvability of the algebraic equation obtained for the case of two
distinct roots of the characteristic equation.

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\end{document}