# Periodic Solutions in a Simple Delay Differential Equation

Anatoli Ivanov  Sergiy Shelyag

###### Abstract

Simple form scalar differential equation with delay and nonlinear negative
periodic feedback is considered. The existence of several types of slowly
oscillating periodic solutions is shown with the same and double periods of
the feedback coefficient. The periodic solutions are built explicitly in the
case of piecewise constant nonlinearities involved. The periodic dynamics are
shown to persist under small perturbations of the equation which make it
smooth. The theoretical results are verified by extensive numerical
simulations.

###### keywords:

delay differential equations; periodic negative feedback; slowly oscillating
solutions; periodic solutions; piecewise constant nonlinearities; explicit
piecewise affine solutions; reduction to interval maps

[inst1]organization=Department of Mathematics,addressline=Pennsylvania State
University, country=USA

[inst2]organization=College of Science and Engineering,addressline=Flinders
University, country=Australia

##  1 Introduction

The simple form delay differential equation (DDE)

$x^{\prime}(t)=-\mu x(t)+f(x(t-\tau))$ (1)

represents a broad variety of complex dynamical behaviors that can happen in
delay differential models. It is also one of the most studied equations in the
field exhibiting the range of diverse dynamics from global stability of
equilibria to instability and existence of periodic solutions and chaotic
behaviours (in contrast to analogous ordinary differential equations). Basic
and fundamental facts about the DDE (1) can be found in many sources, see e.g.
monographs [2, 3, 10, 19], among others.

Equation (1) is also extensively used as a mathematical model of a variety of
real world phenomena, most notably in biological applications; see e.g. the
monographs [2, 6, 14, 19] and further references therein. Among the well known
and most studied models of type (1) are the Mackey-Glass physiological
equations [6, 16], Wazewska-Lasota blood cell model [21], Nicholson’s blowfly
equations [7, 17], and several others [2, 14, 19].

Some of the more accurate and adequate mathematical models are explicitly
time-dependent, which take into account certain intrinsic periodicity factors,
such as e.g. circadian rhythms and seasonal changes. Examples of such
differential delay models can be found in publications [1, 5, 6, 14, 15, 22].
Therefore, a natural generalization of equation (1) is the following DDE

$x^{\prime}(t)=-\mu(t)x(t)+a(t)f(x(t-\tau)),$ (2)

where $\mu(t)$ and $a(t)$ are $T$-periodic functions, and the nonlinearity
$f(x)$ satisfies appropriate feedback conditions. An important theoretical and
applied question then arises whether the DDE (2) admits periodic solutions,
which are induced by the periodicity in the coefficients $\mu$ and $a$.
Therefore, the mathematical problem to address is to derive conditions under
which equation (2) possesses a nontrivial $T$-periodic solution. This research
work is our initial attempt to answer this question for some specific partial
cases of the equation under consideration.

This paper deals with the problem of existence of periodic solutions for the
simplest possible scalar differential delay equation of the type (2) when
$\mu(t)\equiv 0$,

$x^{\prime}(t)=a(t)f(x(t-\tau)),$ (3)

and where $f(x)$ is a continuous function satisfying the negative feedback
assumption $x\cdot f(x)<0,\;\forall x\neq 0,$ the coefficient $a(t)\geq 0$ is
a continuous periodic function with the period $T$ satisfying $T>\tau,$ and
$\tau>0$ is a delay. The principal problem we are addressing in this work is
to derive conditions on the parameters and nonlinearities of equation (3)
which would yield the existence of periodic solutions with the same period $T$
as the coefficient $a(t).$ Though there are some results on the existence of
periodic solutions for the DDEs with periodic coefficients none of them is
applicable to the simplest form equation (3), to the best of our assessment
and knowledge. In many cases that are published in the research literature the
generalization from the autonomous case to the period one leads to the loss of
the steady positive equilibrium. In some others the equilibrium state
persists, however, the sufficient conditions for the existence of periodic
solutions cannot distinguish between the nontrivial periodic solutions and the
steady state. See more details on such relevant results and further references
in e.g. [1, 4, 5, 15, 22].

We are stating and dealing with the problem of existence of nontrivial
periodic solutions in the simple form DDE (3). Due to the negative feedback
assumption on $f$ this equation always admits the trivial constant solution
$x(t)\equiv 0$. The mathematical problem is to find periodic solutions which
are distinct from the trivial one. This is a difficult problem which is
largely not addressed on the general theoretical level. One way to pave a path
for a systematic analysis in this direction is to consider model equations of
the form given by equation (3) with simple $f$ and $a$.

We first construct the periodic solutions in an explicit form by using
piecewise constant functions (both the nonlinearity $f$ and the coefficient
$a$). Then we modify both functions to be continuous and “close” to the
piecewise constant ones, by “smoothing” them in a small neighborhood of the
discontinuity set (they can also be made of the $C^{\infty}$ type). The
periodic dynamics and their stability are shown to persist under such small
perturbations.

The paper is structured as follows. First section “Preliminaries” contains
basics of the DDE (3) and preliminary results from our recent conference
papers [11, 12] necessary for the exposition in subsequent sections. Section 3
deals with periodic solutions of equation (3) explicitly constructed for the
initially piecewise constant nonlinearity $f$ and the coefficient $a$. The
dynamics on a set of slowly oscillating solutions are reduced to that for
induced one-dimensional maps, thus proving both the existence of periodic
solutions and the types of their stability. In Section 3 it is shown that the
dynamics of the interval maps and the periodic solutions from section 2
persist when the defining functions $f$ and $a$ of equation (3) are replaced
by close to them continuous or smoothed functions. The arguments of small
regular perturbations of dynamics by DDE (3) are used, which are behind such
“small modifications”. The theoretical derivations and exact analytical
calculations are verified by extensive numerical simulations.

##  2 Preliminaries

We focus in this work on the particular case $\mu=0$ and $\tau=1$ when
equation (1) becomes

$x^{\prime}(t)=a(t)f(x(t-1)).$ (4)

Note that the case of general delay $\tau>0$ is easily reduced to the standard
consideration $\tau=1$ by the time scaling $t=\tau s$.

Given an initial function $\varphi(s)\in C([-1,0],\mathbb{R}):=\mathbb{X}$ in
the standard phase space $\mathbb{X}$ the corresponding solution
$x=x(t,\varphi)$ to equation (4) is easily found for all $t\geq 0$ by
consecutive forward integration (the step method). Let $S^{t}(\varphi)$ be the
shift in $\mathbb{X}$ by time $t$ along the solutions of delay differential
equation (4), that is $S^{t}(\varphi)=x(t+s,\varphi),s\in[-1,0]$. It is a
straightforward observation that a fixed point $\varphi_{0}$ of the shift by
period $T$, $S^{T}(\varphi_{0})=\varphi_{0}$, gives rise to a periodic
solution $x(t,\varphi_{0})$ of the DDE (4). Under such an interpretation, the
zero solution $x(t)\equiv 0$ is a result of the trivial fixed point
$\varphi_{0}(s)\equiv 0,s\in[-1,0]$, which always exists for DDE (4), due to
the negative feedback assumption. The mathematical problem is to derive
conditions for the existence of non-trivial fixed points of the shift operator
$S^{T}$, which is a principal objective of this work.

A solution is called oscillating if it has an infinite sequence of zeros
$\\{t_{n}\\}\to\infty,x(t_{n})=0,n\in\mathbb{N}$, and $x(t)$ is not an
identical zero eventually (on any interval of the form $[t_{0},\infty$)). An
oscillating solution is called slowly oscillating, if the sequence of zeros is
such that the distance between consecutive zeros is greater than the delay,
i.e. $t_{n+1}-t_{n}>1,\forall n\in\mathbb{N}.$

Sufficient conditions for the oscillation of all solutions to equation (4) are
well known, see e.g. [8] and further references therein. With the negative
feedback assumption $xf(x)<0,x\neq 0,$ and the non-negativity of the
coefficient $a(t)\geq 0,a(t)\not\equiv 0,$ they are that either
$f^{\prime}(0)$ is sufficiently large, $|f^{\prime}(0)|\geq f_{1}>0,$ and the
coefficient $a$ is separated from zero, $a(t)\geq a_{1}>0$, or
$f^{\prime}(0)\neq 0$ and $a_{1}>0$ is large enough (in both cases the product
$|f_{1}\cdot a_{1}|$ must be sufficiently large). These conditions are
satisfied for our considerations to follow in next sections, where the
functions $f$ and $a$ are smoothed continuous ones derived from the initial
piecewise constant functions (the piecewise constant case can be viewed as a
limit of the continuous one).

We deal in this paper with the slowly oscillating solutions to equation (4).
The slowly oscillating solutions are obtained when the initial functions
belong to one of the two sign definite cones. Introduce the following two
standard subsets of initial functions

$\displaystyle
K_{-}:=\\{\varphi\in\mathbb{X}\;|\;\varphi(s)<0\;\leavevmode\nobreak\ \forall
s\in[-1,0]\\}\text{ and}$ $\displaystyle
K_{+}:=\\{\varphi\in\mathbb{X}\;|\;\varphi(s)>0\;\leavevmode\nobreak\ \forall
s\in[-1,0]\\}.$

Assuming that all solutions to equation (4) oscillate, it is a straightforward
calculation to verify that for arbitrary $\varphi\in K_{-}$ the corresponding
solution $x(t;\varphi)$ has an increasing sequence of zeros
$0<t_{1}<t_{2}<t_{3}<\cdots$ such that $t_{k+1}-t_{k}>\tau=1,k\in\mathbb{N},$
and

$x(t)>0\;\;\forall t\in(t_{2k-1},t_{2k})\quad\text{and}\quad x(t)<0\;\;\forall
t\in(t_{2k},t_{2k+1}).$ (5)

In the case when $\varphi\in K_{+},\varphi(0)=h>0,$ the above inequalities on
the consecutive intervals $(t_{2k-1},t_{2k})$ and $(t_{2k-1},t_{2k})$ are
reversed into opposite. Therefore, the solution $x(t;\varphi)$ is slowly
oscillating. The consecutive intervals $(t_{j},t_{j+1}),j\in\mathbb{N}$, are
called semicycles.

Based on the well-known facts [8] and the reasoning above we have the
following statement.

###### Proposition 1

Suppose that the nonlinearity $f\in C(\mathbb{R,\mathbb{R}})$ satisfies the
negative feedback assumption $x\cdot f(x)<0,\forall x\neq 0,$ and
$f^{\prime}(0)<0$. Let the coefficient $0<a(t)\in C([0,\infty),\mathbb{R})$ be
$T$\- periodic and such that $\min\\{a(t),t\in[0,T]\\}\geq m_{0}>0$. Assume
that $|f^{\prime}(0)|\cdot m_{0}>{1}/{e}$. Then for every initial function
$\varphi\in\mathbb{K_{-}}$ with $\varphi(0)=h<0$ the corresponding solution
$x(t,h),t\geq 0,$ is slowly oscillating with the sequence of simple zeros
$\\{t_{n}\\}$ satisfying the inequalities (5) on the consecutive semicycles.
Likewise, for every initial function $\psi\in K_{+}$ with $\psi(0)=h>0$ the
corresponding solution $x(t,\psi),t\geq 0,$ is slowly oscillating satisfying
the inequalities opposite to those in (5).

Additional related theoretical basics on delay differential equations can be
found in e.g. monographs [3, 10].

##  3 Periodic Solutions

In this section the periodic solutions of interest are explicitly constructed
based on model piecewise constant nonlinearity $f$ and the periodic
coefficient $a$. Let $f(x)=f_{0}(x)=-\text{sign}(x)$ and define
$a=a(t,p_{1},p_{2},a_{1},a_{2}):=A_{0}(t)$ as follows

$a(t)=A_{0}(t)=\begin{cases}a_{1}>0,\;\text{if}\;0\leq t<p_{1}\\\
a_{2}>0,\;\text{if}\;p_{1}\leq t<p_{1}+p_{2}\\\ \text{periodic extension
outside}\;[0,p_{1}+p_{2})\;\text{for all}\;t\in\mathbb{R}.\end{cases}$ (6)

Let an initial function $\varphi(s)\in\mathbb{X}$ be such that either
$\varphi(s)<0$ or $\varphi(s)>0,\forall s\in[-1,0],$ and $\varphi(0)=h\neq 0$.
Clearly, the corresponding solution $x=x(t,\varphi)$ depends on the value $h$
only, and does not depend of the particular values of $\varphi(s)\neq 0$ for
$s\in[-1,0)$. In addition, it is uniquely determined by the parameters
$p_{1},p_{2},a_{1},a_{2}$ of the function $a(t)$. The solution $x$ is slowly
oscillating and piecewise linear (affine) for $t\geq 0$.

Like in the case of continuous $f$ and $a$ the corresponding solutions to the
DDE (4) with $f=f_{0}$ and $a=a_{0}$ are slowly oscillating with the
consecutive zeros $\\{t_{n}\\}$ satisfying the inequalities (5). In this case
we do not need any additional assumptions on $f_{0}$ or $a_{0}$ as in
Proposition 1, as the lower rates of growth or decay of any such solution are
bounded away from zero (the rates of growth/decay are also bounded from above,
due to the boundedness of $f$ and $a$). However, since the constructed
solutions are slowly oscillating, we assume that necessarily the assumption
$p_{1}+p_{2}>1$ must be in place.

In this section we follow and expand on ideas and explicit constructions from
our recent work [11, 12]. In particular, functions $f$ and $a$ are of the same
form as in the above papers.

###  3.1 Periodic solutions with coefficient’s period (Type I)

In this subsection we explicitly construct piecewise affine periodic solutions
to equation (4) which are of the same period $T>1$ as the periodic coefficient
$a(t)$ is. Some of the exposition elements are close and based on results in
our recent conference paper [12].

We presume that the desired periodic solutions have the shape as shown in
Figure 1. That is, the initial value $\varphi(0)=h<0$ is determined by an
initial function $\varphi(s)\in K_{-}$. The corresponding solution
$x=x(t,\varphi),t\geq 0,$ has exactly two zeros $\\{t_{1},t_{2}\\}$ on the
initial period $[0,T],T=p_{1}+p_{2},$ with $t_{1}<t_{2}=t_{1}+2<T$, for
particular assumptions on the parameters $p_{1},p_{2},a_{1},a_{2}$. Such
assumptions are verified to be valid by the analytical calculations that
follow, as well as they are confirmed by corresponding numerical simulations.

Figure 1: Slowly oscillating piecewise affine solution (Type I)

It is straightforward to find desired values of the solution $x=x(t,h)$ for
any $t\in[0,T]$. We have the following calculations:

$t_{1}=-\frac{h}{a_{1}},\quad x_{1}=x(p_{1})=-h+2a_{1}-a_{1}p_{1},$
$x_{2}=x(t_{1}+3)=(\frac{a_{2}}{a_{1}}-1)h+a_{1}(2-p_{1})+a_{2}(p_{1}-3),$
$x_{3}=x(T)=x(p_{1}+p_{2})=(2\frac{a_{2}}{a_{1}}-1)h+a_{1}(2-p_{1})+a_{2}[(2p_{1}+p_{2})-6]:=G(h),$

where the coefficients of the affine function $G(h):=mh-b$ are given by

$m=2\frac{a_{2}}{a_{1}}-1,\quad\text{and}\quad
b=a_{1}(p_{1}-2)+a_{2}[6-(2p_{1}+p_{2})].$ (7)

It is elementary to deduce that the range for the slope $m$ is the open
interval $(-1,\infty)$ with $|m|<1$ when $0<a_{2}<a_{1}$ holds, and $m>1$ when
$0<a_{1}<a_{2}$ is valid.

A fixed point $h_{*}<0$ of the map $G(h)=mh-b$ gives rise to a slowly
oscillating periodic solution $x=x(t,h_{*})$ of equation (4). Moreover, by the
affine type of $G$ and the construction, such periodic solution is
asymptotically stable if $|m|<1$, which is equivalent to $0<a_{2}<a_{1}$. The
periodic solution is unstable when $m>1$ , which is equivalent to
$0<a_{1}<a_{2}$. The unique fixed point $x=h_{*}$ is easily found as

$h_{*}=b/(m-1)=\frac{a_{1}[a_{1}(p_{1}-2)+a_{2}[6-(2p_{1}+p_{2})]]}{2(a_{2}-a_{1})}.$
(8)

Therefore, we arrive at the following statement.

###### Theorem 1

Let the parameters $a_{1},a_{2},p_{1},p_{2}$ be given with the values $m$ and
$b$ defined by (7). Then

  * (a)

In case when $|m|<1\,\leavevmode\nobreak\ (0<a_{2}<a_{1})$ and $b>0$ are
valid, the DDE (4) has an asymptotically stable slowly oscillating
$T$-periodic solution.

  * (b)

In the complimentary case when $m>1\,\leavevmode\nobreak\ (0<a_{2}<a_{1})$ and
$b<0$ are valid the equation has an unstable slowly oscillating $T$-periodic
solution.

The periodic solutions are generated by the initial function $\varphi(s)\equiv
h_{*},\leavevmode\nobreak\ s\in[-1,0],$ where $h_{*}$ is given by (8).

###### Remark 1

  * (a)

The local stability case $|m|<1$ is equivalent to the relationship
$a_{1}>a_{2}\geq 0$. At the same time, in order for the fixed point to be
negative, one must assume that $b=a_{1}(p_{1}-2)+a_{2}[6-(2p_{1}+p_{2})]>0$.
The inequality $p_{1}>2$ is valid by the construction. For arbitrary fixed
$p_{1}>2,p_{2}>0,a_{2}>0$ there exists $a_{1}^{0}$ such that the inequalities
$|m|<1$ and $b>0$ are satisfied for all $a_{1}\geq a_{1}^{0}.$ Thus, the
required assumptions are met for all sufficiently large $a_{1}.$ This case is
treated in more detail in our recent paper [12].

  * (b)

The instability case $m>1$ is equivalent to $0<a_{1}<a_{2}$ (note that the
case $m<-1$ is not possible for any choices of positive $a_{1}$ and $a_{2}$).
In order to have the fixed point $h_{*}$ negative, one must assume that $b<0$
is satisfied, that is $a_{1}(p_{1}-2)+a_{2}[6-(2p_{1}+p_{2})]<0$. Since
$p_{1}>2$ the assumption $p_{2}\geq 2$ would imply that $6-(2p_{1}+p_{2})<0$.
Therefore, given fixed values of $a_{1}>0,p_{1}>2,p_{2}\geq 2$ there exists
$a_{2}^{0}$ such that $b<0$ for all $a_{2}\geq a_{2}^{0}.$ Thus, the required
assumptions are met for all sufficiently large $a_{2}.$

  * (c)

It is clear that due to the continuous dependence of $m$ and $b$ on the
parameters $a_{1},a_{2},p_{1},p_{2}$ the stable or unstable periodic solutions
of Theorem 1 exist on an open box in $\mathbb{R}^{4}$, each side of which is a
corresponding open interval about a specific value of each of the four
parameters corresponding to a fixed point $h_{*}$. Those boxes can be
multiple, as our numerical insight indicates, and they do not overlap for the
stable and unstable solutions due to the respective opposite relationship
between $a_{1}$ and $a_{2}$.

$\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ a_{1}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $ | $\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ a_{2}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $ | $\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ p_{1}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $ | $\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ p_{2}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $ | $\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ h_{*}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $ | $\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ T\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $
---|---|---|---|---|---
1 | 0.25 | 2.5 | 1.5 | -0.25 | 4
2 | 0.5 | 2.5 | 2 | -1/3 | 4.5
2 | 0.25 | 2.5 | 1 | -4/7 | 3.5
1 | 0.5 | 3 | 1 | -0.5 | 4
2 | 1 | 3 | 1.5 | -0.5 | 4.5
2.5 | 0.5 | 3 | 4 | -0.31 | 7
3 | 0.5 | 3 | 4.5 | -0.45 | 7.5
4 | 1 | 3.5 | 2 | -2 | 5.5
5 | 0.5 | 3 | 3 | -1.94 | 6
5 | 1 | 3 | 2 | -1.88 | 5
$\sqrt{10}$ | $1/\sqrt{5}$ | $\pi$ | $e+1$ | -1.0602 | $\pi+e+1$
Table 1: Examples of parametric values for which stable $T$-periodic solutions
of equation (4) with coefficient $a$ defined by (6) have been demonstrated
numerically.

The parametric range for which Theorem 1 guarantees the existence of slowly
oscillating periodic solutions, either stable or unstable, is large.

Note that due to the symmetry property of the nonlinearity $f$ (oddness) and
the procedure of construction of the periodic solutions $x_{h_{*}}$, under the
assumption of Theorem 1 there also exists the symmetric to $x_{h_{*}}$
periodic solution generated by the initial function $\phi(s)\equiv-h_{*}>0$.
The two periodic solutions are related by $x_{-h_{*}}(t)\equiv-x_{h_{*}}(t)$,
thus, they are symmetric to each other.

A large sample of parametric values has been derived for which numerical
periodic solutions have been obtained and confirmed to be of the described
form. Table 1 is a sample selection of parametric values resulting in
asymptotically stable slowly oscillating periodic solutions given by Theorem
1. The values $h_{*}<0$ for these periodic solutions are easily found from the
explicit formula (8). They are also verified to exist numerically.

The next Table 2 is a sample selection of unstable slowly oscillating periodic
solutions implied by Theorem 1. The values $h_{*}$ for these periodic
solutions are found from formula (8) as well. The unstable periodic solutions
cannot be verified numerically as respective initial functions $\varphi\equiv
h_{*}<0$ actually generate approximate close nearby solutions (not the exact
ones). Those solutions, when calculated in forward time, are repelled for
large times by the actual unstable ones and are attracted by stable periodic
solutions of double period $2T$. The latter are studied analytically and
numerically and described in detail in the next subsection.

$\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ a_{1}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $ | $\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ a_{2}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $ | $\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ p_{1}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $ | $\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ p_{2}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $ | $\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ h_{*}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $ | $\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ T\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $
---|---|---|---|---|---
0.5 | 5 | 5 | 1 | -1.31 | 6
1 | 5 | 4 | 1 | -1.625 | 5
1 | 7 | 4 | 1 | -1.5 | 5
1 | 7 | 3 | 2 | -1.0833 | 5
1.5 | 7 | 2.5 | 2.5 | -1.3295 | 5
1.5 | 7 | 3.5 | 1.5 | -2.0795 | 5
1.5 | 7 | 4 | 3 | -4.3636 | 7
2 | 7 | 3 | 2 | -2.4 | 5
2 | 7 | 3 | 3 | -4.6 | 6
2 | 7 | 3.5 | 2.5 | -4.3 | 6
2.5 | 7 | 2.5 | 2.5 | -1.2614 | 5
2.5 | 7 | 3 | 2 | -1.5682 | 5
$1/\sqrt{5}$ | $3\pi/2$ | $2e$ | $\pi/3$ | -1.382 | $2e+\pi/3$
Table 2: Examples of parametric values for which unstable periodic solutions
exist to equation (4) with the coefficient defined by (6). The values $h_{*}$
are given by (8) with $b<0$ and $m>1$.

###  3.2 Periodic solutions with double coefficient’s period (Type II)

In this subsection we explicitly construct piecewise affine periodic solutions
to equation (4) which are of double period $2T$ of the $T$-periodic
coefficient $a(t)$. Some of the exposition is close and based on the results
of our recent conference paper [11].

We presume that the desired periodic solutions have the initial shape as shown
in Figure 2. That is, the initial value $\varphi(0)=h<0$ is determined by an
initial function $\varphi(s)\in K_{-}$. The corresponding solution
$x=x(t,\varphi),t\geq 0,$ has exactly one zero $t_{1}$ on the initial periodic
interval $[0,T],T=p_{1}+p_{2},$ with $t_{1}<p_{1}$ and $x(t)<0\;\forall
t\in[0,t_{1})$ and $x(t)>0\;\forall t\in(t_{1},T]$. Such assumptions are
verified to be valid by the analytical calculations that follow, as well as
are confirmed by associated numerical simulations.

Figure 2: Slowly oscillating piecewise affine solution (Type II) on initial
periodic interval

It is straightforward to calculate the desired values of
$t_{1},x_{1},x_{2},x_{3}$ as follows:

$t_{1}=-\frac{h}{a_{1}},\quad x_{1}=x(p_{1})=h+a_{1}p_{1},\quad
x_{2}=x(t_{1}+1)=(1-\frac{a_{2}}{a_{1}})h+p_{1}(a_{1}-a_{2})+a_{2},$
$x_{3}=x(p_{1}+p_{2})=(1-2\frac{a_{2}}{a_{1}})h+a_{1}p_{1}+a_{2}(2-2p_{1}-p_{2}):=F_{1}(h)=kh+d,$

where

$k=1-2\frac{a_{2}}{a_{1}}\quad\text{and}\quad
d=a_{1}p_{1}+a_{2}(2-2p_{1}-p_{2})$ (9)

are uniquely defined by the parameters $p_{1},p_{2},a_{1},a_{2}$.
Likewise, when $\psi\in K_{+}$ and $\psi(0)=h>0$ analogous calculations yield

$x_{3}=x(p_{1}+p_{2},\psi)=\left(1-2\frac{a_{2}}{a_{1}}\right)\,h-a_{1}p_{1}-a_{2}(2-2p_{1}-p_{2}):=F_{2}(h)=kh-d,$

with the same $k$ and $d$ as in the expression for $F_{1}$ given by formula
(9). It is easy to see that for arbitrary positive $a_{1},a_{2}$ the
respective values of $k$ can only belong to the range $(-\infty,1)$ with
$|k|<1$ if $0<a_{2}<a_{1}$ and $k<-1$ if $0<a_{1}<a_{2}.$

We are interested in parametric values $p_{1},p_{2},a_{1},a_{2}$ and initial
values $h<0$ such that $h_{1}=F_{1}(h)>0$ and $h_{2}=F_{2}(h_{1})<0$. This
will be the case when $d>0$, at least in some vicinity $|h|<\delta_{1},h\neq
0,$ of the discontinuity point $h=0$. The through map $F_{0}=F_{2}\circ F_{1}$
is given by $F_{0}(h)=k^{2}h+(k-1)d$. Its only fixed point is found as
$h_{*}=-d/(k+1)$, and it is negative when $d>0$ and $|k|<1.$ The fixed point
$h_{*}$ generates an asymptotically stable periodic solution of period $2T$,
when the initial function for equation (4) is chosen as $\varphi(s)\equiv
h_{*}.$ The periodic solution can also be described in terms of a cycle of
period $2$ of an appropriate one-dimensional map $F$ defined by the maps
$F_{1}$ and $F_{2}$ introduced above. Namely, define the piecewise affine map
$F$ for $h\in\mathbb{R}\setminus\\{0\\}$ as follows:

$F(h)=\begin{cases}&F_{1}(h)=kh+d,\quad\text{if}\quad h<0\\\
&F_{2}(h)=kh-d,\quad\text{if}\quad h>0.\end{cases}$

The map $F$ is discontinuous at $h=0$ and is symmetric (odd) with respect to
$h=0$, $F(-h)=-F(h),\;\forall h\neq 0$. Assuming $d>0$ one can see that the
negative feedback condition is satisfied at least locally in a vicinity of
$h=0$. The global dynamics of map $F$ is very simple in this case, as
described by the following statement:

###### Proposition 2

Suppose that $d>0$. Then

  * (i)

If $0<a_{2}<a_{1}$ is satisfied, the map $F$ has a unique globally attracting
cycle of period $2$, given by $\\{h_{*},-h_{*}\\}$, where $h_{*}=-{d}/(k+1)$.
The $2$-cycle attracts all the initial values $h\in\mathbb{R}\setminus\\{0\\}$
in the case when $a_{1}>2a_{2}$ holds, and all the initial values from the
interval $h\in(-d/k,d/k)$ in the case when $2a_{2}>a_{1}>a_{2}$ holds;

  * (ii)

If $0<a_{1}<a_{2}$ is satisfied, the map $F$ has no cycles of period 2;
moreover, any iterative sequence $F^{n}(h),h\neq 0,$ diverges,
$\lim_{n\to\infty}|F^{n}(h)|=\infty$.

Therefore, by translating the properties of the interval map $F$ to the slowly
oscillating solutions of DDE (4), we have the following statement

###### Theorem 2

Let the parameters $a_{1},a_{2},p_{1},p_{2}$ be given with values $k$ and $d$
defined by (9) and $d>0$. Then, in case when $|k|<1\,(0<a_{2}<a_{1})$ is valid
DDE (4) has an asymptotically stable slowly oscillating $2T$-periodic
solution. The periodic solution is generated by the initial function
$\varphi(s)\equiv h_{*},s\in[-1,0],$ where $h_{*}=-{d}/(k+1)$. In the case
when $k<-1\,(0<a_{2}<a_{1})$ is valid the equation does not have any
$2T$-periodic slowly oscillating solutions.

Table 3 provides sample selections of parametric values for which the stable
$2T$-periodic solutions exist. Besides using formula (8) to calculate
respective $h_{*}$, their existence was also confirmed numerically.

$\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ a_{1}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $ | $\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ a_{2}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $ | $\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ p_{1}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $ | $\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ p_{2}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $ | $\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ h_{*}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $ | $\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ 2T\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $
---|---|---|---|---|---
4 | 1 | 0.5 | 2.5 | -0.33 | 6
4 | 1 | 1 | 3.5 | -0.33 | 9
5 | 1 | 1 | 3.5 | -0.938 | 9
5 | 1 | 0.5 | 3 | -0.313 | 7
6 | 1 | 1 | 3.5 | -1.5 | 9
6 | 1 | 1 | 4.5 | -0.9 | 11
6 | 1.5 | 1.0 | 3.5 | -0.5 | 9
7 | 1.5 | 1.5 | 3.5 | -2.39 | 10
7 | 2.5 | 2.5 | 2.5 | -2.92 | 10
7 | 2.5 | 2 | 3 | -1.17 | 10
$1/\sqrt{5}$ | $3\pi/2$ | $3\pi/2$ | $e/2$ | -1.12 | $3\pi+e$
Table 3: Examples of parametric values for which stable periodic solutions
with double period of coefficient $a(t)$ exist to equation (4). The values of
$h_{*}$ are given by formula (9).

###  3.3 Coexistence of periodic solutions

Any slowly oscillating solution considered above in subsections 3.1 and 3.2 is
uniquely defined by the sign definite initial function/values
$\varphi\in\mathbb{X}$ such that $\varphi(0)=h\neq 0$, and $\varphi(s)\neq
0\;\forall s\in[-1,0],$ and by the parameters $p_{1},p_{2},a_{1},a_{2}$ of the
periodic coefficient $a(t)$ (note that $f=f_{0}(x)=-\text{sign}\,(x)$ is
fixed). One solves the initial value problem for $t\geq 0$:

$x^{\prime}(t)=a_{0}(t)\,f_{0}(x(t-1)),\;t\geq
0,\;\varphi\in\mathbb{X},\varphi(0)=h\neq 0,\varphi(s)\neq 0\;\forall
s\in[-1,0].$ (10)

Its solution $x_{\varphi}(t)=x_{\varphi}(t,h,p_{1},p_{2},a_{1},a_{2})$ is
easily found to exist for all $t\geq 0$; it is uniquely determined by the set
of the parameters $p_{1},p_{2},a_{1},a_{2}$ and the initial value of $h$ at
$t_{0}=0$.

Instead of the initial time value $t_{0}=0$ as in the case above, one can
consider a similar initial value problem for any $t_{0}>0$. In particular, in
the case of the initial time $t_{0}=p_{1}$ one is solving the following
initial value problem for $t\geq p_{1}$:

$x^{\prime}(t)=a_{0}(t)\,f_{0}(x(t-1)),\;t\geq p_{1},\;\psi(p_{1})=h\neq
0,\leavevmode\nobreak\ \psi(s)\neq 0\;\leavevmode\nobreak\ \forall
s\in[p_{1}-1,p_{1}].$ (11)

Similarly, its solution $x_{\psi}(t)=x_{\psi}(t,h,p_{1},p_{2},a_{1},a_{2})$ is
uniquely defined by the set of the parameters $p_{1},p_{2},a_{1},a_{2}$ and by
the initial value $h$ at $t_{0}=p_{1}$.

Now, due to the $T$-periodicity of the coefficient $a(t)$, one can easily see
that the solutions of the two initial value problems (10) and (11) are closely
related, with an interchange in the set of the parameters and a time shift.
Namely, they are just a time shift by $p_{1}$ with the parameter interchange
in the following sense:

$x_{\varphi}(t,h,p_{2},p_{1},a_{2},a_{1})=x_{\psi}(t+p_{1},h_{1},p_{1},p_{2},a_{1},a_{2}),\quad\forall
t\geq 0,$ (12)

with an appropriate $h_{1}=h_{1}(h).$ This simple observation allows to
conclude the simultaneous existence of periodic solutions of both types I and
II, which are described in Theorems 1 and 2. Since the value of the ratio
$a_{1}/a_{2}$ with respect to $1$ determines the stability of those periodic
solutions, their stability types must be opposite to each other. We have the
following statement

###### Theorem 3

There is an open set of parameters $p_{1},p_{2},a_{1},a_{2}$ such that the DDE
(4) has simultaneously an unstable slowly oscillating $T$-periodic solution of
type I and an asymptotically stable $2T$-periodic solution of type II. Such
two periodic solutions can be paired in the following sense: small
perturbations of the unstable solution are repelled by it and attracted by a
respective stable solution, as $t\to\infty.$

Such situation of coexisting two periodic solutions is depicted in Figure 3.
The unstable periodic solution $x_{u}(t)$ of period $T=4$ is given by Theorem
1 when $p_{1}=3,p_{2}=1,a_{1}=1,a_{2}=6$; the respective value of $h$ is
$h_{*}=-0.5$. The stable periodic solution $x_{s}(t)$ of period $2T=8$ is
given by Theorem 2 for the parameter values $p_{1}=1,p_{2}=3,a_{1}=6,a_{2}=1$,
so that the respective values of $p_{1},p_{2}$ and $a_{1},a_{2}$ are
interchanged for these two solutions. In addition, the solution $x_{s}(t)$ is
shifted by $p_{1}$ to the right (the graph (b) in Figure 3 shows the solution
$x_{s}(t+p_{1})$ for longer time intervals).

Tables 4 and 5 provide a sample selection of such dual cases of the
coexistence of two types of stable/unstable solutions. Table 5 is a mirror
image of Table 4 for the stable periodic solutions with double periods.
Namely, each row in Table 5 represents a stable periodic solution of the
corresponding unstable solution in the same row number from Table 4.

$\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ a_{1}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $ | $\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ a_{2}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $ | $\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ p_{1}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $ | $\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ p_{2}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $ | $\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ h_{*}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $ | $\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ T\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $
---|---|---|---|---|---
0.5 | 2.5 | 3 | 0.5 | -0.09 | 3.5
0.5 | 3 | 5 | 1 | -1.35 | 6
0.5 | 5 | 4 | 0.5 | -0.64 | 4.5
0.5 | 7 | 3 | 2 | -0.52 | 5
1 | 6 | 3 | 1 | -0.5 | 4
1 | 7 | 5 | 1 | -2.67 | 6
2 | 7 | 2 | 3 | -1.4 | 5
3 | 7 | 4 | 1 | -5.62 | 5
$\pi/6$ | $e$ | $\pi$ | 1/2 | -0.18 | $\pi+1/2$
Table 4: Examples of parametric values for which unstable and stable periodic solutions coexist to DDE (4) with the coefficient defined by (6). The parametric data is shown for the unstable solutions. $\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ a_{1}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $ | $\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ a_{2}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $ | $\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ p_{1}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $ | $\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ p_{2}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $ | $\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ h_{*}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $ | $\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ 2T\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $
---|---|---|---|---|---
2.5 | 0.5 | 0.5 | 3 | -0.156 | 7
3 | 0.5 | 1 | 5 | -0.3 | 12
5 | 0.5 | 0.5 | 4 | -0.56 | 9
7 | 0.5 | 2 | 3 | -6.19 | 10
6 | 1 | 1 | 3 | -1.8 | 8
7 | 1 | 1 | 5 | -1.17 | 12
7 | 2 | 3 | 2 | -6.3 | 10
7 | 3 | 1 | 4 | -4.375 | 10
$e$ | $\pi/6$ | 1/2 | $\pi$ | -1.92 | $2\pi+1$
Table 5: Examples of parametric values for which stable periodic solutions
exist to equation (4) with the coefficient defined by equation (6). The
parametric data is shown for the stable solutions. Figure 3: An example of
numerical solution with $a_{1}=1$, $a_{2}=6$, $p_{1}=3$, $p_{2}=1$. Left panel
- unstable solution with single period, which is destroyed after two first
periods. Right panel - stable solution with double period. Numerical solution
of equation 4 with $a_{1}=6$, $a_{2}=1$, $p_{1}=1$, $p_{2}=3$, whose solution
has double period of the $T$-periodic coefficient, is overplotted as a red
dashed line in the right panel.

##  4 Smoothed nonlinearities

In this section, we follow a standard procedure of replacing the discontinuous
piecewise constant functions $f_{0}$ and $A_{0}$ by close to them continuous
functions $f_{\delta}$ and $A_{\delta}$ for small $\delta>0$. This procedure
has been used in numerous cases; typical examples can be found in papers [9,
13, 18, 20]. In a $\delta$-neighborhood of every discontinuity point for both
$f_{0}$ and $A_{0}$ the jump discontinuity is replaced by an affine function
by connecting the two constant levels by a line segment. That is, define
$f_{\delta}(x)$ and $A_{\delta}(t)$ as follows

$f(x)=f_{\delta}(x)=\begin{cases}+1\;\text{if}\;x\leq-\delta\\\
-1\;\text{if}\;x\geq\delta\\\
-({1}/{\delta})x\;\text{if}\;x\in(-\delta,\delta),\end{cases}$ (13)

and

$a(t)=A_{\delta}(t)=\begin{cases}a_{2}+\frac{a_{1}-a_{2}}{2\delta}(t+\delta)\;\text{if}\;t\in(-\delta,\delta)\\\
a_{1}\;\text{if}\;t\in[\delta,p_{1}-\delta]\\\
a_{1}+\frac{a_{2}-a_{1}}{2\delta}[t-(p_{1}-\delta)]\;\text{if}\;t\in(p_{1}-\delta,p_{1}+\delta)\\\
a_{2}\;\text{if}\;t\in[p_{1}+\delta,p_{1}+p_{2}-\delta]\\\
a_{2}+\frac{a_{1}-a_{2}}{2\delta}[t-(p_{2}-\delta)]\;\text{if}\;t\in(p_{1}+p_{2}-\delta,p_{1}+p_{2}+\delta)\\\
\text{periodic extension on}\;\mathbb{R}\;\text{outside
interval}\;[0,p_{1}+p_{2}).\end{cases}$ (14)

We shall show next that the values of the piecewise affine slowly oscillating
solutions constructed in Section 3 for $t\geq 0$ do not change on the entire
periodic interval $[0,T]$, except in a small $\varepsilon$-vicinity of the
corner points ($\varepsilon$ is a multiple of $\delta$).

Indeed, the case of a symmetric corner solution, as it passes through a
$\delta$-neighborhood of $x=0$ for the nonlinearity $f_{\delta}$, is shown in
Figure 4 (which corresponds to e.g. the Type I solution on the first positive
semi-cycle depicted in Figure 1). The $\delta$-neighborhood of $x=0$ for $f$
corresponds to the $\varepsilon=\delta/a_{1}$ neighborhood for $t$ at
$t=t_{1}+1$. It is immediately clear that the replacement of the corner type
solution on the interval $[t_{1}+1-\varepsilon,t_{1}+1+\varepsilon]$ by the
respective parabolic segment, as a result of the corresponding integration now
with a linear nonlinearity $f$, results in a symmetric parabola with the
vertex at $t_{1}+1$ and the same equal values $x(t_{1}+1-\varepsilon)$ and
$x(t_{1}+1-\varepsilon)$ as for the initial piecewise solution with the
discontinuous $f_{0}$.

Figure 4: Sketches of (a) continuous solution $x$ with a corner-type
discontinuity for $x^{\prime}$ at $t_{1}+1$ for functions $A_{0}(t)$ and
$f_{0}(x)$. (b) $C^{1}$-smooth solution with parabolic matching on
$[t_{1}+1-\varepsilon,t_{1}+1+\varepsilon],\varepsilon a_{1}=\delta,$ for
functions $A_{\delta}(t)$ and $f_{\delta}(x)$ with small $\delta>0$.

The general case of a corner type solution (piecewise affine) is shown in
Figure 5, which corresponds to e.g. point $(p_{1},x_{1})$ on the graph of a
solution of type II (see Figure 2). We shall show now that a smooth parabolic
segment solution connecting points $(p_{1}-\varepsilon,x((p_{1}-\varepsilon))$
and $(p_{1}+\varepsilon,x((p_{1}+\varepsilon))$ is the precise replacement of
the corresponding piecewise affine segment solution which is obtained for the
piecewise constant functions $f_{0}$ and $A_{0}$. The values of the solution
$x$ and its derivatives for the piecewise affine solution are given as

$x(p_{1}-\varepsilon)=x_{1}-a_{1}\varepsilon,\quad
x^{\prime}(p_{1}-\varepsilon)=a_{1},\quad
x(p_{1}+\varepsilon)=x_{1}+a_{2}\varepsilon,\quad
x^{\prime}(p_{1}+\varepsilon)=a_{2},\;$

Looking for a parabolic connection $y=P(t)=A(t-p_{1})^{2}+B(t-p_{1})+C$ as a
solution of the DDE (4) on the interval
$[p_{1}-\varepsilon,p_{1}+\varepsilon]$, starting at the point
$(p_{1}-\varepsilon,x_{1}-a_{1}\varepsilon)$ and satisfying the necessary
initial-boundary conditions:

$x(p_{1}-\varepsilon)=x_{1}-a_{1}\varepsilon,\quad
x^{\prime}(p_{1}-\varepsilon)=a_{1},\quad
x^{\prime}(p_{1}+\varepsilon)=a_{2},\;$

one easily finds that the values of the parameters $A,B,$ and $C$ are:

$A=\frac{a_{2}-a_{1}}{4\varepsilon},\quad B=\frac{a_{1}+a_{2}}{2},\quad
C=\frac{a_{2}-a_{1}}{4}\,\varepsilon+x_{1}.$

But then, by checking the value of the parabolic segment of the solution
$x(t)=P(t)$ at $t=p_{1}+\varepsilon$, one verifies that

$\displaystyle P(p_{1}+\varepsilon)$
$\displaystyle=A(t-p_{1})^{2}+B(t-p_{1})+C|_{t=p_{1}+\varepsilon}$
$\displaystyle=\frac{a_{2}-a_{1}}{4\varepsilon}\varepsilon^{2}+\frac{a_{1}+a_{2}}{2}\varepsilon+\frac{a_{2}-a_{1}}{4}\varepsilon+x_{1}=x_{1}+a_{2}\varepsilon=x_{1}+a_{2}(t-p_{1})|_{t=p_{1}+\varepsilon},$

which is the same value as the piecewise affine solution at that point.

Thus, the solutions to equation (4) with continuous $f=f_{\delta}$ and
$a=A_{\delta}$, are obtained from the solutions when $f=f_{0}$ and $a=A_{0}$
by simple cutting out and replacement of the small corner segments by
respective parabolic segments, in their $\varepsilon$-vicinity
($\varepsilon=c\cdot\delta$). An immediate consequence of this fact is that
the periodic solutions and their stability persist under such small
$\delta$-perturbations of $f_{0}$ and $a_{0}$ in DDE (4).

Indeed, the easiest way to see this is to consider the new starting point for
$t,t=\varepsilon,$ with $\varphi(\varepsilon)=H\neq 0$. Then the corresponding
shift map $S^{T}$ by the period $T$ is dynamically equivalent (conjugate) to
the respective map constructed in the case of $f_{0}$ and $a_{0}$. But the
latter is equivalent to a composition of respective maps
$G,F,F_{1}.F_{2},F_{0}$ derived in Section 3. If one would like to stay with
the initial value $H:=\varphi(0)\neq 0$ in the case of continuous
$f_{\delta},a_{\delta},\delta>0,$ then a small adjustment in the
considerations should be made. Namely, the new $h:=H\pm c\varepsilon$ should
be introduced, where $c\varepsilon$ is the difference between the two
solutions of the differential delay equation (4) in the two cases when
$\delta=0$ and small $\delta>0$. Such difference is shown in the graph in
Figure 4, part (b) and in the graph in Figure 5, part (b) (in both graphs it
is positive). In general, such a difference is independent of the value
$x_{1}$ (or equivalent values in other analogous corner situations).
Therefore, all the resulting maps $G,F,F_{0},F_{1},F_{2}$ derived in Section 3
will become finite compositions of the affine maps of the form
$h\mapsto\tilde{F},$ where
$\tilde{F}(h)=G_{*}(h+c_{1}\varepsilon)+c_{2}\varepsilon$ and $G_{*}(h)$ is
one of the elementary maps considered in Subsections 3.1 and 3.2. Therefore,
we are in a position to state the following

###### Theorem 4

Suppose that the parametric values $a_{1},a_{2},p_{1},p_{2}$ are such that the
assumptions of either one of Theorems 1 or 2 are satisfied. Then there exists
$\delta_{0}>0$ such that for every $0\leq\delta\leq\delta_{0}$ the delay
differential equation (4) with functions $f=f_{\delta}(x)$ and
$a=A_{\delta}(t)$ has a $C^{1}$-smooth periodic solution with the same type of
stability as in the respective Theorems 1 or 2. Such a solution converges in
uniform metric as $\delta\to+0$ to the respective periodic solution with the
corner type discontinuity for $x^{\prime}$ (when $f=f_{0},a=A_{0}$).

Note that the smoothing of both functions $f_{0}$ and $a_{0}$ in a
$\delta$-neighborhood of their discontinuity set can be made such that the
resulting continuous nonlinearities $f_{\delta}$ and $a_{\delta}$ are
continuously differentiable or even of the $C^{\infty}$ class, and Theorem 4
still remains valid. Such $C^{1}$-smoothness is required in some cases when
e.g. one would need to study Floquet multipliers of the periodic solutions,
and consider the corresponding linearized equation along the periodic
solutions.

Indeed, it is an elementary fact that there are many solutions for the
following approximation problem. Find a smooth function $F(t)$ defined on an
interval $[t_{1},t_{2}]$ such that it satisfies the given boundary conditions:
$F(t_{1})=x_{1},F^{\prime}(t_{1})=m_{1}$ and
$F(t_{2})=x_{2},F^{\prime}(t_{1})=m_{2}$. One of the possible solutions can be
suggested as a polynomial of degree 3 or higher. In our case of $C^{\infty}$
smooth connection for $f_{0}(x)$ on the interval $[-\delta,\delta]$ one can
use the following function

$f^{0}(x)=e^{\frac{\delta
x}{x-\delta}}-1,\;\text{for}\;x\in[0,\delta],\quad\text{and}\quad
f^{0}(x)=-f^{0}(x),\;\text{for}\;x\in[-\delta,0].$

It is straightforward to verify that such connecting function is decreasing on
the interval $[-\delta,\delta],$ is of class $C^{\infty}$ there, with all the
matching derivatives at $x=0,x=-\delta,x=\delta$. Analogous
$C^{\infty}$-connection can be used for the coefficient $a(t)$ at points
$0,p_{1},T=p_{1}+p_{2}$.

Figure 5: Sketches of (a) continuous coefficient $A_{\delta}(t)$ in
$\delta$-vicinity of $t_{1}=p_{1}$; (b) $C^{1}$-smooth solution with parabolic
matching on $[p_{1}+1-\varepsilon,p_{1}+1+\varepsilon]$ for functions
$A_{\delta}(t)$ and $f_{\delta}(x)$ with small $\delta>0$.

##  5 Acknowledgements

The authors thank the mathematical research institute MATRIX in Australia
where part of this research was performed. Its final version resulted from
collaborative activities of the authors during the workshop “Delay
Differential Equations and Their Applications” (https://www.matrix-
inst.org.au/events/delay-differential-equations-and-their-applications/) held
in December 2023. The authors are also grateful for the financial support
provided for these research activities by Simons Foundation (USA), Flinders
University (Australia), and the Pennsylvania State University (USA). A.I.’s
research was also supported in part by the Alexander von Humboldt Stiftung
(Germany) during his visit to Justus-Liebig-Universität, Giessen, in June-
August 2023.

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