# Periodic convergence for a class of nonlinear time-delay systems

Alexander Aleksandrov, Denis Efimov, Xubin Ping

► To cite this version:

Alexander Aleksandrov, Denis Efimov, Xubin Ping. Periodic convergence for a class of nonlinear time-delay systems. IEEE Control Systems Letters, 2025, 10.1109/LCSYS.2025.3564586. hal-05041742

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https://inria.hal.science/hal-05041742v1

Submitted on 22 Apr 2025

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# Periodic convergence for a class of nonlinear time-delay systems

A. Aleksandrov<fnref n="1" />,<fnref n="2" />, D. Efimov<fnref n="3" />,<fnref n="4" />, X. Ping<fnref n="5" />

*Abstract*—The new existence conditions for periodic steady-state solution in time-delay convergent systems are presented. The main advantage of this result is that highly nonlinear (without meaningful linear approximation) dynamics are allowed for analysis. These conditions are developed for Persidskii and Lotka-Volterra time-delay systems. The efficiency of the approach is demonstrated on academic examples of these models.

## I. INTRODUCTION

Nonlinear systems may demonstrate rather complex nonlinear behavior ranging from (almost) globally attracting equilibrium (equilibria), to periodic or close to periodic oscillations, and chaotic trajectories [1]–[4]. Appearances of time-varying components in the models, or time-delays, make the resulting behaviors even less predictable and more difficult for analysis. That is why different dynamics reduction approaches become popular, based for instance on timescale separation [5] or the convergence concept [6], [7].

The latter property means that the system has a bounded globally defined in time solution, which uniformly attracts all other trajectories in the system. Therefore, such a convergence significantly simplifies the asymptotic analysis of dynamic properties of complex dynamical models. Different conditions appeared in the literature for capturing the converging dynamics. The original ideas [6] used the boundedness and uniform stability of Jacobian of the system, other results were based on application of the Lyapunov function method [8], [9], different metrics for contraction analysis [10], relaxed concepts of differential contraction [11] and incremental stability [12]. The last notion was shown to be equivalent to the convergence if the system has a forward invariant compact set [13]. There are also extensions of these results to time-delay systems through the Lyapunov-Razumikhin approach [14].

These general results have been exemplified and adapted to various important canonical representations of nonlinear systems such as, for example, port-Hamiltonian [15], Lur'e [16] or Persidskii [17] systems (see also the references in the surveys [18], [19]). The latter results received recent generalizations to analysis of convergence and stability in time-delay models separately [20], [21].

The aim of this work is, first, to develop the conditions of existence of periodic steady-state solution in non-autonomous time-delay systems, and second, to apply these conditions for Persidskii and Lotka-Volterra models. The latter represent a large class of populations dynamics, where appearance of delay is important for modeling the underlying physical phenomena [22]–[25], while making their investigation rather difficult [26], [27]. The former is a rather general class of systems that includes Lur'e [28] and Lotka-Volterra ones (under a change of coordinates). This class of models is widely used for diagonal stability analysis [29], [30], sliding mode control [31]–[33], opinion dynamics [34], neural networks [35]–[37], and in numerous other domains. Following the ideas of [8], [9], our proof is conceptually different from the existing arguments [13], [14] and based on different assumptions, which usually require some boundedness of the Jacobian of the system, and well adapted to the models that are close to linear ones. The latter is frequently not the case in Persidskii and Lotka-Volterra systems. We use the Lyapunov-Krasovskii method for the analysis. The examples of efficiency of our approach are provided.

The outline of this paper is as follows. Preliminaries are formulated in Section II. The conditions of existence of periodic solutions in time-delay time-varying systems are given in Section III. These conditions are developed for Persidskii and Lotka-Volterra models in sections IV and V, respectively.

## II. PRELIMINARIES

The real numbers are denoted by $\mathbb{R}$, $\mathbb{R}_+ = \{s \in \mathbb{R} : s \ge 0\}$. Euclidean norm for a vector $x \in \mathbb{R}^n$ is defined as $|x|$, for a matrix $|\cdot|$ denotes the induced norm. We denote by $C([a, b], \mathbb{R}^n)$, $-\infty < a < b < +\infty$ the Banach space of continuous functions $\phi : [a, b] \to \mathbb{R}^n$ with the uniform norm $||\phi|| = \sup_{a \le s \le b} |\phi(s)|$.

For a locally Lipschitz continuous function $V : \mathbb{R} \to \mathbb{R}_+$ the upper Dini derivative is defined as follows:

$$ \partial^+ V(t) = \limsup_{h \to 0^+} \frac{V(t+h) - V(t)}{h} $$

for any $t \in \mathbb{R}$.

A continuous function $\sigma : \mathbb{R}_+ \to \mathbb{R}_+$ belongs to class $\mathcal{K}$ if it is strictly increasing and $\sigma(0) = 0$; it belongs to class $\mathcal{K}_\infty$ if it is also unbounded. A continuous function $\beta : \mathbb{R}_+ \times \mathbb{R}_+ \to \mathbb{R}_+$ belongs to class $\mathcal{KL}$ if $\beta(\cdot, r) \in \mathcal{K}$ for all $r \in \mathbb{R}_+$ and $\beta(r, \cdot)$ is a strictly decreasing to zero for any fixed $r > 0$.

<footnote n="1">¹Saint Petersburg State University, 7-9 Universitetskaya nab., 199034 Saint Petersburg, Russia.</footnote>

<footnote n="2">²Institute for Problems in Mechanical Engineering, Russian Academy of Sciences, Saint Petersburg 199178, Russia.</footnote>

<footnote n="3">³Inria, Univ. Lille, CNRS, UMR 9189 - CRISTAL, F-59000 Lille, France.</footnote>

<footnote n="4">⁴ITMO University, 49 av. Kronverkskiy, 197101 Saint Petersburg, Russia</footnote>

<footnote n="5">⁵Xidian University, Xi’an, Shaanxi, China.</footnote>

<footnote>The research was partially supported by the Ministry of Science and Higher Education of the Russian Federation (project no. 124041500008-1).</footnote>
### A. Convergence and incremental stability

Consider a generic non-autonomous functional differential equation of retarded type [38]:

$$ \dot{x}(t) = f(t, x_t), \quad t \in \mathbb{R}, \qquad (1) $$

where $x(t) \in \mathbb{R}^n$ and $x_t \in C([-\tau, 0], \mathbb{R}^n)$ is the state function, $x_t(s) = x(t+s)$, $-\tau \le s \le 0$ and $\tau > 0$ is a finite delay; $f: \mathbb{R} \times C([-\tau, 0], \mathbb{R}^n) \to \mathbb{R}^n$ is a continuous functional, $f(\cdot, 0) = 0$, and it provides (at least local) existence and uniqueness of solutions in forward time for the system (1). Denote such a unique solution satisfying the initial condition $x_0 \in C([-\tau, 0], \mathbb{R}^n)$ at time $t_0 \in \mathbb{R}$ by $x(t, t_0, x_0)$ with $t \in [t_0 - \tau, T)$ for some $T \in (t_0, +\infty)$, then $x_t(s, t_0, x_0) = x(t+s, t_0, x_0)$ is defined for $t \in [t_0, T)$ and $-\tau \le s \le 0$.

The standard notions of uniform stability and (global) uniform asymptotic stability for the zero solution of (1) can be found in [39].

**Definition 1.** [14] The system (1) is said to be uniformly incrementally stable if there exists $\beta \in \mathcal{KL}$ such that, for any $t_0 \in \mathbb{R}$, any $x_1, x_2 \in C([-\tau, 0], \mathbb{R}^n)$, and all $t \ge t_0$,

$$ |x(t, t_0, x_1) - x(t, t_0, x_2)| \le \beta(\|x_1 - x_2\|, t - t_0). $$

Uniformity in this definition is understood as independence of $\beta$ in the initial time $t_0$. The sufficient condition of this property formulated in terms of Lyapunov-Razumikhin approach can be found in [14].

**Definition 2.** [14] The system (1) is said to be uniformly convergent if for all $t_0 \in \mathbb{R}$ and $x_0 \in C([-\tau, 0], \mathbb{R}^n)$ the respective solutions $x(t, t_0, x_0)$ are defined for all $t \ge t_0$, and there exist $\beta \in \mathcal{KL}$ and a unique and bounded solution $\bar{x}: \mathbb{R} \to \mathbb{R}^n$ such that, for any $t_0 \in \mathbb{R}$, any $x_0 \in C([-\tau, 0], \mathbb{R}^n)$, and all $t \ge t_0$,

$$ |x(t, t_0, x_0) - \bar{x}(t)| \le \beta(\|x_0 - \bar{x}_{t_0}\|, t - t_0). $$

In comparison with incremental stability, this property requires existence of solutions for $t_0 \in \mathbb{R}$, and that they all are bounded approaching the steady-state solution $\bar{x}(t)$. It is demonstrated in [14] that convergence and incremental stability happen to be equivalent if solutions evolve on a compact set.

### III. EXISTENCE CONDITIONS

Consider a nonlinear system with pointwise constant delay:

$$ \dot{x}(t) = F(t, x(t), x(t - \tau)), \quad t \ge t_0, t_0 \in \mathbb{R}, \qquad (2) $$

where $x(t) \in \mathbb{R}^n$ is the current value of the state function, vector function $F(t, x, y)$ is continuous for any $t \in \mathbb{R}$, $x \in \mathbb{R}^n$, $y \in \mathbb{R}^n$ and periodic with respect to $t$ with a period $T > 0$, $\tau > 0$ is the delay. Let initial functions for (2) belong to the space $C([-\tau, 0], \mathbb{R}^n)$.

Moreover, we need a stronger regularity of $F$:

**Assumption 1.** The function $F(t, x, y)$ is locally Lipschitz continuous with respect to $x, y \in \mathbb{R}^n$.

And that the system admits boundedness and incremental attractiveness of all solutions, a property slightly weaker than the convergence or incremental stability (uniform stability with respect to initial deviations of solutions is not required):

**Assumption 2.** Any solution $x(t, t_0, \psi)$ of (2) is defined and bounded for all $t \in [t_0, +\infty)$, and all $\psi \in C([-\tau, 0], \mathbb{R}^n)$ and $t_0 \in \mathbb{R}$.

**Assumption 3.** For any $r > 0$,

$$ |x(t, t_0, \tilde{\psi}) - x(t, t_0, \bar{\psi})| \to 0 $$

as $t - t_0 \to +\infty$ uniformly with respect to $t_0 \in \mathbb{R}$, where $\tilde{\psi}, \bar{\psi} \in C([-\tau, 0], \mathbb{R}^n)$ with $\|\tilde{\psi}\| < r$, $\|\bar{\psi}\| < r$.

Our main result claims that under these conditions the system is in fact convergent with a periodic common asymptotic behavior.

**Theorem 1.** Let assumptions 1–3 be satisfied. Then the system (2) admits a unique solution that is bounded for all $t \in \mathbb{R}$, T-periodic and globally uniformly asymptotically stable.

*Proof.* Choose and fix a function $\psi^{(0)} \in C([-\tau, 0], \mathbb{R}^n)$ with $t_0 \in \mathbb{R}$, and consider the solution $\hat{x}(t) = x(t, t_0, \psi^{(0)})$ of (2) that is defined and bounded for $t \ge t_0$ due to Assumption 2.

Since $F(t, x, y)$ is T-periodic, for any integer $k$, $\hat{x}(t + kT)$ is also a solution of (2) that is defined for $t \ge t_0 - kT$, and we have

$$ \hat{x}(t + kT) = x(t + kT, t_0, \psi^{(0)}) = x(t, t_0 - kT, \psi^{(0)}) $$

for any $t \ge t_0 - kT$.

Construct the sequence of functions $\psi^{(k)}(\xi) = x(\xi + t_0 + kT, t_0, \psi^{(0)}), \xi \in [-\tau, 0], k = 1, 2, \dots$. The sequence is uniformly bounded and equicontinuous (due to continuity of $F$). Applying Arzela–Ascoli theorem, we obtain that there exists a subsequence $\psi^{(k_n)}(\xi) = x(\xi + t_0 + k_n T, t_0, \psi^{(0)})$ that converges uniformly with respect to $\xi \in [-\tau, 0]$ to a function $\phi \in C([-\tau, 0], \mathbb{R}^n)$. Denote the solution of (2) initiated at $\phi$ by $\tilde{x}(t) = x(t, t_0, \phi)$.

According to Assumption 3, it follows that

$$ \begin{aligned} & |x(t + T, t_0, \psi^{(0)}) - x(t, t_0, \psi^{(0)})| \\ & = |x(t, t_0, \psi^{(1)}) - x(t, t_0, \psi^{(0)})| \to 0 \end{aligned} $$

as $t \to +\infty$. Hence, for any $\xi \in [-\tau, 0]$,

$$ \begin{aligned} & |x(\xi + t_0 + k_n T + T, t_0, \psi^{(0)}) - x(\xi + t_0 + k_n T, t_0, \psi^{(0)})| \\ & = |x(\xi + t_0 + T, t_0, \psi^{(k_n)}) - \psi^{(k_n)}(\xi)| \to 0 \end{aligned} $$

as $n \to \infty$, and

$$ x(\xi + t_0 + T, t_0, \phi) = \phi(\xi) $$

for $\xi \in [-\tau, 0]$ since $\psi^{(k_n)}(\xi) \to \phi(\xi)$, $x(\xi + t_0 + T, t_0, \psi^{(k_n)}) \to x(\xi + t_0 + T, t_0, \phi)$ as $n \to +\infty$ (due to integral continuity of solutions under Assumption 1). Therefore, $x(\xi + t_0 + T, t_0, \phi) = \phi(\xi)$ for $\xi \in [-\tau, 0]$. As a result, we obtain that $x(t, t_0 - T, \phi) = x(t + T, t_0, \phi) = x(t, t_0, \phi)$ for $t \in [t_0 - T, +\infty)$.

In a similar way it can be verified that $x(t, t_0 - lT, \phi) = x(t + lT, t_0, \phi) = x(t, t_0, \phi)$ for $t \in [t_0 - lT, +\infty)$,
l = 2, 3, ..., Thus, the union of solutions $x(t, t_0 - lT, \phi)$,
l = 2, 3, ..., gives us the periodic solution of the system (2).
Assumption 3 implies that (2) admits only one bounded for
$t \in \mathbb{R}$ solution.

The uniform stability follows from the integral continuity
of the solutions, then Assumption 3 implies global uniform
asymptotic stability of the periodic solution passed by $\phi$. The
proof is completed.
□

Remark 1. Considering a more general version of the system (2):

$$
\dot{x}(t) = F(t, x_t),
$$

where the functional $F : \mathbb{R} \times C([-\tau, 0], \mathbb{R}^n) \to \mathbb{R}^n$ is $T$-periodic in the first argument and locally Lipschitz continuous in the second one, then the result of Theorem 1 keeps its meaning (all arguments can be directly repeated).

IV. PERSIDSKII-TYPE SYSTEMS

Consider a special case of the system in (2):

$$
\dot{x}(t) = Af(x(t)) + Bf(x(t-\tau)) + H(t), \quad (3)
$$

where $A \in \mathbb{R}^{n \times n}$ is a constant Metzler matrix, $B \in \mathbb{R}^{n \times n}$ is a constant nonnegative matrix, $H(t)$ is a continuous for $t \in \mathbb{R}$ and $T$-periodic vector function, $\tau$ is a constant positive delay, the nonlinearity $f: \mathbb{R}^n \to \mathbb{R}^n$, is locally Lipschitz continuous (Assumption 1 is verified for (3)) and diagonal:

$$
f(x) = f(x_1, \dots, x_n) = (f_1(x_1), f_2(x_2), \dots, f_n(x_n))^{\top}
$$

and assumed to satisfy a sector-type constraint: $x_i f_i(x_i) > 0$ for $x_i \in \mathbb{R} \setminus \{0\}$, $i = 1, \dots, n$. The latter property is the characteristic one for the Persidskii systems [29]. Moreover, we need also monotonicity and unboundedness of the nonlinearities:

**Assumption 4.** The functions $f_i(x_i)$ for $i = 1, \dots, n$ possess the following properties:

(i) $f_i(x_i) \to -\infty$ as $x_i \to -\infty$, and $f_i(x_i) \to +\infty$ as $x_i \to +\infty$;

(ii) $f_i(x_i)$ is strictly increasing function for $x_i \in \mathbb{R}$.

We assume that the delay-free counterpart of (3) is asymp-
totically stable for the case $H \equiv 0$:

**Assumption 5.** The matrix $A+B$ is Hurwitz.

Under these restrictions we can prove that the system is
convergent with a periodic globally attractive solution:

**Theorem 2.** Let assumptions 4, 5 be satisfied. Then the system (3) admits a unique solution that is bounded for all $t \in \mathbb{R}$, $T$-periodic and globally asymptotically stable.

*Proof.* We should verify assumptions 2 and 3. From Assumption 5 it follows the existence of a positive vector $\nu = (\nu_1, \dots, \nu_n)^\top$ such that [29] (here the relation $<$ between two vectors is understood elementwise)

$$
(A + B)^{\top} \nu < 0. \tag{4}
$$

Define a Lyapunov–Krasovskii functional by the formula

$$
V(x_t) = \sum_{i=1}^{n} \nu_i |x_i(t)| + \sum_{i=1}^{n} \mu_i \int_{t-\tau}^{t} |f_i(x_i(\xi))| d\xi \\
+ \varepsilon \sum_{i=1}^{n} \int_{t-\tau}^{t} (\xi - t + \tau) |f_i(x_i(\xi))| d\xi,
$$

where $\mu_i$ and $\varepsilon$ are positive constants. Considering the upper-
right Dini derivative $\partial^+ V(t)$ of $V(t) = V(x_t)$, we obtain

$$
\begin{aligned}
\partial^+ V(t) \le {}& [|f_1(x_1(t))| \dots |f_n(x_n(t))|](A^\top \nu + \mu) \\
& + [|f_1(x_1(t-\tau))| \dots |f_n(x_n(t-\tau))|](B^\top \nu - \mu) \\
& - \varepsilon \sum_{i=1}^n \int_{t-\tau}^t |f_i(x_i(\xi))| d\xi + \varepsilon \tau \sum_{i=1}^n |f_i(x_i(t))| + M,
\end{aligned}
$$

where $M$ is a positive constant related with the amplitude of $H$. Let $\mu = B^\top \nu + \delta 1_n$, where $1_n = (1, \dots, 1)^\top$ and $\delta$ is a positive constant. From the inequality (4) it follows that if $\delta$ and $\varepsilon$ are sufficiently small, then

$$
\partial^+ V(t) \leq -\gamma \sum_{i=1}^{n} |f_i(x_i(t))| - \varepsilon \sum_{i=1}^{n} \int_{t-\tau}^{t} |f_i(x_i(\xi))| d\xi + M,
$$

where $\gamma$ is a positive constant. Let

$$
G_1 = \left\{
\begin{array}{l}
x_t \in C([-\tau, 0], \mathbb{R}^n) : \gamma \sum_{j=1}^{n} |f_j(x_j(t))| \\
\qquad + \varepsilon \sum_{i=1}^{n} \int_{t-\tau}^{t} |f_i(x_i(\xi))| d\xi < 2M
\end{array}
\right\}
$$

$$
N = \sup_{x_t \in G_1} V(x_t), \quad G_2 = \{x_t \in C([-\tau, 0], \mathbb{R}^n) : V(x_t) \le N\}.
$$

We obtain $0 < N < +\infty$, $G_1 \subset G_2$ and if $x_t \notin G_2$ for $t \in [t_1, t_2]$, then

$$
\partial^+ V(t) \leq -M,
$$

hence,

$$
V(x_{t_2}) \leq V(x_{t_1}) - M(t_2 - t_1).
$$

This implies that if $x_{\bar{t}} \in G_2$, then $x_t \in G_2$ for all $t \ge \bar{t}$.
Moreover, from $t_0 \in \mathbb{R}$, $\|x_{t_0}\| \le R$, where $R > 0$, it follows
that $V(x_t) \le \max\{N, K\}$ for $t \ge t_0$ and $x_t \in G_2$ for $t \ge$
$t_0 + \mathcal{T}$, where

$$
K = \sup_{\|\varphi\| \le R} V(\varphi), \quad \mathcal{T} \ge \frac{K - N}{M}.
$$

Therefore, there exists a constant $c > 0$, independent of
$t_0$, $R$, such that $|x(t)| \le c \max\{N, K\}$ for all $t \ge t_0$. Thus,
solutions of (3) are uniformly ultimately bounded (or input-
to-state practically stable [40], [41]). Hence, Assumption 2 is
satisfied.

To verify Assumption 3, let a number $r > 0$ be given, then
one can find a number $g > 0$ such that, if $t_0 \in \mathbb{R}$, $\|\tilde{\psi}\| < r$,
$\|\bar{\psi}\| < r$, then $|x(t)| \le g$, $|y(t)| \le g$ for all $t \ge t_0$, where
$x(t) = x(t, t_0, \tilde{\psi})$ and $y(t) = x(t, t_0, \bar{\psi})$ are solutions of the
system (3). Let $z(t) = y(t) - x(t)$ be the difference between
these solutions, then

$$
\begin{aligned}
\dot{z}(t) ={}& A(f(z(t) + x(t)) - f(x(t))) \\
& + B(f(z(t - \tau) + x(t - \tau)) - f(x(t - \tau))).
\end{aligned}
$$
Define a Lyapunov-Krasovskii functional by the formula

$$ \tilde{V}(z_t, x_t) = \sum_{i=1}^{n} \nu_i |z_i(t)| + \sum_{i=1}^{n} \mu_i \int_{t-\tau}^{t} |f_i(z_i(\xi) + x_i(\xi)) - f_i(x_i(\xi))| d\xi, $$

where $\mu_i$ is a nonnegative constant. Consider the upper-right Dini derivative of $\tilde{V}(t) = \tilde{V}(z_t, x_t)$:

$$ \begin{aligned} \partial^+ \tilde{V}(t) \le & | |f_1(z_1(t) + x_1(t)) - f_1(x_1(t))| \\ & \dots |f_n(z_n(t) + x_n(t)) - f_n(x_n(t))| |(A^\top \nu + \mu) \\ & + | |f_1(z_1(t - \tau) + x_1(t - \tau)) - f_1(x_1(t - \tau))| \\ & \dots |f_n(z_n(t - \tau) + x_n(t - \tau)) - f_n(x_n(t - \tau))| |(B^\top \nu - \mu). \end{aligned} $$

Similarly to the first part of the proof, if $\mu = B^\top \nu$, then the inequality

$$ \partial^+ \tilde{V}(t) \le -\gamma \sum_{i=1}^{n} |f_i(z_i(t) + x_i(t)) - f_i(x_i(t))| $$

holds, where $\gamma$ is a positive constant. Denote

$$ W(z) = \min_{|x| \le g} \sum_{i=1}^{n} |f_i(z_i + x_i) - f_i(x_i)|, $$

then the function $W(z)$ is continuous and positive definite. Consequently, we obtain

$$ \partial^+ \tilde{V}(t) \le -\gamma W(z(t)). $$

Note that the functions $f_i(x_i)$ are locally Lipschitz continuous, therefore,

$$ \alpha_1 |z(t)| \le \tilde{V}(z_t, x_t) \le \alpha_2 \|z_t\| $$

for all $\|z_t\| \le 2g$ and $\|x_t\| \le g$, where $\alpha_1 > 0$, $\alpha_2 > 0$. Next, repeating the conventional arguments [39], for any $\varepsilon \in (0, r)$ and $\delta \in (0, r)$, there exists $\mathbb{T}(\varepsilon, \delta) \ge 0$ such that if $\|z_{t_0}\| < \delta$, then $|z(t)| < \varepsilon$ for $t \ge t_0 + \mathbb{T}(\varepsilon, \delta)$. Indeed, assume that there is no such a time $\mathbb{T}(\varepsilon, \delta)$, then there exists a growing unbounded sequence $\{t_k\}$ such that $|z(t_k)| \ge \varepsilon$ for $k = 1, 2, \dots$. Let us note that $|\dot{z}(t)| \le \beta$, where the positive constant $\beta$ depends only on chosen value of $r$. Hence, $|z(t)| \ge \varepsilon/2$ for $t \in [t_k - \varepsilon/(2\beta), t_k + \varepsilon/(2\beta)]$. Without loss of generality, we may assume that the intervals $[t_k - \varepsilon/(2\beta), t_k + \varepsilon/(2\beta)]$ do not overlap (by considering a subsequence with sufficiently big distances between $t_k$ and $t_{k+1}$), then

$$ \begin{aligned} \alpha_1 |z(t)| \le \tilde{V}(z_t, x_t) \le \tilde{V}(z_{t_0}, x_{t_0}) & - \gamma \frac{\varepsilon}{\beta} k \min_{\delta/2 \le |z| \le 2g} W(z) \\ \le 2\alpha_2 r & - \gamma \frac{\varepsilon}{\beta} k \min_{\delta/2 \le |z| \le 2g} W(z) \end{aligned} $$

for $k = 1, 2, \dots$. As a result, if $k$ is sufficiently large, then we arrive at a contradiction since the value of $\tilde{V}(z_t, x_t)$ is not growing for all $t \ge t_0$. Therefore, one can find $\mathbb{T}(\varepsilon, \delta)$ such that $|z(t)| < \varepsilon$ for $t \ge t_0 + \mathbb{T}(\varepsilon, \delta)$, and Assumption 3 is satisfied. $\square$

<figure><img src="image_2.png" /><figcaption>Figure 1. Results of simulation for a Persidskii system: the variables $x_1, x_2, x_3$ are plotted versus time $t$ for three different initial conditions</figcaption></figure>

**Example 1.** For a Persidskii system, let $n = 3, \tau = 0.5$,

$$ A = \begin{bmatrix} -4 & 1 & 0 \\ 2 & -5 & 1 \\ 1 & 0 & -2 \end{bmatrix}, \quad B = \begin{bmatrix} 0 & 1 & 1 \\ 1 & 1 & 0 \\ 0 & 1 & 0 \end{bmatrix}, \\ f(x) = [x_1^3 \ x_2 \ |x_3|x_3]^\top, \quad H(t) = [\sin(t) \ \cos(2t) \ \sin^2(t)], $$

then all required conditions are satisfied, and the system is convergent. The results of simulation of this system for three different initial conditions with the use of Euler explicit method and the discretization step $h = 0.01$ are shown in Fig. 1.

## V. CONVERGENCE CONDITIONS FOR A LOTKA–VOLTERRA MODEL

Let the system

$$ \dot{x}_i(t) = x_i(t) \left( \sum_{j=1}^{n} a_{ij} f_j(x_j(t)) + \sum_{j=1}^{n} b_{ij} f_j(x_j(t-\tau)) + h_i(t) \right), \quad i = 1, \dots, n, \quad (5) $$

be given, where $t \in \mathbb{R}$, $x_i(t) \in \mathbb{R}_+$, the functions $f_j(x_j)$ are defined for $x_j \in \mathbb{R}_+$, $a_{ij}$, $b_{ij}$ are constant coefficients, the functions $h_i(t)$ are continuous for $t \in \mathbb{R}$ and $T$-periodic, $T > 0$, $\tau$ is a constant positive delay.

The system (2) is a generalized Lotka–Volterra model with delay describing the interaction of species in a biological community [42]–[44]. In this model, $x_i(t)$ is the population density of the $i$-th species, the function $h_i(t)$ characterizes the intrinsic growth rate of the $i$-th population, the terms $a_{ii}x_i(t)f_i(x_i(t))$ and $b_{ii}x_i(t)f_i(x_i(t-\tau))$ reflect the self-interaction in the $i$-th population, whereas the terms $a_{ij}x_i(t)f_j(x_j(t))$ and $b_{ij}x_i(t)f_j(x_j(t-\tau))$ for $j \ne i$ describe the influence of population $j$ on dynamics of population $i$. Taking into account the biological meaning of the model, we will consider the system (5) in $\mathcal{X} = \text{int} \mathbb{R}_+^n$ (i.e., then all populations stay alive), then initial functions $\varphi$ for solutions $x(t) = x(t, t_0, \varphi)$ of (5) belong to the space $C([-\tau, 0], \mathcal{X})$.
Following the assumptions common in the system biology [42]–[44], we consider the case where functions $f_i(x_i)$ possess the following properties:

1) $f_i(x_i)$ are continuous, strictly increasing and locally Lipschitz for $x_i \in \mathbb{R}_+$;

2) $f_i(x_i) \ge 0$ for $x_i \in \mathbb{R}_+$, and $f_i(0) = 0$;

3) $f_i(x_i) \to +\infty$ as $x_i \to +\infty$.

An actual problem that arises when analyzing the dynamics of biological models is to find the conditions ensuring sustainable coexistence of species [42]–[44]. A sustainable biological system should possess the following two important properties: ultimate boundedness and persistence. The first one means the existence of a compact domain in the state space of the system such that each motion enters into it in a finite time and remains in this domain thereafter. The property of persistence means that in the process of evolution, the species do not die out, and, moreover, no matter how small their initial quantities are, starting from a certain time instant, the quantities of species will exceed some fixed positive values. If a system admits both of these properties, then it is called permanent [45]:

**Definition 3.** The system (5) is called permanent if there exist numbers $\delta_1$ and $\delta_2$ satisfying the inequalities $0 < \delta_1 < \delta_2$ and, for any solution $x(t, t_0, \varphi) = (x_1(t, t_0, \varphi), \dots, x_n(t, t_0, \varphi))^{\top}$ with $t_0 \in \mathbb{R}$, $\varphi \in C([-\tau, 0], \mathcal{X})$, one can choose $\mathcal{T} \ge t_0$ such that $\delta_1 \le x_i(t, t_0, \varphi) \le \delta_2$, $i = 1, \dots, n$, for $t \ge \mathcal{T}$.

**Definition 4.** The system (5) will be called convergent in $\mathcal{X}$ if it admits a unique periodic solution $\Omega(t)$ in $\mathcal{X}$ such that:

(a) $\Omega(t)$ is uniformly stable in $\mathcal{X}$;

(b) for any $0 < r_1 < r_2$,

$$|x(t, t_0, \psi) - \Omega(t)| \to 0$$

as $t - t_0 \to +\infty$ uniformly with respect to $t_0 \in \mathbb{R}$, $\psi \in C([-\tau, 0], \mathcal{X})$, $\|\psi\| < r_2$, $\psi_i(0) \ge r_1$, $i = 1, \dots, n$.

**Assumption 6.** Let $h_i(t) \ge \bar{h} = \text{const} > 0$ for $t \in \mathbb{R}$, $i = 1, \dots, n$.

From this property and the characteristics of nonlinearities, it follows that $\mathcal{X}$ is an invariant set for (5).

**Assumption 7.** The matrix $A = \{a_{ij}\}_{i,j=1}^n$ is Metzler, the matrix $B = \{b_{ij}\}_{i,j=1}^n$ is nonnegative and the matrix $A+B$ is Hurwitz.

**Theorem 3.** Under assumptions 6 and 7, the system (5) is convergent in $\mathcal{X}$.

*Proof.* From the results of [45] it follows that, under assumptions 6 and 7, the system (5) is uniformly permanent. This is a stronger property than one given in Definition 3, which means that there exist numbers $\delta_1$ and $\delta_2$, $0 < \delta_1 < \delta_2$, such that for any $0 < r_1 < r_2$, one can find $\mathcal{T} > 0$ such that if $t_0 \in \mathbb{R}$, $\psi \in C([-\tau, 0], \mathcal{X})$, $\|\psi\| < r_2$, $\psi_i(0) \ge r_1$, $i = 1, \dots, n$, then $\delta_1 \le x_i(t, t_0, \psi) \le \delta_2$, $i = 1, \dots, n$, for $t - t_0 \ge \mathcal{T}$.

Let $u_i(t) = \log x_i(t)$, $i = 1, \dots, n$. If $x(t) \in \mathcal{X}$, then $u(t) \in \mathbb{R}^n$. We obtain

$$\dot{u}(t) = Ap(u(t)) + Bp(u(t - \tau)) + H(t), \quad (6)$$

where $p_i(u_i) = f_i(e^{u_i}) - f_i(1)$ and $H(t) = [h_1(t) \dots h_n(t)]^\top + (A + B)f(1_n)$ is a new T-periodic

<figure><img src="image_3.png" /><figcaption>Figure 2. Results of simulation for a Lotka–Volterra system: the variables $x_1, x_2, x_3$ are plotted versus time $t$ for three different initial conditions</figcaption></figure>

time function. The system (6) is a counterpart of the system (3), but here $p_i(u_i) \to -\infty$ as $u_i \to -\infty$. If (5) is uniformly permanent in $\mathcal{X}$, then solutions of (6) are uniformly ultimately bounded in $\mathbb{R}^n$ and Assumption 2 can be verified in the permanence set. The final part of the proof is similar to that of Theorem 2. $\square$

**Example 2.** Let $n = 3$ for a Lotka–Volterra system, and $\tau = 0.5$,

$$A = \begin{bmatrix} -3 & 1 & 1 \\ 0 & -4 & 1 \\ 1 & 0 & -5 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & 0 & 1 \\ 1 & 0 & 1 \\ 0 & 1 & 1 \end{bmatrix},$$

$$f(x) = [x_1 \ x_2^2 \ \frac{x_3^3}{1 + x_3^2}]^\top,$$

$$H(t) = [2 + \sin(2t) \ 1 + \cos^2(t) \ 1 + \sin^2(2t)],$$

then again all required conditions are verified, and the system is convergent. The results of simulation of this system for three different initial conditions with the use of Euler explicit method and the discretization step $h = 0.01$ are shown in Fig. 2.

## VI. CONCLUSION

The paper studied the existence of periodic steady-state solutions within time-delay convergent systems. The main advantage of the newly established conditions is the ability to analyze highly nonlinear dynamics without the necessity for a meaningful linear approximation. This is particularly relevant in systems where traditional methods may fail to capture the complexity of the behavior. The conditions were specifically developed for two types of systems: Persidskii and Lotka-Volterra ones. These models are notable for their applications in various fields, including ecology and economics, where understanding the dynamics of interactions over time is crucial, and where periodic steady-states are frequently observed. The efficiency of these new existence conditions has been demonstrated through academic examples of the aforementioned models. Extension of these conditions to the class of time-delay systems with distributed delays can be considered as a direction for future research.
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