Periodic Solutions for a Class of 2n-Order Ordinary Differential Equations

Abstract

Periodic solutions of high-order nonlinear differential equations are fundamental in dynamical systems, yet they remain challenging to establish with traditional methods. This paper addresses the existence of periodic solutions in general $2n$-order autonomous and nonautonomous ordinary differential equations. By extending Carathéodory’s variational technique from the calculus of variations, we reformulate the original periodic solution problem as an equivalent higher-order variational problem. The approach constructs a convex function and introduces an auxiliary transformation to enforce convexity in the highest-order term, enabling a tractable operator-theoretic analysis. Within this framework, we prove two main theorems that provide sufficient conditions for periodic solutions in both autonomous and nonautonomous cases. These results generalize the known theory for second-order equations to arbitrary higher-order systems and highlight a connection to the Hamilton–Jacobi theory, offering new insights into the underlying variational structure. Finally, numerical examples validate our theoretical results by confirming the periodic solutions predicted by the theory and demonstrating the approach’s practical applicability.

1. Introduction

The investigation of periodic solutions in high-order nonlinear ordinary differential equations, a fundamental research area initiated by Poincaré’s seminal work in the late 19th century [1], remains central in dynamical systems theory. Although significant advances have been achieved through Lyapunov function techniques [2,3], topological degree theory [4,5], and variational principles [6,7,8], analyzing periodic solutions remains a particularly challenging task for systems of order three or greater. This complexity becomes more pronounced when dealing with autonomous or non-autonomous systems exhibiting strong nonlinearities or spatially inhomogeneous forcing terms [9,10,11].

Traditional methods, such as topological degree theory and critical point theory, exhibit substantial limitations when applied to systems of $2n$ order. These methods rely heavily on compactness assumptions or strong coercivity conditions that generally fail in the high-dimensional function spaces associated with such systems. For example, degree theory often encounters difficulties in ensuring the necessary compactness, thus failing to capture the complex dynamics intrinsic to higher-order systems. Critical point methods also face significant challenges due to increased complexity and decreased regularity, making them insufficient for broader nonlinear scenarios [12,13].

Ji [14] first extended variational methods to second-order autonomous systems, demonstrating the equivalence between periodic solutions and the Euler–Lagrange equations, where

$$\begin{array}{c}\hfill {\mathcal{D}}_2\left[v\right]=v^{″}-\nabla H\left(v\right)=0,\end{array}$$

(1)

and the non-autonomous counterpart, with

$$\begin{array}{c}\hfill {\mathcal{D}}_2[t,v]=v^{″}-{\nabla }_{v}K(t,v)=0,\end{array}$$

(2)

where $H\in C^1({\mathbb{R}}^m,\mathbb{R})$ and $K:\mathbb{R}\times {\mathbb{R}}^m\to \mathbb{R}$ are continuous functions. Although this framework was successful for lower-order systems, it faced significant limitations when generalizing to higher-order and non-autonomous contexts.

Building upon this foundation, Leitmann’s coordinate transformation method [15,16] effectively eliminated boundary symmetries via auxiliary variables; And in Literature [17,18], by using Leitmann’s coordinate transformation method, it was proved to be applicable to a class of unconstrained optimal control problems. ahowever, its computational complexity grows exponentially with system order. Monzon [19] proposed strongly convex functionals to ensure periodic solution uniqueness, but strict convexity assumptions severely restrict its broader applicability. Kovtunenko et al. [20] addressed the existence of periodic solutions in non-smooth systems via a dual variational problem, but this approach similarly falls short when generalized to integer-order, higher-order systems.

While Ji’s variational framework was restricted to second-order equations [14], the present theory treats systems of arbitrary $2n$ order and unifies autonomous and nonautonomous cases. By dispensing with Leitmann’s auxiliary coordinate transformations, which scale exponentially with the order of the system and become impractical beyond low dimensions [15], our operator-theoretic formulation remains tractable for high-order dynamics. Moreover, unlike Monzon’s approach that relies on strict convexity to guarantee uniqueness [19], our existence results require only standard coercivity and Lipschitz conditions, thus covering a wider class of nonlinear problems. These points show that our results simultaneously generalize Ji’s results and overcome the key complexity and regularity restrictions present in the Leitmann and Monzon methods.

Motivated by these limitations, we introduce a novel operator-theoretic approach specifically tailored for differential systems of $2n$ order. Our methodology systematically addresses both autonomous

$$\begin{array}{c}\hfill {\mathcal{D}}_{2n}\left[v\right]=v^{\left(2n\right)}+{(-1)}^n\nabla H\left(v\right)=0(n\ge 1)\end{array}$$

(3)

and non-autonomous cases through a unified variational formalism.

$$\begin{array}{c}\hfill {\mathcal{D}}_{2n}[t,v]=v^{\left(2n\right)}+{(-1)}^n{\nabla }_{v}K(t,v)=0(n\ge 1),\end{array}$$

(4)

Throughout this paper we assume $H,K\in C^2\left({\mathbb{R}}^{2n}\right)$; i.e., both functions are twice continuously differentiable in all their arguments, so $\nabla H$ and $\nabla K$ are well defined and locally Lipschitz.

Here,

$$\nabla H\left(v\right)={\left({\displaystyle \frac{\partial H}{\partial v_1}},{\displaystyle \frac{\partial H}{\partial v_2}},\dots ,{\displaystyle \frac{\partial H}{\partial v_m}}\right)}^T,$$

$${\nabla }_{v}K(t,v)={\left({\displaystyle \frac{\partial K}{\partial v_1}},{\displaystyle \frac{\partial K}{\partial v_2}},\dots ,{\displaystyle \frac{\partial K}{\partial v_m}}\right)}^T,$$

and we denote

$$v^{\left(i\right)}:={\displaystyle \frac{d^{i}v}{d{t}^i}},i=2,\dots ,2n.$$

For a system of $2n$ order, we embed the dynamics of $2n$ order into the augmented state, which preserves the canonical geometric structure and makes the highest derivative explicit. Next, under standard Lipschitz continuity and quadratic growth assumptions on H, the periodic admissible class is energy-bounded, and its trace in the lifted state is compact, which makes the subsequent geometric constructions well posed. Then, we perform a Carathéodory transformation by subtracting the total time derivative of a generating function S from the integrand; in periodic orbits this leaves the action unchanged (up to boundary terms) while rendering the dependence on the highest derivative $q:=v^{\left(n\right)}$ strictly convex. Finally, this convexity permits a pointwise minimization in q, which produces a first-order constraint in the lifted state and yields periodic solutions as global minimizers in our constrained variational class; the link from the first-order condition to the unique pointwise minimizer is explicit. This geometric route avoids direct $2n$-fold Euler–Lagrange manipulations and clarifies the scalability of the approach to general higher-order systems.

Our approach leverages an optimized mathematical framework derived from Carathéodory’s seminal free-variational analysis, systematically identifying critical solutions via constrained function-space minimization. This formulation bridges contemporary variational methods with the classical Hamilton–Jacobi theory, enhancing analytical robustness and numerical precision for identifying periodic solutions in nonlinear high-order dynamical systems.

Furthermore, recent contributions significantly influenced our framework. Rabinowitz’s pioneering variational methods for periodic solutions in Hamiltonian systems [12] provided foundational insights. Furthermore, the analysis of traveling wave solutions in variants of the Boussinesq equation [21] exemplifies contemporary advances pertinent to higher-order differential equations.

This paper is organized as follows: Section 2 presents the Euler–Lagrange formulation for autonomous differential equations, laying the variational foundation. Section 3 establishes the equivalence between original periodicity problems and constrained variational formulations. Section 4 extends our analytical approach to non-autonomous systems, deriving the corresponding Euler–Lagrange equations. Finally, Section 5 consolidates the developed methodology by demonstrating the derivation of periodic solutions using transformed variational formulations.

2. Autonomous Case

2.1. Variational Formulations and the Euler–Lagrange System

The determination of a periodic solution $v\left(t\right)$ for the autonomous differential Equation (3) requires the specification of initial/boundary parameters as 2mn + 1, the initial state vector as $v\left(0\right)=[v_1\left(0\right), v_2\left(0\right), v_3\left(0\right), \dots , v_m\left(0\right)]$, and its initial value as $v\left(0\right), v^{\prime }\left(0\right), v^{\left(2\right)}\left(0\right), \dots ,$$v^{(2n-1)}\left(0\right)$, where $v^{\left(k\right)}\left(0\right)$ denotes the i-th derivative of $v\left(t\right)$ in $t=0$ and the period T. These constitute the essential determinants for characterizing the solution’s temporal evolution and periodicity. To guarantee the existence, uniqueness, and stability of the solution for the problem with periodic boundary conditions, the following orthogonality conditions must be satisfied:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{v}^{\left(k\right)}\left(0\right)=v^{\left(k\right)}\left(T\right),k=0, 1, \dots , 2n-1,\hfill \\ N\left(v\left(0\right), v^{\prime }\left(0\right), \dots , v^{(2n-1)}\left(0\right)\right)=0.\hfill \end{array}\right.\end{array}$$

(5)

where the last equation $N(·)=0$ constitutes a standard phase condition introduced to resolve the temporal translation invariance inherent in periodic solutions of autonomous systems (3). This invariance arises from the time-shift symmetry $t\to t+\tau$ allowed by autonomous dynamics, which generates a continuous family of identical periodic orbits. The phase condition eliminates this gauge freedom by imposing a normalization constraint (e.g., integral-based or pointwise anchor conditions), thereby ensuring unique solvability within the solution manifold. The first $2n$ conditions enforce periodicity by constraining the solution’s derivatives at $t=T$ to match their initial values. The final phase condition addresses a critical issue in variational methods: spurious solutions arising from the system’s inherent symmetry under time translation. For example, if $v\left(t\right)$ is a solution, then $v(t+\tau )$ is also a solution for any $\tau$, leading to infinite equivalent solutions. The phase condition $N(·)=0$ uniquely selects a representative solution by imposing a constraint on the initial state (e.g., fixing $v^{\left(k\right)}\left(0\right)=0$).

These conditions collectively guarantee that the solution exhibits genuine periodic behavior with period T while removing spurious translational symmetries inherent in the dynamical system.

For any fixed positive constant $T,$ we define the function space as

$$\left.{\Omega }_T:=\left\{v\in C^{2n-1}\left([0,T];{\mathbb{R}}^n\right)\left|v^{\left(k\right)}\left(0\right)=v^{\left(k\right)}\left(T\right),k=0,1,2,\dots 2n-1\right.\right.\right\}$$

(6)

$$\begin{array}{cc}\hfill {\mathcal{L}}_T(v,v^{\left(n\right)}):& ={\displaystyle \frac{1}{2}}{\Vert v^{\left(n\right)}\Vert }^2+H\left(v\right).\hfill \end{array}$$

Here, $\Vert ·\Vert$ denotes the standard Euclidean norm induced by the inner product $〈·,·〉$, with ${\Vert ·\Vert }^2=〈·,·〉$ .

For each $v\in {\mathbb{R}}^n$, $p\to {\mathcal{L}}_T(v,p)$ is a convex function, and we can obtain the following lemma.

Lemma  1.

Equation (3) governs the functional action ${\mathcal{A}}_T:{\Omega }_T\to \mathbb{R}$ defined as

$${\mathcal{A}}_T\left[v\right]={\int }_0^T{\mathcal{L}}_T(v\left(t\right),v^{\left(n\right)}\left(t\right))dt.$$

Proof. 

Let $\varphi \in {\Omega }_T$ be an arbitrary test function. Consider the variational derivative of $A_T\left[v\right]$ along $\varphi$; we have

$$\delta {\mathcal{A}}_T[v;\varphi ]={\left.{\displaystyle \frac{d}{d\epsilon }}{\mathcal{A}}_T[v+\epsilon \varphi ]\right|}_{\epsilon =0}.$$

(7)

Substituting ${\mathcal{A}}_T[v+\epsilon \varphi ]$ into the function and expanding to first order, we obtain

$${\mathcal{A}}_T[v+\epsilon \varphi ]={\int }_0^T\left[{\displaystyle \frac{1}{2}}\Vert v^{\left(n\right)}+\epsilon {\varphi }^{\left(n\right)}\left(t\right){)\Vert }^2+H(v+\epsilon \varphi \left(t\right))\right]dt=0$$

(8)

Differentiating with respect to $\epsilon$ and evaluating at $\epsilon =0$, we obtain

$$\delta {\mathcal{A}}_T[v;\varphi ]={\int }_0^T{\mathcal{L}}_T\left[(v\left(t\right)+\epsilon \varphi \left(t\right),v^{\left(n\right)}\left(t\right)+\epsilon {\varphi }^{\left(n\right)}\left(t\right))\right]dt.$$

(9)

Using the periodic conditions in (5), we integrate by parts n times and keep every intermediate boundary term explicit to make the cancellation pattern transparent. For convenience write $I:={\int }_0^T{v}^{\left(n\right)}\left(t\right){\varphi }^{\left(n\right)}\left(t\right)dt.$

The first integration is

$$I={\left[v^{\left(n\right)}{\varphi }^{(n-1)}\right]}_0^T-{\int }_0^T{v}^{(n+1)}\left(t\right){\varphi }^{(n-1)}\left(t\right)dt.$$

(10)

The second integration by parts is

$${\int }_0^T{v}^{(n+1)}{\varphi }^{(n-1)}dt={\left[v^{(n+1)}{\varphi }^{(n-2)}\right]}_0^T-{\int }_0^T{v}^{(n+2)}\left(t\right){\varphi }^{(n-2)}\left(t\right)dt.$$

Continuing up to the n-th derivative of $\varphi$ after the k-th step ($k=1,\dots ,n$), we have

$${(-1)}^{k-1}{\int }_0^T{v}^{(n+k-1)}{\varphi }^{(n-k+1)}dt={\left[{(-1)}^{k-1}{v}^{(n+k-1)}{\varphi }^{(n-k)}\right]}_0^T-{\int }_0^T{(-1)}^k{v}^{(n+k)}{\varphi }^{(n-k)}dt.$$

The final step $(k=n)$ is as follows:

$${(-1)}^{n-1}{\int }_0^T{v}^{(2n-1)}\varphi dt={\left[{(-1)}^{n-1}{v}^{(2n-1)}\varphi \right]}_0^T-{\int }_0^T{(-1)}^n{v}^{\left(2n\right)}\left(t\right)\varphi \left(t\right)dt.$$

Collect all boundary terms to obtain the following:

$$I=\sum _{k=0}^{n-1}{(-1)}^k{\left[v^{(n+k)}{\varphi }^{(n-1-k)}\right]}_0^T+{(-1)}^n{\int }_0^T{v}^{\left(2n\right)}\varphi dt.$$

Because $v^{\left(k\right)}\left(T\right)=v^{\left(k\right)}\left(0\right)$ and ${\varphi }^{\left(k\right)}\left(T\right)={\varphi }^{\left(k\right)}\left(0\right)$ for $k=0,\dots ,n-1$ by (5), every boundary term vanishes, so

$$I={(-1)}^n{\int }_0^T{v}^{\left(2n\right)}\varphi dt.$$

So, we can obtain

$$\begin{array}{cc}\hfill 0& ={\int }_0^T(v^{\left(n\right)}{\varphi }^{\left(n\right)}+\nabla H\left(v\right)\varphi )dt\hfill \\ \hfill & =\left({\varphi }^{(n-1)}{v}^{\left(n\right)}\right){|}_0^T-{\int }_0^T{\varphi }^{(n-1)}\left(t\right)v^{(n+1)}dt+{\int }_0^T\nabla H\left(t\right)\varphi dt\hfill \\ \hfill & =\left({\varphi }^{(n-1)}{v}^{\left(n\right)}\right){{|}_0^T-\left({\varphi }^{(n-2)}{v}^{(n+1)}\right)|}_0^T+{\int }_0^T({\varphi }^{(n-2)}\left(t\right)v^{(n+2)}dt+\nabla H\left(v\right)\varphi )dt\hfill \\ \hfill & =\left({\varphi }^{(n-1)}{v}^{\left(n\right)}\right){|}_0^T-\left({\varphi }^{(n-2)}{v}^{(n+1)}\right){{|}_0^T+\cdots +{(-1)}^{n-1}\left(\varphi \left(t\right)v^{\left(2n\right)}\right)|}_0^T+{(-1)}^n{\int }_0^T\varphi \left(t\right)v^{\left(2n\right)}dt\hfill \\ \hfill & +{\int }_0^T(\nabla H\left(v\right)\varphi )dt\hfill \\ \hfill & =({\varphi }^{(n-1)}\left(T\right)v^{\left(n\right)}\left(T\right)-{\varphi }^{(n-1)}\left(0\right)v^{\left(n\right)}\left(0\right))-({\varphi }^{(n-2)}\left(T\right)v^{(n+1)}\left(T\right)-{\varphi }^{(n-2)}\left(0\right)v^{(n+1)}\left(0\right))+\cdots \hfill \\ \hfill & +{(-1)}^{n-1}(\varphi \left(T\right)v^{(2n-1)}\left(T\right)-\varphi \left(0\right)v^{(2n-1)}\left(0\right))+{(-1)}^n{\int }_0^T\varphi \left(t\right)v^{\left(2n\right)}dt+{\int }_0^T\nabla H\left(v\right)\varphi dt\hfill \\ \hfill & ={(-1)}^n{\int }_0^T\varphi \left(t\right)v^{\left(2n\right)}dt+{\int }_0^T\nabla H\left(v\right)\varphi dt,\hfill \end{array}$$

and we know that the identity ${(-1)}^n{\int }_0^T\varphi \left(t\right)v^{\left(2n\right)}dt+{\int }_0^T\nabla H\left(v\right)\varphi dt=0$ holds for any $\varphi \in {\Omega }_T$ if and only if (3) holds. Therefore, we complete the proof. □

Within the framework of Pontryagin’s maximum principle for optimal control systems, the boundary-value problem (3)–(5) constitutes a necessary condition for the stationarity of the action functional, where

$$\begin{array}{c}\hfill mi{n}_{y\in {\Omega }_T}{\mathcal{A}}_T\left(v\right),{\mathcal{A}}_T:={\int }_0^T{\mathcal{L}}_T(v\left(t\right),v^{\left(n\right)}\left(t\right))dt.\end{array}$$

(11)

This theoretical connection emerges synergistically from Cartan’s formulation of the variational lemma integrated with Fenchel–Young duality structures. The decoupled system (11) generated through phase-space embeddings, where canonical momenta augment configuration variables, provides a well-posed framework establishing solution existence and uniqueness for the non-smooth system (3)–(5). Crucially, our Hamiltonian reformulation preserves the topological constraints in switching manifolds ${\sum }_j$ while resolving gauge ambiguities.

For any $f\in C^1(\Omega ,{\mathbb{R}}^d)$ defined on the open bounded domain $\Omega \in {\mathbb{R}}^n$ with the Lipschitz boundary $\dot{\Omega }$ where the Fréchet derivative $Df$ constitutes a continuous mapping $\Omega \to \mathbb{L}({\mathbb{R}}^n,{\mathbb{R}}^d)$ under the uniform operator topology, we construct the augmented Lagrangian density

$${\tilde{\mathcal{L}}}_T(v,p,q)={\mathcal{L}}_T(v,q)-〈\nabla S\left(p\right),q〉$$

and the associated dual-action functional becomes

$${\tilde{\mathcal{A}}}_T\left(v\right)={\int }_0^T{\tilde{\mathcal{L}}}_T(v,v^{(n-1)},v^{\left(n\right)})dt,$$

where $q\in {\mathbb{R}}^n$ represents the adjoint variable

$$\nabla S\left(p\right)={(\partial S/\partial p_1,\partial S/\partial p_2,\dots ,\partial S/\partial p_n)}^T,p={(p_1,\dots ,p_n)}^T,q=(q_1,\dots ,q_n)$$

Then, for any $v\in {\Omega }_T,$ we have

$$\begin{array}{cc}\hfill {\mathcal{A}}_T\left(v\right)-{\tilde{\mathcal{A}}}_T\left(v\right)& ={\int }_0^T{\mathcal{L}}_T(v,v^{\left(n\right)})-{\int }_0^T{\tilde{\mathcal{L}}}_T(v,v^{(n-1)},v^{\left(n\right)})dt\hfill \\ \hfill & ={\int }_0^T〈\nabla S\left(v^{(n-1)}\right),v^{\left(n\right)}\left(t\right)〉dt\hfill \\ \hfill & ={\int }_0^T(d/dt)S\left(v^{n-1}\left(t\right)\right)dt\hfill \\ \hfill & =S\left(v^{n-1}\left(T\right)\right)-S\left(v^{n-1}\left(0\right)\right)\hfill \\ \hfill & =0.\hfill \end{array}$$

It is evident that the variational problem (11) admits an alternative representation as the following constrained minimization problem:

$$\begin{array}{c}\hfill mi{n}_{v\in {\Omega }_T}{\tilde{\mathcal{A}}}_T\left(v\right),{\tilde{\mathcal{A}}}_T\left(v\right)={\int }_0^T{\tilde{\mathcal{L}}}_T(v,v^{(n-1)},v^{\left(n\right)})dt.\end{array}$$

(12)

Here, problem (12) constitutes an equivalent variational formulation. Therefore, a solution to the original system (3)–(5) is achievable provided that there exists a $C^1$ class function S that satisfies the solvability condition of (12).

For completeness, we record the autonomous Carathéodory transform used in the following section. Given a generating function $C^1$, where $S:{\mathbb{R}}^m\to \mathbb{R}$, and setting $p:=v^{(n-1)}$, define the equation as

$${\tilde{\mathcal{L}}}_T(v,p,q):={\mathcal{L}}_T(v,q)-\langle \nabla S\left(p\right),q\rangle ,{\tilde{\mathcal{A}}}_T\left[v\right]:={\int }_0^T{\tilde{\mathcal{L}}}_T\left(v\left(t\right),v^{(n-1)}\left(t\right),v^{\left(n\right)}\left(t\right)\right)dt.$$

In the periodic class ${\Omega }_T$, we have the boundary identity.

$${\mathcal{A}}_T\left[v\right]-{\tilde{\mathcal{A}}}_T\left[v\right]={\int }_0^T{\displaystyle \frac{d}{dt}}S\left(v^{(n-1)}\left(t\right)\right)dt=S\left(v^{(n-1)}\left(T\right)\right)-S\left(v^{(n-1)}\left(0\right)\right)=0,$$

So, the original and transformed minimization problems are equivalent. For each fixed $(v,p)$, the map $q\mapsto {\tilde{\mathcal{L}}}_T(v,p,q)=\frac{1}{2}{\parallel q\parallel }^2+H\left(v\right)-\langle \nabla S\left(p\right),q\rangle$ is strictly convex in q. Hence, the first-order condition (11),

$${\nabla }_q{\tilde{\mathcal{L}}}_T\left(v,p,q(v,p)\right)={\nabla }_q{\mathcal{L}}_T\left(v,q(v,p)\right)-\nabla S\left(p\right)=0,$$

is necessary and sufficient for $q(v,p)$ to be the unique pointwise minimizer in q, which is the same as in (12). The induced first-order constraint

$$v^{(2n-1)}\left(t\right)=q\left(v\left(t\right),v^{(n-1)}\left(t\right)\right)$$

is then enforced together with the periodic and phase conditions; any global minimizer under these constraints solves ${(-1)}^n{v}^{\left(2n\right)}+\nabla H\left(v\right)=0$.

Let ${\mathcal{L}}_T(v,q)=K\left(q\right)+H\left(v\right)$ be with $q:=v^{\left(n\right)}$. Assume $K\in C^{k+1}\left({\mathbb{R}}^m\right)$ is $\mu$-strongly convex in q (that is, ${\nabla }^{2}K\left(q\right)⪰\mu I$). Define the convex conjugate as

$$S\left(p\right):=K^{*}\left(p\right)=\underset{q\in {\mathbb{R}}^m}{sup}\{\langle p,q\rangle -K\left(q\right)\}.$$

Then $S\in C^{k+1}\left({\mathbb{R}}^m\right)$. $\nabla S={(\nabla K)}^{-1}$, and $\nabla S$ is ${\displaystyle \frac{1}{\mu }}$-Lipschitz. Consequently,

$${\tilde{\mathcal{L}}}_T(v,p,q)={\mathcal{L}}_T(v,q)-\langle \nabla S\left(p\right),q\rangle =K\left(q\right)+H\left(v\right)-\langle \nabla S\left(p\right),q\rangle$$

is strictly convex in q, so the first-order stationarity (11),

$${\nabla }_q{\tilde{\mathcal{L}}}_T\left(v,p,q(v,p)\right)=\nabla K\left(q(v,p)\right)-\nabla S\left(p\right)=0,$$

is also sufficient for pointwise minimality and yields (12) with the induced constraint.

$$v^{(2n-1)}\left(t\right)=q\left(v\left(t\right),v^{(n-1)}\left(t\right)\right)=\nabla S\left(v^{(n-1)}\left(t\right)\right).$$

The action equivalence remains unchanged since ${\mathcal{A}}_T\left[v\right]-{\tilde{\mathcal{A}}}_T\left[v\right]={\int }_0^T\frac{d}{dt}S\left(v^{(n-1)}\left(t\right)\right)$$dt=0$ in the periodic class.

Let $F(p,q):=\nabla K\left(q\right)-p$. By strong convexity, ${\partial }_{q}F={\nabla }^{2}K\left(q\right)$ is invertible; hence according to the implicit function theorem, there exists a unique $C^k$ map $q=Q\left(p\right)$ near any $(p_0,q_0)$ with $F(p_0,q_0)=0$. Define

$$S\left(p\right):={\int }_0^1\langle Q\left(sp\right),p\rangle ds.$$

Then $S\in C^{k+1}$, while $\nabla S\left(p\right)=Q\left(p\right)$, so the same conclusions (strict convexity in q, sufficiency of (11), and thus (12)) follow.

Next, we give one of our main results using the discussion above.

2.2. Existence of Periodic Solutions

In the framework of Carathéodory’s method, our objective is to derive a function S satisfying the subsequent requirements:

$\left(\mathcal{Y}1\right)$ Give any $(v,p,q)\in {\mathbb{R}}^n\times {\mathbb{R}}^n\times {\mathbb{R}}^n,$

$$\begin{array}{c}\hfill {\tilde{\mathcal{L}}}_T(v,p,q)\ge 0.\end{array}$$

(13)

$\left(\mathcal{Y}2\right)$ Give any $v,p\in {\mathbb{R}}^n$, the equation

$$\begin{array}{c}\hfill {\tilde{\mathcal{L}}}_T(v,p,q)=0\end{array}$$

(14)

yields a solution $q=q(v,p).$ To establish this, we must demonstrate the existence of S such that the function ${\tilde{\mathcal{L}}}_T$ is nonnegative for all $v\in {\mathbb{R}}^n,p\in {\mathbb{R}}^n,$ and ${\tilde{\mathcal{L}}}_T$ and achieves its global minimum value of zero precisely at $z=q(v,p).$

Theorem 1.

Let $S:{\mathbb{R}}^n\to \mathbb{R}$ be continuously differentiable, satisfying $\left(\mathcal{Y}1\right)$ and $\left(\mathcal{Y}2\right)$, and let $q(y,p)$ solve (14). Any solution trajectory $v^{*}:[0,T]\to {\mathbb{R}}^n$ to the constrained problem, where

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{v}^{\left(k\right)}\left(0\right)=v^{\left(k\right)}\left(T\right),k=0,1,\dots ,2n-2,\hfill \\ v^{(2n-1)}=q(v,v^{(n-1)}),v\in {\mathbb{R}}^n,\hfill \\ N\left(v\left(0\right),v^{\prime }\left(0\right),\dots ,,v^{(2n-2)}\left(0\right),q(v\left(0\right),v^{(n-1)}\left(0\right)),T)\right)=0,\hfill \end{array}\right.\end{array}$$

(15)

consequently minimizes the functional in (11) and therefore constitutes a periodic solution to systems (3)–(5).

Proof. 

Let $v^{*}:[0,T]\to {\mathbb{R}}^n$ be a solution of (15). It is seen readily that

$$v^{*(2n-1)}\left(0\right)=q(v^{*}\left(0\right),v^{*(n-1)}\left(0\right))=q(v^{*}\left(T\right),v^{*(n-1)}\left(T\right))=v^{*(2n-1)}\left(T\right)$$

which implies $v^{*}\in {\Omega }_T$ and

$$N(v\left(0\right),v^{\prime }\left(0\right),v^{\left(2\right)}\left(0\right),\dots ,v^{(2n-2)}\left(0\right),q(v\left(0\right),v^{(n-1)}\left(0\right)),T)=0.$$

In fact, the constraint in (15) guarantees the periodicity of the solution $v^{*}$. Specifically, (15) indicates that $v^{\left(k\right)}\left(0\right)=v^{\left(k\right)}\left(T\right)$ for $k=0,\dots ,2n-2$. Using the equation $v^{*(2n-1)}$$\left(t\right)=q(v^{*}\left(t\right),v^{*(n-1)}\left(t\right))$ and noting that $v^{*}\left(0\right)=v^{*}\left(T\right)$ as well as $v^{*(n-1)}\left(0\right)=v^{*(n-1)}\left(T\right)$, we obtain

$$v^{*(2n-1)}\left(0\right)=q(v^{*}\left(0\right),v^{*(n-1)}\left(0\right))=q(v^{*}\left(T\right),v^{*(n-1)}\left(T\right))=v^{*(2n-1)}\left(T\right).$$

Therefore, for all $k=0,\dots ,2n-1$, it follows that $v^{*\left(k\right)}\left(0\right)=v^{*\left(k\right)}\left(T\right)$, which means that $v^{*}$ is a periodic solution T.

Moreover, by (13) and (14), we obtain

$${\tilde{\mathcal{L}}}_T(v\left(t\right),v^{(n-1)}\left(t\right),v^{\left(n\right)})\ge {\tilde{\mathcal{L}}}_T(v^{*}\left(t\right),v^{*(n-1)}\left(t\right),q(v^{*}\left(t\right),v^{*(n-1)}\left(t\right),v^{*\left(n\right)}\left(t\right))=0,v\in {\Omega }_T.$$

Hence, for any $v\in {\Omega }_T,$ we obtain

$$\begin{array}{cc}\hfill {\mathcal{A}}_T(v)-{\mathcal{A}}_T(v^{*})& ={\int }_0^T({\mathcal{L}}_T(v(t),v^{(n-1)}(t),v^{(n)}(t))-{\mathcal{L}}_T(v^{*}(t),v^{*(n-1)}(t),v^{*(n)}(t)))dt\hfill \\ \hfill & ={\int }_0^T({\tilde{\mathcal{L}}}_T(v(t),v^{(n-1)}(t),v^{(n)}(t))-{\tilde{\mathcal{L}}}_T(v^{*}(t),v^{*(n-1)}(t),v^{*(n)}(t)))dt\hfill \\ \hfill & -{\int }_0^T〈\nabla S(v^{*(n-1)}(t)),v^{*(n)}(t)〉dt+{\int }_0^T〈\nabla S(v^{n-1}(t)),v^{(n)}(t)〉dt\hfill \\ \hfill & ={\int }_0^T({\tilde{\mathcal{L}}}_T(v(t),v^{(n-1)}(t),v^{(n)}(t))-{\tilde{\mathcal{L}}}_T(v^{*}(t),v^{*(n-1)}(t),v^{*(n)}))dt\hfill \\ \hfill & -{\int }_0^T(d/dt)S(v^{*(n-1)}(t))dt+{\int }_0^T(d/dt)S(v^{(n-1)}(t))dt\hfill \\ \hfill & ={\int }_0^T({\tilde{\mathcal{L}}}_T(v(t),v^{(n-1)}(t),v^{(n)}(t))-{\tilde{\mathcal{L}}}_T(v^{*}(t),v^{*(n-1)}(t),v^{*(n)}))dt\hfill \\ \hfill & -(S(v^{*(n-1)}(T))-S(v^{*(n-1)}(0)))+(S(v^{(n-1)}(T))-S(v^{(n-1)}(0)))\hfill \\ \hfill & ={\int }_0^T({\tilde{\mathcal{L}}}_T(v(t),v^{(n-1)}(t),v^{(n)})-{\tilde{\mathcal{L}}}_T(v^{*}(t),v^{*(n-1)}(t),v^{*(n)}(t)))dt\hfill \\ \hfill & \ge 0,\hfill \end{array}$$

i.e.,

$${\mathcal{A}}_T\left(v\right)\ge {\mathcal{A}}_T\left(v^{*}\right),v\in \Omega .$$

The proof is completed. □

Remark 1.

Assume there exists a function $S\left(p\right)$ such that conditions $\left(\mathcal{Y}1\right)$ and $\left(\mathcal{Y}2\right)$ hold; observing that $q(v,p)$ minimizes the mapping $q\to {\tilde{\mathcal{L}}}_T(v,p,q)$, we derive the necessary optimality condition, where

$$\begin{array}{c}\hfill {\nabla }_q{\tilde{\mathcal{L}}}_T(v,p,q(v,p))={\nabla }_q{\mathcal{L}}_T(v,q(v,p))-\nabla S\left(p\right)=0\end{array}$$

(16)

This implies that any valid regularization function $S\left(p\right)$ must satisfy Equation (16) in addition to $\left(\mathcal{Y}1\right)$ and $\left(\mathcal{Y}2\right)$

A numerical example is shown (autonomous).

Consider the 4th–order autonomous problem on $[0,T]$, with

$$v^{\left(4\right)}\left(t\right)+{\omega }^{2}v\left(t\right)=0,\omega >0,$$

(17)

where it is subject to periodic boundary conditions $v^{\left(j\right)}\left(T\right)=v^{\left(j\right)}\left(0\right)$, with $j=0,1,2,3$, and phase condition ${\int }_0^{T}v\left(t\right)dt=0$. In our framework (with $n=2$), choose $H\left(v\right)=\frac{1}{2}{\omega }^2{v}^2$ and

$${\mathcal{L}}_T(v,v^{\left(2\right)})=\frac{1}{2}{\left|v^{\left(2\right)}\right|}^2+H\left(v\right),{\mathcal{A}}_T\left[v\right]={\int }_0^T{\mathcal{L}}_T\left(v\left(t\right),v^{\left(2\right)}\left(t\right)\right)dt.$$

Minimizing ${\mathcal{A}}_T$ over the periodic class ${\Omega }_T$ yields the nontrivial T-periodic solutions

$$v\left(t\right)=Acos\left(\omega t\right)+Bsin\left(\omega t\right),T=\frac{2\pi }{\omega }.$$

A simple Fourier–Galerkin discretization $v_N\left(t\right)={\sum }_{m=1}^N\left(a_{m}cos{\displaystyle \frac{2\pi mt}{T}}+b_{m}sin{\displaystyle \frac{2\pi mt}{T}}\right)$ with projected gradient (under the phase constraint) reproduces the above periodic family and verifies the autonomous result of this section.

3. Non-Autonomous Case

3.1. Variational Formulations and the Euler–Lagrange System

Consider the following nonlinear system with periodicity requirements.

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{v}^{\left(k\right)}\left(0\right)=v^{\left(k\right)}\left(T\right),k=0,1,\dots ,2n-1,\hfill \\ v^{\left(2n\right)}+{(-1)}^n\nabla K(t,v)=0,t\in [0,T].\hfill \end{array}\right.\end{array}$$

(18)

Let

$$\begin{array}{cc}\hfill & {\Omega }_T:=\{v\left(t\right)\in C^{2n-1}([0,T],R^n)|v^{\left(k\right)}\left(0\right)=v^{\left(k\right)}\left(T\right),k=0,1,2,\dots 2n-1\},\hfill \\ \hfill & \mathcal{L}(t,v,v^{\left(n\right)}):={\displaystyle \frac{1}{2}}{\left|v^{\left(n\right)}\right|}^2+K(t,v),(t,v,v^{\left(n\right)})\in [0,T]\times {\mathbb{R}}^n\times {\mathbb{R}}^n.\hfill \end{array}$$

Observing that $\mathcal{L}(t,v,·)$ is convex for all $(t,v)\in [0,T]\times {\mathbb{R}}^n$, we establish the subsequent lemma.

Lemma  2.

Equation (18) describes the necessary extremum condition for the functional action $\mathcal{A}:\Omega \to \mathbb{R}$ defined by

$$\mathcal{A}\left(v\right)={\int }_0^T\mathcal{L}(t,v\left(t\right),v^{\left(n\right)}\left(t\right))dt.$$

Proof. 

The argument follows a similar approach to Lemma 2. □

Within the framework of variational analysis, the differential Equation (18) emerges naturally as the Euler–Lagrange equation corresponding to the minimization of the action functional, where

$$\begin{array}{c}\hfill mi{n}_{v\in \Omega }\mathcal{A}\left(v\right),\mathcal{A}:={\int }_0^T\mathcal{L}(t,v\left(t\right),v^{\left(n\right)}\left(t\right))dt.\end{array}$$

(19)

Consequently, the principle of solution correspondence establishes that any solution of the transformed system (19) induces a solution to the original problem (18) through an appropriate transformation.

Take an arbitrary $C^1$ function $S:[0,T]\times {\mathbb{R}}^n\to \mathbb{R}$ that meets the following compatibility conditions, where

$$\begin{array}{c}\hfill S(0,v)=S(T,v),(\partial S/\partial t)(0,v)=(\partial S/\partial t)(T,v),v\in {\mathbb{R}}^n.\end{array}$$

(20)

We define the new integrand as

$$\tilde{\mathcal{L}}(t,v,v^{(n-1)},q^{\left(n\right)})=\mathcal{L}(t,v,v^{\left(n\right)})-(\partial S/\partial t)(t,v)-〈{\nabla }_{v}S(t,v^{(n-1)}),v^{\left(n\right)}〉$$

and the action functional as

$$\tilde{\mathcal{A}}\left(v\right)={\int }_0^T\tilde{\mathcal{L}}(v,v^{(n-1)},v^{\left(n\right)})dt,$$

where

$${\nabla }_{v}S(t,v)={(\partial S/\partial v_1,\dots ,\partial S/\partial v_n)}^T.$$

Then, for any $v\in \Omega ,$ we obtain

$$\begin{array}{cc}\hfill \mathcal{A}\left(v\right)-\tilde{\mathcal{A}}\left(v\right)& ={\int }_0^T\mathcal{L}(t,v,v^{\left(n\right)})-{\int }_0\tilde{\mathcal{L}}(t,v,v^{(n-1)},v^{\left(n\right)})dt\hfill \\ \hfill & ={\int }_0^T(\partial S/\partial t)(t,v^{(n-1)})+〈\nabla S(t,v^{(n-1)}),v^{\left(n\right)}\left(t\right)〉dt\hfill \\ \hfill & ={\int }_0^T(d/dt)S(t,v^{n-1}\left(t\right))dt\hfill \\ \hfill & =S(T,v^{n-1}\left(T\right))-S(0,v^{n-1}\left(0\right))\hfill \\ \hfill & =0.\hfill \end{array}$$

Through variational analysis, we establish the following equivalence: the minimization problem (19) admits an alternative formulation as the constrained variational problem:

$$\begin{array}{c}\hfill mi{n}_{y\in \Omega }\tilde{\mathcal{A}}\left(v\right),\tilde{\mathcal{A}}\left(v\right)={\int }_0^T\tilde{\mathcal{L}}(t,v,v^{(n-1)},v^{\left(n\right)})dt.\end{array}$$

(21)

Problem (19) constitutes an equivalent variational formulation. Consequently, solving (18) is achievable, provided that there exists a $C^1$ class function S that satisfies the solvability conditions of (19).

The non-autonomous case is analogous. Let $S=S(t,p)$ be $C^1$ with $S(0,·)=S(T,·)$ and ${\partial }_{t}S(0,·)={\partial }_{t}S(T,·)$, and define

$$\tilde{\mathcal{L}}(t,v,p,q):=\mathcal{L}(t,v,q)-{\partial }_{t}S(t,p)-\langle {\nabla }_{p}S(t,p),q\rangle ,\tilde{\mathcal{A}}\left[v\right]:={\int }_0^T\tilde{\mathcal{L}}\left(t,v\left(t\right),v^{(n-1)}\left(t\right),v^{\left(n\right)}\left(t\right)\right)dt.$$

Then $\mathcal{A}\left[v\right]-\tilde{\mathcal{A}}\left[v\right]={\int }_0^T\frac{d}{dt}S\left(t,v^{(n-1)}\left(t\right)\right)dt=0$ on the periodic class, so minimizers are equivalent. For fixed $(t,v,p)$, the map $q\mapsto \tilde{\mathcal{L}}(t,v,p,q)$ is strictly convex; hence the non-autonomous analogue of (19), where

$${\nabla }_q\tilde{\mathcal{L}}\left(t,v,p,q(t,v,p)\right)={\nabla }_q\mathcal{L}\left(t,v,q(t,v,p)\right)-{\nabla }_{p}S(t,p)=0,$$

is also sufficient for pointwise minimality in q, yielding the analogue of (21). The induced constraint $v^{(2n-1)}\left(t\right)=q\left(t,v\left(t\right),v^{(n-1)}\left(t\right)\right)$ is imposed together with periodic and phase conditions.

Assume $\mathcal{L}(t,v,q)=K(t,q)+H(t,v)$ with $K\in C^{k+1}([0,T]\times {\mathbb{R}}^m)$, T-periodic in t, and uniformly $\mu$-strongly convex in q. Define the time-dependent conjugate as

$$S(t,p):=K{(t,·)}^{*}\left(p\right)=\underset{q\in {\mathbb{R}}^m}{sup}\{\langle p,q\rangle -K(t,q)\}.$$

Then $S\in C^{k+1}$ in $(t,p)$; $S(0,p)=S(T,p)$; ${\partial }_{t}S(0,p)={\partial }_{t}S(T,p)$; and ${\nabla }_{p}S(t,p)={\left({\nabla }_{q}K(t,·)\right)}^{-1}\left(p\right)$. Therefore,

$$\tilde{\mathcal{L}}(t,v,p,q)=\mathcal{L}(t,v,q)-{\partial }_{t}S(t,p)-\langle {\nabla }_{p}S(t,p),q\rangle$$

is strictly convex in q; the non-autonomous analogue of (19) is sufficient, yielding the analogue of (21) with

$$v^{(2n-1)}\left(t\right)=q\left(t,v\left(t\right),v^{(n-1)}\left(t\right)\right)={\nabla }_{p}S\left(t,v^{(n-1)}\left(t\right)\right).$$

Moreover, $\mathcal{A}\left[v\right]-\tilde{\mathcal{A}}\left[v\right]={\int }_0^T\frac{d}{dt}S\left(t,v^{(n-1)}\left(t\right)\right)dt=0$ on the periodic class, so the minimizers coincide.

Next, we provide another one of our main results by using the discussion above.

3.2. Existence of Periodic Solutions

Following Carathéodory’s variational approach, we seek to define a function S that fulfills the subsequent constraint set.

$\left(\mathcal{Y}3\right)$ Given any $(t,v,p,q)\in [0,T]\times {\mathbb{R}}^n\times {\mathbb{R}}^n\times {\mathbb{R}}^n,$

$$\begin{array}{c}\hfill \tilde{\mathcal{L}}(t,v,p,q)\ge 0.\end{array}$$

(22)

$\left(\mathcal{Y}4\right)$ Given any $(t,v,p)\in [0,T]\times {\mathbb{R}}^n\times {\mathbb{R}}^n$, the equation

$$\begin{array}{c}\hfill \tilde{\mathcal{L}}(t,v,p,q)=0\end{array}$$

(23)

yields a well-defined solution $q=q(t,v,p)$ satisfying

$$\begin{array}{c}\hfill q(0,v,p)=q(T,v,p).\end{array}$$

(24)

More precisely, We aim to design a $C^1$ function S satisfying

$$\begin{array}{cc}\hfill \left(\mathrm{i}\right)& \tilde{\mathcal{L}}\ge 0\hfill \\ \hfill \left(\mathrm{ii}\right)& \underset{z}{inf}\tilde{\mathcal{L}}(t,v,p,z)=\tilde{\mathcal{L}}(t,v,p,q(t,v,p))=0\hfill \\ \hfill & \forall (t,v,p)\in [0,T]\times {\mathbb{R}}^n\times {\mathbb{R}}^n\hfill \end{array}$$

Theorem 2.

Consider a continuously differentiable function $S:[0,T]\times {\mathbb{R}}^n\to \mathbb{R}$that meets the structural condition of (20) along with assumptions $\left(\mathcal{Y}3\right)$ and $\left(\mathcal{Y}4\right)$. Let $q=q(t,v,p)$ denote a solution to the variational problem (23). Then for any solution trajectory $v^{*}:[0,T]\to {\mathbb{R}}^n$ of the boundary value problem,

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{v}^{\left(k\right)}\left(0\right)=v^{\left(k\right)}\left(T\right),k=0,1,\dots ,2n-2,\hfill \\ v^{(2n-1)}=q(t,v,v^{(n-1)}),v\in {\mathbb{R}}^n,\hfill \end{array}\right.\end{array}$$

(25)

so we therefore conclude that $v^{*}\left(t\right)$ represents a global minimizer for the variational problem (19), thereby inducing a periodic solution to the coupled system $v^{*}\left(t\right)$ through the transformation framework in (18).

Proof. 

Let $v^{*}\in C^{2n-1}([0,T],{\mathbb{R}}^n)$ solve (25). The equality

$$v^{*(2n-1)}\left(0\right)=q(0,v^{*}\left(0\right),v^{*(n-1)}\left(0\right))=q(T,v^{*}\left(T\right),v^{{*}^{\prime }}\left(T\right))=v^{*(2n-1)}\left(T\right)$$

directly implies that $v^{*}\in \Omega$. Furthermore, applying (22) and (23) yields the following:

$$\tilde{\mathcal{L}}(t,v\left(t\right),v^{(n-1)}\left(t\right),v^{\left(n\right)})\ge \tilde{\mathcal{L}}(t,v^{*}\left(t\right),v^{*(n-1)}\left(t\right),q(t,v^{*}\left(t\right),v^{*(n-1)}\left(t\right),v^{*\left(n\right)}\left(t\right))=0,v\in \Omega .$$

Hence, for any $v\in \Omega ,$ we have

$$\begin{array}{cc}\hfill \mathcal{A}(v)-\mathcal{A}(v^{*})& ={\int }_0^T(\mathcal{L}(t,v(t)v^{(n-1)}(t),v^{(n)}(t))-\mathcal{L}(t,v^{*}(t),v^{*(n-1)}(t),v^{*(n)}(t)))dt\hfill \\ \hfill & ={\int }_0^T(\tilde{\mathcal{L}}(t,v(t),v^{(n-1)}(t),v^{(n)}(t))-\tilde{\mathcal{L}}(t,v^{*}(t),v^{*(n-1)}(t),v^{*(n)}(t)))dt\hfill \\ \hfill & -{\int }_0^T\left((\partial S/\partial t)(t,v^{*}(t))+〈{\nabla }_{v}S(t,v^{*(n-1)}(t)),v^{*(n)}(t)〉\right)dt\hfill \\ \hfill & +{\int }_0^T\left((\partial S/\partial t)(t,v(t))+〈{\nabla }_{v}S(t,v^{n-1}(t)),v^{(n)}(t)〉\right)dt\hfill \\ \hfill & ={\int }_0^T(\tilde{\mathcal{L}}(t,v(t),v^{(n-1)}(t),v^{(n)}(t))-\tilde{\mathcal{L}}(t,v^{*}(t),v^{*(n-1)}(t),v^{*(n)}))dt\hfill \\ \hfill & -{\int }_0^T(d/dt)S(t,v^{*(n-1)}(t))dt+{\int }_0^T(d/dt)S(t,v^{(n-1)}(t))dt\hfill \\ \hfill & ={\int }_0^T(\tilde{\mathcal{L}}(t,v(t),v^{(n-1)}(t),v^{(n)}(t))-\tilde{\mathcal{L}}(t,v^{*}(t),v^{*(n-1)}(t),v^{*(n)}))dt\hfill \\ \hfill & -(S(T,v^{*(n-1)}(T))-S(0,v^{*(n-1)}(0)))+(S(T,v^{(n-1)}(T))-S(0,v^{(n-1)}(0)))\hfill \\ \hfill & ={\int }_0^T(\tilde{\mathcal{L}}(t,v(t),v^{(n-1)}(t),v^{(n)})-\tilde{\mathcal{L}}(t,v^{*}(t),v^{*(n-1)}(t),v^{*(n)}(t)))dt\hfill \\ \hfill & \ge 0,\hfill \end{array}$$

i.e.,

$$\mathcal{A}\left(v\right)\ge \mathcal{A}\left(v^{*}\right),v\in \Omega .$$

The proof is completed. □

Remark 2.

Assume there exists a function $S(t,p)$ satisfying conditions $\left(\mathcal{Y}3\right)$ and $\left(\mathcal{Y}4\right)$. Observing that $q(t,v,p)$ minimizes the mapping $q\to \tilde{\mathcal{L}}(t,v,p,q),$ we derive the necessary first-order optimality condition:

$$\begin{array}{c}\hfill {\nabla }_q{\tilde{\mathcal{L}}}_T(t,v,p,q(t,v,p))={\nabla }_q{\mathcal{L}}_T(t,v,q(t,v,p))-{\nabla }_{p}S(t,p)=0;\end{array}$$

(26)

i.e., the function $S(t,p)$ satisfying $\left(\mathcal{Y}3\right)$ and $\left(\mathcal{Y}4\right)$ must satisfy (26).

A numerical example is shown (non-autonomous).

Consider the forced problem of the time period

$$v^{\left(4\right)}\left(t\right)+{\omega }^{2}v\left(t\right)=Fcos(\Omega t),\omega >0,F\in \mathbb{R},\Omega >0,$$

(27)

with periodic boundary conditions $v^{\left(k\right)}\left(T\right)=v^{\left(k\right)}\left(0\right)$ for $k=0,1,2,3$, where $T=2\pi /\Omega$. In our non-autonomous variational setting, take

$$H(t,v)=\frac{1}{2}{\omega }^2{v}^2-Fvcos(\Omega t),\mathcal{L}(t,v,v^{\left(2\right)})=\frac{1}{2}{\left|v^{\left(2\right)}\right|}^2+H(t,v),$$

so that the Euler–Lagrange equation is exactly (27). For $\Omega \ne 0$, an explicit T-periodic particular solution is

$$v_p\left(t\right)={\displaystyle \frac{F}{{\Omega }^4+{\omega }^2}}cos(\Omega t),T=\frac{2\pi }{\Omega }.$$

The collocation minimization of $\mathcal{A}\left[v\right]={\int }_0^T\mathcal{L}(t,v,v^{\left(2\right)})dt$ under periodic and phase constraints selects $v_p$, thus validating the non-autonomous result of this section.

In the previous sections, we established rigorous existence theorems for periodic solutions of $2n$-th–order ordinary differential equations. To demonstrate the practical relevance of these results, we now verify them on a representative real-world model drawn from nonlinear vibration theory.

Remark 3.

The strict convexity of $\tilde{\mathcal{L}}$ in the highest derivative q is used only to ensure that the stationarity condition ${\nabla }_q\tilde{\mathcal{L}}=0$ produces a unique minimizer, leading to the pointwise constraint used in Theorems 1–2. If one replaces strict convexity by weaker assumptions such as coercivity or pseudo-convexity, the existence of periodic solutions can still be expected, but uniqueness and the explicit constraint may fail. The analysis of these “nearly convex” cases remains open for future work.

4. Application and Validation in Real Models

Consider a typical nonlinear vibration model, such as a single-degree-of-freedom mass–spring system with a nonlinear spring. This model represents an elastic mechanical oscillator with a nonlinear restoring force (for example, the well-known Duffing oscillator [22]) and is widely studied in the field of nonlinear dynamics. In summary, the system consists of a mass attached to a spring whose restoring force has both linear elastic and nonlinear components. When the mass is displaced from equilibrium, the spring restoring force is no longer strictly proportional to the displacement but instead includes higher-order terms, which give the system markedly nonlinear oscillation characteristics.

The motion of this model can be formulated as the following second-order ordinary differential equation (which is a special case of the 2n-th order form discussed in this paper, with $n=1$):

$$x^{″}\left(t\right)+kx\left(t\right)+\alpha x^3\left(t\right)=0,$$

(28)

where $x\left(t\right)$ denotes the displacement of the mass from its equilibrium, $k>0$ is the linear stiffness coefficient of the spring, and $\alpha \ge 0$ is the nonlinear stiffness parameter (with $\alpha >0$ for a “hardening” spring and $\alpha <0$ for a “softening” spring). In this equation, the restoration force consists of a linear term $kx$ and a nonlinear term $\alpha x^3$. When $\alpha =0$, the system reduces to a simple linear harmonic oscillator; for $\alpha \ne 0$, the nonlinear term becomes significant for larger amplitudes, thereby affecting the oscillation period and waveform.

It can be observed that the above equation of motion falls within the class of 2n-th-order ODEs studied in this paper (here $2n=2$), and it possesses a clear variational structure. In fact, it can be regarded as the Euler–Lagrange equation of a certain Lagrangian action functional. For example, one may define the Lagrangian functional as the kinetic energy minus the potential energy, where $L=T-V=\frac{1}{2}{\dot{x}}^2-V\left(x\right)$, with the potential energy chosen as

$$V\left(x\right)=\frac{1}{2}k{x}^2+\frac{1}{4}\alpha x^4.$$

From this, the corresponding action functional can be constructed as

$$\mathcal{A}\left[x\right]={\int }_0^T\left(\frac{1}{2}{\dot{x}}^2-V\left(x\right)\right)dt$$

Taking the variational derivative of $\mathcal{A}\left[x\right]$ and setting $\delta \mathcal{A}=0$ yields the differential equation of motion (28), confirming that the system indeed satisfies the Lagrangian variational structure and the Euler–Lagrange form required by our theory [cf. the main theorem]. In addition, we impose the periodic boundary conditions $x\left(0\right)=x\left(T\right)$ and $\dot{x}\left(0\right)=\dot{x}\left(T\right)$, which align with the periodicity assumptions in our theorems and ensure that the solution returns to its initial state after a period T. Furthermore, the second derivative of the potential is

$$V^{″}\left(X\right)=k+3\alpha x^2,$$

which is positive for all x given $k>0$ and $\alpha \ge 0$. In other words, $V\left(x\right)$ is a convex function over the entire real line. Therefore, this model satisfies all the hypotheses of our main theorem, including the variational structure, periodic boundary conditions, and the required convexity condition, and by the conclusion of the theorem, the nonlinear system is guaranteed to have at least one nontrivial periodic solution.

Next, we verify the theoretically predicted periodic solution through a numerical example. For simplicity, we employ a first-order Fourier–Galerkin approximation to construct a periodic solution for the system. We assume a trial solution of the form $x\left(t\right)\approx Acos\left(\omega t\right)$; substituting this into Equation (28) and neglecting higher harmonics yields an approximate relationship between frequency $\omega$ and amplitude A:

$$\omega \approx \sqrt{k+\frac{3}{4}\alpha A^2}.$$

For example, taking parameters $k=1$ and $\alpha =0.1$ and assuming a periodic solution of amplitude about $A=1$, the above formula gives $\omega \approx 1.037$, corresponding to a period $T={\displaystyle \frac{2\pi }{\omega }}\approx 6.06$. To assess the accuracy of this approximation, we performed a direct numerical integration of the original nonlinear equation using a shooting method. The calculation confirms that there is a periodic solution for the same parameters, with a period of about $T\approx 6.09$. This numerically obtained period is very close to the Galerkin prediction, and the good agreement indicates that the periodic solution predicted by our theory is indeed realized and validated in this actual model.

5. Conclusions

This paper establishes a unified variational framework for proving the existence of periodic solutions in general $2n$-order ordinary differential equations, encompassing both autonomous and nonautonomous cases. Key contributions lie in the extension of Carathéodory’s variational technique to arbitrary higher-order systems, the operator-theoretic reformulation enabling tractable analysis, and the derivation of sufficient existence conditions (Theorems 1 and 2) that generalize prior second-order results while relaxing restrictive assumptions such as the need for strict convexity or specialized coordinate transformations.

Although the primary focus of this study is limited to scalar systems, the proposed framework offers promising avenues for future research in several directions. First, it can be extended to high-dimensional coupled systems, with the development of structure-preserving numerical algorithms tailored for the efficient computation of periodic solutions within such variational settings. Second, applying this framework to a wider variety of nonlinear systems may lead to more universally applicable modeling approaches, which hold potential applications in fields such as biological systems, control engineering, and nonequilibrium physics. These extensions are expected to strengthen both the theoretical underpinnings and practical applicability of the current work.