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In this study, a template is proposed for designing ordinary differential equation-based nonlinear oscillators. The template is a two-dimensional system with two control parameters and an energy function. By choosing the appropriate energy function, it is possible to obtain globally stable systems. These systems can be a gradient system, a Hamiltonian system, or a system with a stable limit cycle depending on the choice of control parameters. Hamiltonian and limit cycle cases can be used as a nonlinear oscillator for various applications. An example system is demonstrated by choosing a simple energy function and the obtained system is simulated to verify its dynamics. Hardware verification of the simulated system is performed with a field programmable gate array (FPGA) implementation. [ABSTRACT FROM AUTHOR]

In previous studies, we developed two techniques aimed at expanding the scope of constructing a periodic orbit dividing surface within a Hamiltonian system with three or more degrees of freedom. Our approach involved extending a periodic orbit into a torus or cylinder, thereby elevating it into a higher-dimensional entity within the energy surface (see [0, Katsanikas & Wiggins, 0, 2021a, 0, 2021b, 0, 2023a, 0, 2023b]). Recently, we introduced two alternative methods for creating dividing surfaces, distinct from the utilization of periodic orbits, by employing 2D surfaces (geometric entities) or 3D surfaces within a Hamiltonian system with three degrees of freedom (refer to [0, Katsanikas & Wiggins, 0, 2024a, 0, 2024b, 0, 2024c]). In these studies, we applied these surfaces in a quadratic normal form Hamiltonian system with three degrees of freedom. In this series of two papers, we extend our results to 2D generating surfaces for quartic Hamiltonian systems with three degrees of freedom. This paper presents the first method of constructing 2D generating surfaces. [ABSTRACT FROM AUTHOR]

A global center for a vector field in the plane is a singular point p having R 2 filled of periodic orbits with the exception of the singular point p. Polynomial differential systems of degree 2 have no global centers. In this paper we classify the global nilpotent centers of planar cubic polynomial Hamiltonian systems symmetric with respect to the y-axis. [ABSTRACT FROM AUTHOR]

This paper expands the concept of periodic orbit dividing surfaces within rotating Hamiltonian systems possessing three degrees of freedom. Initially, we detail the implementation of our second method for constructing these surfaces, as outlined in [Katsanikas & Wiggins, 2021b, 2023b], for such systems. Subsequently, we analyze the configuration of these surfaces within a rotating quadratic normal-form Hamiltonian system characterized by three degrees of freedom. [ABSTRACT FROM AUTHOR]

Finding vibration mode functions satisfying both high-order partial differential governing equations and various non-Lévy-type boundary conditions is extremely challenging for the free vibration problems of functionally graded rectangular plates, making it difficult to analytically solve the problem with traditional methods. Herein, the newly developed Hamiltonian system-based symplectic superposition method was extended and successfully applied to analytical solutions to the free vibration problems of functionally graded rectangular plates. For the solution methodology, the original vibration problem was divided into sub-problems and the physical neutral plane was introduced to eliminate the stretching-bending coupling effect caused by the transversely non-uniform materials. The sub-problems were analytically solved with some mathematical techniques, i.e., the variable separation and the symplectic eigenvector expansion, which are not applicable in the traditional Lagrangian system. The final solution to an original vibration problem was obtained through the superposition of sub-problems. The symplectic superposition method has the advantage of not requiring the pre-defined solution forms, which overcomes the limitations of traditional semi-inverse methods and allows for obtaining analytical solutions to more complex problems. Comparison of the obtained solutions with the numerical solutions proves the accuracy of the presented method. On this basis, quantitative parameter analyses on the natural frequencies were conducted to reveal the effects of boundary conditions, material distributions and aspect ratios. [ABSTRACT FROM AUTHOR]

Good Boussinesq equations are considered in this work. First, we apply three combined compact schemes to approximate spatial derivatives of good Boussinesq equations. Then, three fully discrete schemes are developed based on a symplectic scheme in the time direction, which preserves the symplectic structure. Meanwhile, the convergence and conservation of the fully discrete schemes are analyzed. Finally, we present numerical experiments to confirm our theoretical analysis. Both our analysis and numerical tests indicate that the fully discrete schemes are efficient in solving the spatial derivative mixed equation. [ABSTRACT FROM AUTHOR]

In this paper, by using the bifurcation theory for dynamical system, we construct traveling wave solutions of a high-order nonlinear Schrödinger equation with a quintic nonlinearity. Firstly, based on wave variables, the equation is transformed into an ordinary differential equation. Then, under the parameter conditions, we obtain the Hamiltonian system and phase portraits. Finally, traveling wave solutions which contains solitary, periodic and kink wave solutions are constructed by integrating along the homoclinic or heteroclinic orbits. In addition, by choosing appropriate values to parameters, different types of structures of solutions can be displayed graphically. Moreover, the computational work and it's figures show that this technique is influential and efficient. [ABSTRACT FROM AUTHOR]

L-shaped moderately thick plates have widespread applications in diverse engineering structures. Exploring the benchmark analytic solutions for the free vibration of L-shaped moderately thick plates is important in order to accurately analyze and efficiently design structures. Nevertheless, analytical solutions, which serve as the benchmarks, have been rarely documented in previous literature due to the challenge of finding suitable solutions that satisfy both the governing higher-order partial differential equations (PDEs) and the boundary conditions of the plates. The symplectic superposition method was employed in our recently published study to present the benchmark vibration solutions of clamped rectangular moderately thick plates. In this study, we expand upon this method to solve the free vibration problem of L-shaped moderately thick plates. By employing domain decomposition, we construct an irregular domain by combining multiple rectangular domains. First, the construction of the superposition system is carried out, followed by the import of the sub-problems into the Hamiltonian system, utilizing the fundamental governing equations of the plate. Then, the sub-problems are resolved through the application of the symplectic geometry methodology in an analytical manner. Ultimately, the analytical solutions for frequencies and mode shapes are derived through ensuring the equivalence between the initial problem and the combination of sub-problems. The finite element method is used to validate and present a comprehensive analysis of the natural frequencies and mode shapes obtained from this method. This method possesses the benefits of rapid convergence and accurate precision, rendering it well suited for the analytic modeling of a broader range of plate-related problems. [ABSTRACT FROM AUTHOR]

Abstract The second-order periodic system with p-Laplacian and unbounded time-dependent perturbation terms is investigated. Using the principle integral method, it is shown that under certain assumptions on the unbounded and periodic terms, all solutions to the equation possess boundedness.

We present an example of a three-degrees-of-freedom polynomial Hamilton function with a critical point characterized by indefinite quadratic part with a Morse index 2. This function generates a Hamiltonian system wherein all eigenvalues equal $ \pm \mathrm{i} $, but it lacks small-amplitude periodic solutions with a period $ \approx 2\pi. $

The second-order periodic system with p-Laplacian and unbounded time-dependent perturbation terms is investigated. Using the principle integral method, it is shown that under certain assumptions on the unbounded and periodic terms, all solutions to the equation possess boundedness. [ABSTRACT FROM AUTHOR]

Beyond Hölder’s type, this paper mainly concerns the persistence and the remaining regularity of an individual frequency-preserving KAM torus in a finitely differentiable Hamiltonian system, even allowing the non-integrable part to be critically finitely smooth. To achieve this goal, besides investigating the Jackson approximation theorem towards only the modulus of continuity, we demonstrate an abstract regularity theorem adapting to the new iterative scheme. Via these tools, we obtain a KAM theorem with sharp differentiability hypotheses, asserting that the persistent torus keeps the prescribed universal Diophantine frequency unchanged. Further, the non-Hölder regularity for the invariant KAM torus as well as the conjugation is explicitly shown by introducing asymptotic analysis. To our knowledge, this is the first approach to KAM on these aspects in a continuous sense, and we also provide two systems, which cannot be studied by previous KAM but by ours. [ABSTRACT FROM AUTHOR]

In prior studies [0, Katsanikas & Wiggins, 0, 2021a, 0, 2021b, 0, 2023a, 0, 2023b], we introduced two methodologies for constructing Periodic Orbit Dividing Surfaces (PODS) tailored specifically for Hamiltonian systems with three or more degrees of freedom. These approaches, as described in the aforementioned papers, were applied to a quadratic Hamiltonian system in its normal form with three degrees of freedom. Within this framework, we provide a more intricate geometric characterization of this entity within the family of 4D toratopes which elucidates the structure of the dividing surfaces discussed in these works. Our analysis affirmed the nature of this construction as a dividing surface with the property of no-recrossing. These insights were derived from analytical findings tailored to the Hamiltonian system discussed in these publications. In this series of papers, we extend our previous findings to quartic Hamiltonian systems with three degrees of freedom. We establish the no-recrossing property of the PODS for this class of Hamiltonian systems and explore their structural aspects. Additionally, we undertake the computation and examination of the PODS in a coupled scenario of quartic Hamiltonian systems with three degrees of freedom. In the initial paper [Gonzalez Montoya et al., 2024], we employed the first methodology for constructing PODS, while in this paper, we utilize the second methodology for the same purpose. [ABSTRACT FROM AUTHOR]

In this paper we consider the persistence of elliptic lower-dimensional invariant tori with prescribed frequencies in Hamiltonian systems with small parameters. Under the Brjuno nondegeneracy condition, if the prescribed frequencies satisfy a Diophantine condition, by the KAM technique we prove that for most of small parameters in the sense of Lebesgue measure, the Hamiltonian systems admit a lower-dimensional invariant torus whose frequency vector is a dilation of the prescribed frequencies. [ABSTRACT FROM AUTHOR]

The motivation for this paper comes from a blowup problem for a semi-linear wave equation and a heat equation. Following an idea of J. Leray, we study a radially symmetric self-similar solution which has singularities on the characteristic cone. This naturally leads to the study of a Hamiltonian system called a profile equation. The novelty of this paper is that we focus on the movable singularity of the Hamiltonian system and we use Borel summability in constructing a singular solution. By "movable singularity" we mean that the singularity does not appear in the coefficients of the equation and depends on the respective solution. In the proof of our theorem we reduce the Hamiltonian system to a simpler form by a method similar to the so-called Birkhoff reduction. We obtain the parametrization of a singular solution by an elementary function. We also give applications to a semi-linear wave equation and a heat equation. [ABSTRACT FROM AUTHOR]

A Hamiltonian system is developed for the three-dimensional (3D) problem of two-dimensional (2D) decagonal piezoelectric quasicrystals via the variational principle. Based on the full state vector and the properties of the Hamiltonian operator matrix, the superposition principle of solutions obtains the symplectic analytical solutions of the problem under simply supported boundary conditions. Numerical examples are illustrated to display the effects of the stacking sequences and material constants on the stresses, displacements, electric potential, and electric displacements under the mechanical and electric displacement loadings. The symplectic analytical solutions presented in the article can be used as a reference for further numerical research. [ABSTRACT FROM AUTHOR]

The symplectic superposition method was used to study the vibration problem of the orthotropic rectangular thin plate point-supported at a corner and clamped at its opposite edges. Firstly, based on the boundary conditions, the original vibration problem was decomposed into 2 subproblems with 2 opposite edges simply supported. Next, the series expansion solutions to the 2 sub-vibration problems were obtained based on the separation variable method in the Hamiltonian system. Then the symplectic superposition solution to the original vibration problem was obtained with the superposition method. To determine the terms of the series expansion of the obtained symplectic superposition solution in specific calculations, the convergence analysis of the solution for calculating orthotropic rectangular thin plates was performed. The symplectic superposition solution was also used to calculate the vibration frequencies of the isotropic and orthotropic rectangular thin plate point-supported at a corner and clamped at its opposite edges, respectively, and to give the modes corresponding to the 1st 8 vibration frequencies of an orthotropic square thin plate. [ABSTRACT FROM AUTHOR]

During this century, an increasing interest appeared in studying the planar piecewise differential systems. This is due to their numerous applications for modelling many natural phenomena. For understanding the dynamics of the planar differential systems we must control the existence or non-existence of periodic orbits and limit cycles. So many papers have been published studying the existence or non-existence of periodic orbits and limit cycles for continuous or discontinuous piecewise differential systems. But until now very few papers have studied the periodic orbits and limit cycles of piecewise differential systems where two differential systems of the piecewise differential system are continuous and discontinuous respectively. We study the periodic orbits and limit cycles of the planar continuous-discontinuous piecewise differential systems separated by two parallel straight lines, such that either in one of these straight lines the piecewise differential system is continuous and in the other one discontinuous. In two pieces of these piecewise differential systems there are arbitrary Hamiltonian systems of degree two and in the third piece there is an arbitrary Hamiltonian system of degree one forming the continuous-discontinuous piecewise differential systems. We determine the limit cycles of these piecewise differential systems by considering two cases. In the first the Hamiltonian system of degree one can be in the middle of the three zones, and in the second it is on one side of the three zones. [ABSTRACT FROM AUTHOR]

This paper studies the summability of the transseries solution of a nonintegrable Hamiltonian system. Since our system has a resonance and is not integrable a general transseries theory does not work well as far as the author knows. In order to construct a formal transseries solution and prove its summability our main idea is to use the superintegrability of a Hamiltonian system in a class of transseries. More precisely we first show the superintegrability of a Hamiltonian system in the category of transseries via the key Lemmas 1 and 4 which follow. By virtue of the superintegrability we show the existence of a formal transseries solution. Then its summability is proved via the superintegrability. We note that the argument based on the superintegrability is elementary. [ABSTRACT FROM AUTHOR]

In this paper, we start from the periodic geodesic of generalized Hermite operators, and analyze their geometric characteristics and analytical properties. For the quantitative study of periodic solutions to the Schrödinger operators with non-polynomial potentials, we systematically discuss the corresponding Hamilton system, and use the harmonic balance method (HBM) and the modified harmonic balance method (mHBM) to approximate and estimate the periodic solution in high accuracy. [ABSTRACT FROM AUTHOR]

New approaches are proposed in the problem of constructing equivalent Hamiltonian systems for linear Lur'e equations (differential equations containing derivatives of even orders only). The approaches are based on the transition to the normal forms of the corresponding Hamiltonian systems with subsequent transformation of the resulting system. The proposed scheme does not require complex and cumbersome transformations of the original equation. It is shown that this problem is uniquely solvable. The appendix provides similar results for systems with two degrees of freedom. The effectiveness of the proposed formulas is illustrated by examples. [ABSTRACT FROM AUTHOR]

In this article, we consider the following elliptic system of Hamiltonian-type on a bounded domain:−Δu=K1(∣y∣)∣v∣p−1v,inB1(0),−Δv=K2(∣y∣)∣u∣q−1u,inB1(0),u=v=0on∂B1(0),\left\{\begin{array}{ll}-\Delta u={K}_{1}\left(| y| ){| v| }^{p-1}v,\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{B}_{1}\left(0),\\ -\Delta v={K}_{2}\left(| y| ){| u| }^{q-1}u,\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{B}_{1}\left(0),\\ u=v=0\hspace{1.0em}& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial {B}_{1}\left(0),\end{array}\right. where K1(r){K}_{1}\left(r) and K2(r){K}_{2}\left(r) are positive bounded functions defined in [0,1]\left[0,1], B1(0){B}_{1}\left(0) is the unit ball in RN{{\mathbb{R}}}^{N}, and (p,q)\left(p,q) is a pair of positive numbers lying on the critical hyperbola 1p+1+1q+1=N−2N.\frac{1}{p+1}+\frac{1}{q+1}=\frac{N-2}{N}. Under some suitable further assumptions on the functions K1(r){K}_{1}\left(r) and K2(r){K}_{2}\left(r), we prove the existence of infinitely many nonradial positive solutions whose energy can be made arbitrarily large. Our proof is based on the reduction method. The most ingredients of the article are using the Green representation and estimating the Green function and its regular part very carefully. For this purpose, some more extra ideas and techniques are needed. We believe that our method and techniques can be applied to other related problems.

In this paper, we consider the following elliptic system of Hamiltonian type on a bounded domain: (0.1) { − Δ u = K 1 (| y |) | v | p − 1 v , in B 1 (0) , − Δ v = K 2 (| y |) | u | q − 1 u , in B 1 (0) , u = v = 0 on ∂ B 1 (0) , where K 1 (r) and K 2 (r) are positive bounded functions, B 1 (0) is the unit ball in R N , (p , q) is a pair of positive numbers lying on the critical hyperbola: 1 p + 1 + 1 q + 1 = N − 2 N. We investigate the local uniqueness and the nondegeneracy of the bubbling solution for problem (0.1). Our proof is based on the local Pohozaev identities, blow-up analysis, and the properties of Greens function. Our results are quite different from single equation, which is mainly caused by the non-cooperative nonlinear terms and (p , q) lying on the critical hyperbola. We believe that the various new ideas and technique computations that we used in this paper would be very useful to deal with other related problems of Hamiltonian type with interact critical exponents. [ABSTRACT FROM AUTHOR]

This paper presents a series expansion solution for the vibration question of an orthotropic rectangular thin plate (RTP) with four corners point-supported based on the superposition method and the separation variable method in the Hamiltonian system. First, the original vibration question is decomposed into two sub-vibration questions by analyzing the boundary conditions. After that, the separation variable method in the Hamiltonian system is used to calculate the general solutions for the two sub-vibration questions. Then, these two general solutions are superimposed to obtain the symplectic superposition solution for the original vibration question. Finally, the gained symplectic superposition solution is applied to calculate the vibration questions of isotropic and orthotropic RTPs with four corners point-supported, and the convergence analysis is conducted for the calculation of the orthotropic RTPs using the symplectic superposition solution. [ABSTRACT FROM AUTHOR]

Some efficient temporal first-, second- and higher-order numerical schemes are constructed for the coupled nonlinear Schrödinger-Boussinesq (CNSB) equations based on discretizing the Hamiltonian formula by an exponential method in a compact representation, discrete gradient method and composition method. The schemes are fully decoupling of the three solution components, which is distinct from the partially decoupled scheme in the literature, and their compact representation reduces the storage requirement and operation count. Rigorous analyses are carried out to show the exact preservation of the discrete energy for the proposed schemes. Numerical experiments verify the theoretical results and confirm the satisfactory solution accuracy and excellent efficiency of the present schemes. [ABSTRACT FROM AUTHOR]

We develop an efficient energy-preserving exponential numerical scheme for the Schrödinger equation in the semiclassical limit with small parameter $ \varepsilon $. The scheme is based on adopting the Fourier pseudo-spectral method in space and applying the compact implicit integrating factor method with the discrete gradient method in time. The compact representation of our scheme saves the storage requirement and operation count. Ample numerical tests for the equation in one dimension suggest that, in order to compute correct physical quantities, the meshing strategies are: $ \tau = \mathcal{O}(\varepsilon) $ and $ h = \mathcal{O}(\varepsilon) $ for the nonlinear equation with strong $ \mathcal{O}(1) $–defocusing or weak $ \mathcal{O}(\varepsilon) $–focusing nonlinearities, and $ \tau $ independent of $ \varepsilon $ and $ h = \mathcal{O}(\varepsilon) $ for the linear equation or nonlinear equation with weak $ \mathcal{O}(\varepsilon) $–focusing nonlinearities. Numerical experiments for the two-/three-dimensional equation show the power of our scheme in simulating Bose-Einstein condensation. [ABSTRACT FROM AUTHOR]

In prior work [Katsanikas & Wiggins, 2021a, 2021b, 2023c, 2023d], we introduced two methodologies for constructing Periodic Orbit Dividing Surfaces (PODS) tailored for Hamiltonian systems possessing three or more degrees of freedom. The initial approach, outlined in [Katsanikas & Wiggins, 2021a, 2023c], was applied to a quadratic Hamiltonian system in normal form having three degrees of freedom. Within this context, we provided a more intricate geometric characterization of this object within the family of 4D toratopes that describe the structure of the dividing surfaces discussed in these papers. Our analysis confirmed the nature of this construction as a dividing surface with the no-recrossing property. All these findings were derived from analytical results specific to the case of the Hamiltonian system discussed in these papers. In this paper, we extend our results for quartic Hamiltonian systems with three degrees of freedom. We prove for this class of Hamiltonian systems the no-recrossing property of the PODS and we investigate the structure of these surfaces. In addition, we compute and study the PODS in a coupled case of quartic Hamiltonian systems with three degrees of freedom. [ABSTRACT FROM AUTHOR]

The present paper investigates static deformation, free vibration, and patch loading response of one-dimensional (1D) hexagonal quasicrystal (QC) plates by the symplectic method. The Hamiltonian system is established by utilizing the variational principle, and then the symplectic analytical solutions of the extended displacement and traction vectors are obtained with simply supported boundary conditions. Numerical examples illustrate the effect of stacking sequence on the bending deformation of multilayered plates made up of QC and crystal materials. Moreover, the effects of the patch region, frequency, and phonon-phason coupling elastic coefficients on the corresponding problems are investigated. The symplectic analytical solutions presented in the article can be further used as benchmarks for numerical methods, such as the finite integral transformation and finite element methods. • The Hamiltonian system for dynamic equations of 1D hexagonal QC is established using the variational principle. • Analytic solutions of 1D hexagonal QC are rationally derived by the symplectic method. • The static deformation, free vibration, and patch loading response of 1D hexagonal QCs plates are investigated. • Some numerical tests are illustrated to justify the symplectic method. [ABSTRACT FROM AUTHOR]

We present an example of a three-degrees-of-freedom polynomial Hamilton function with a critical point characterized by indefinite quadratic part with a Morse index 2. This function generates a Hamiltonian system wherein all eigenvalues equal ±i, but it lacks small-amplitude periodic solutions with a period ≈ 2π. [ABSTRACT FROM AUTHOR]

In this study, we propose an innovative approach for real-time leakage detection in pipelines by integrating conservative Hamiltonian equations and experimental Internet of Things (IoT) technologies. The proposed method combines a hybrid model that utilizes sensors and IoT devices to acquire real-time data and solves the coupled system of Hamiltonian equations using the ODE45 numerical integration method. Spectral frequency analysis is an essential part of this method, as it reveals specific patterns in the pressure and flow signals. The findings highlighted 95% accuracy in leak detection, which was validated through a comparison of the theoretical and experimental data. The novelty of this approach lies in its ability to maintain constant total system energy, thereby enabling continuous monitoring for early leak detection. As an improvement, the proper handling of sensor signals is emphasized, underscoring its contribution to the efficient management of water resources in potable water distribution systems. [ABSTRACT FROM AUTHOR]

In this article, we establish a nonexistence result of nontrivial solutions for the Choquard-type Hamiltonian system by the method of scaling spheres. [ABSTRACT FROM AUTHOR]

Spatial generalized linear mixed models with skew latent variables are usually used to model discrete spatial responses that have some skewness. Since the likelihood function in these models is complex, the Monte Carlo EM algorithms are commonly applied to estimate the model parameters. In this paper, we use an approximately stationary skew Gaussian random field, which is more flexible than the Gaussian one, to analyze the skew discrete spatial responses. We also present a new hybrid EM algorithm using the Hamiltonian Monte Carlo method, which is both faster and more accurate than the Monte Carlo EM algorithm. The performance of the proposed skew model with a hybrid algorithm is evaluated through a simulation study and an application to the earthquake data of Iran. The simulation results indicate that the proposed Hamiltonian algorithm with the skew random field has better performance than the existing models. In addition, spatial prediction is presented through the proposed approach for the earthquake data on the entire map of Iran, where high-risk areas with the probability of large earthquakes are identified. [ABSTRACT FROM AUTHOR]

We investigate the connection between the dynamical properties of a class of 3D systems and the geometric characteristics of the image of the energy-Casimir mapping. By examining the energy-Casimir mapping for such systems, we can explore the stability of the equilibrium states, the distribution of the periodic solutions, and the existence of homoclinic or heteroclinic orbits. We apply our findings to investigate the dynamic behavior of two specific equations, and provide a topological classification of the fibers of the energy-Casimir mapping for the two systems. [ABSTRACT FROM AUTHOR]

We are concerned with the quasi-periodic semilinear Duffing equation $ x''+\omega^2x+g(x,t) = 0, $ where $ \omega $ is a Diophantine number, $ g(x,t) $ is bounded, real analytic in $ x $ and $ t $, and is quasi-periodic in $ t $ with the frequency $ \tilde{\omega} = (1, \alpha) $, where $ \alpha $ is Liouvillean. Without assuming the twist condition and the polynomial-like condition on this equation, we will prove the boundedness of all solutions. [ABSTRACT FROM AUTHOR]

This paper is concerned with the following first-order Hamiltonian system $ \begin{equation} \nonumber \dot{z} = \mathscr{J}H_{z}(t, z), \end{equation} $ where the Hamiltonian function $ H(t, z) = \frac{1}{2}Lz\cdot z+A(\epsilon t)G(|z|) $ and $ \epsilon > 0 $ is a small parameter. Under some natural conditions, we obtain a new existence result for ground state homoclinic orbits by applying variational methods. Moreover, the concentration behavior and exponential decay of these ground state homoclinic orbits are also investigated.

In our earlier research (see [Katsanikas & Wiggins, 2021a, 2021b, 2023a, 2023b, 2023c]), we developed two methods for creating dividing surfaces, either based on periodic orbits or two-dimensional generating surfaces. These methods were specifically designed for Hamiltonian systems with three or more degrees of freedom. Our prior work extended these dividing surfaces to more complex structures such as tori or cylinders, all within the energy surface of the Hamiltonian system. In this paper, we introduce a new method for constructing dividing surfaces. This method differs from our previous work in that it is based on 3D surfaces or geometrical objects, rather than periodic orbits or 2D generating surfaces (see [Katsanikas & Wiggins, 2023a]). To explain and showcase the new method and to present the structure of these 3D surfaces, the paper provides examples involving Hamiltonian systems with three degrees of freedom. These examples cover both uncoupled and coupled cases of a quadratic normal form Hamiltonian system. Our current paper is the first in a series of two papers on this subject. This research is likely to be of interest to scholars and researchers in the field of Hamiltonian systems and dynamical systems, as it introduces innovative approaches to constructing dividing surfaces and exploring their applications. [ABSTRACT FROM AUTHOR]

A high-precision semi numerical and semi analytical method for interfacial fracture problem of piezo- electric quasicrystals (PQCs)/piezoelectric crystals (PZCs)/elastic material composites with notches was de- veloped. Firstly, the Hamiltonian system was introduced and the Hamiltonian dual equations for the 3-material composite were formulated. The higher order partial differential governing equations were transformed into a set of ordinary differential equations. Secondly, the symplectic eigenvalues and eigensolutions were obtained through separation of variables. The physical quantities were expressed with the expansion of symplectic series. Finally, a symplectic isogeometric analysis (IGA) coupling equation was derived through combination of the symplectic series and the IGA. The analytical expressions of the physical quantities near the notch tip and the intensity factors were derived. [ABSTRACT FROM AUTHOR]

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This paper deals with the three-dimensional (3D) elasticity problem of two-dimensional (2D) decagonal quasicrystal (QC) plates by an analytical symplectic approach. Based on the elasticity theory of QCs, a system of Hamiltonian governing equations for a 2D decagonal QC plate is constructed with the fundamental mechanical quantities as the symplectic variables, such that the symplectic eigenvalue problem is formed. Combined with the properties of trigonometric functions of two variables, the adjoint symplectic orthogonality relations among the eigenvectors of the Hamiltonian operator are derived, and the expansion theorem of the double Fourier trigonometric series is proven. The symplectic analytical solutions of stresses and displacements in the phonon and phason fields of the 2D decagonal QC plate with four lateral simply supported edges are given without any assumptions. Furthermore, the numerical results are plotted to verify the correctness of the analytical solutions by comparison with those acquired by the pseudo-Stroh formalism. The proposed approach has broad applicability and high precision and can be extended to the 3D elasticity analysis of more QCs. [ABSTRACT FROM AUTHOR]

In this paper, we consider the generalized isomonodromic deformations of rank-two irregular connections on the Riemann sphere. We introduce Darboux coordinates on the parameter space of a family of rank-two irregular connections by apparent singularities. Using the Darboux coordinates, we describe the generalized isomonodromic deformations as Hamiltonian systems. [ABSTRACT FROM AUTHOR]

In our previous work, we developed two methods for generalizing the construction of a periodic orbit dividing surface for a Hamiltonian system with three or more degrees of freedom. Starting with a periodic orbit, we extend it to form a torus or cylinder, which then becomes a higher-dimensional object within the energy surface (see [Katsanikas & Wiggins, 2021a, 2021b, 2023a, 2023b]). In this paper, we present two methods to construct dividing surfaces not from periodic orbits but by using 2D surfaces (2D geometrical objects) in a Hamiltonian system with three degrees of freedom. To illustrate the algorithm for this construction, we provide benchmark examples of three-degree-of-freedom Hamiltonian systems. Specifically, we employ the uncoupled and coupled cases of the quadratic normal form of a Hamiltonian system with three degrees of freedom. [ABSTRACT FROM AUTHOR]

Numerous branches of science and engineering, including the design of electronic circuits and the analysis of systems with periodic forcing, use the study of limit cycles in the Mathieu equation to solve problems that have significant effects on the stability of periodic systems. In this paper, using the averaging theory of first order. we provide upper bounds for the maximum number of limit cycles bifurcating from the periodic orbits adapted to the Hamiltonian system i = y*-1, 9 - --liq-1 with p. q are positive integers, when perturbed in the particular class of the generalized Mathieu differential equanons. [ABSTRACT FROM AUTHOR]

This paper continues a series of papers in which simultaneous matrix solutions of two scalar evolution equations, which are analogs of time-dependent Schrödinger equations, were constructed. In the constructions in the present paper, these equations correspond to the Hamiltonian system —one of the representatives of the hierarchy of degenerations of the isomonodromic Garnier system. The mentioned hierarchy was described by H. Kimura in 1986. In terms of solutions of linear systems of differential equations in the method of isomonodromic deformations, the consistency condition for which is the Hamiltonian equations of the system, the constructed simultaneous matrix solutions of analogs of time-dependent Schrödinger equations are written out explicitly in this paper. [ABSTRACT FROM AUTHOR]

New approaches are proposed in the problem of constructing equivalent scalar differential equations for multidimensional nonlinear systems of control theory, as well as in the problem of constructing equivalent Hamiltonian systems for nonlinear Lurie equations (scalar differential equations containing derivatives of only even orders). The conditions for the solvability of the corresponding problems are studied, and new formulas for the transition to equivalent equations and systems are proposed. For the Lurie equations, the proposed approaches are based on the transition from the linear part to the normal forms of the corresponding Hamiltonian systems with a subsequent transformation of the resulting system. Calculation formulas and algorithms are obtained, and their efficiency is illustrated by examples. [ABSTRACT FROM AUTHOR]