Article
Mathematics, Applied
Jie Rui, Min Zhang, Yi Wang
Summary: In this study, a Kolmogorov-Arnold-Moser theorem regarding the existence of almost-periodic solutions for certain infinitely dimensional Hamiltonian systems was constructed and applied to an almost-periodically forced nonlinear beam equation. The obtained solutions demonstrate the characteristics of the system with specific conditions on the parameters involved.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2021)
Article
Engineering, Mechanical
Xiufang Ren, Shiji Zhao
Summary: By modifying the coefficients of a one-dimensional quasi-periodic linear Schrodinger equation, the existence of quasi-periodic solutions is obtained and the properties of the reduced system are analyzed. The original result is based on infinite-dimensional KAM theory and bifurcation theory.
NONLINEAR DYNAMICS
(2023)
Article
Mathematics, Applied
Yingte Sun
Summary: This paper considers a class of nonlinear beam equations and proves the existence of many quasi-periodic solutions with non-resonant frequencies.
QUALITATIVE THEORY OF DYNAMICAL SYSTEMS
(2022)
Article
Mathematics, Applied
Chuanfang Ge, Jiansheng Geng, Yingfei Yi
Summary: In this paper, we investigate the presence and stability of quasi-periodic breathers in granular chains of coupled Duffing oscillators with Hertzian interaction potential that has finite smoothness. By employing the Jackson-Moser-Zehnder analytic approximation technique and KAM iterations, we not only establish the existence and linear stability of quasi-periodic breathers, but also provide explicit estimates on the localization rate, etc., depending on the smoothness order and the number of oscillating frequencies.
JOURNAL OF NONLINEAR SCIENCE
(2023)
Article
Mathematics
Weiping Yan, Binlin Zhang
Summary: This article considers the motion of relativistic strings in the Minkowski space R1+n, proving that they can exhibit a more generalized time quasi-periodic motion. These solutions are also timelike solutions.
JOURNAL OF GEOMETRIC ANALYSIS
(2021)
Article
Mathematics, Applied
Livia Corsi, Riccardo Montalto, Michela Procesi
Summary: In this study, almost-periodic solutions for quasi-linear perturbations of the Airy equation are proven to exist, marking the first result of its kind. These solutions are shown to be analytic in both time and space, achieved through a novel approach combining Craig-Wayne, KAM reducibility scheme, and pseudo-differential calculus on T-infinity.
JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Applied
Yuan Zhang, Wen Si, Jianguo Si
Summary: In this paper, we consider quasi-periodically forced perturbations of dissipative Boussinesq systems with an elliptic fixed point in two cases: Hamiltonian case and reversible case. We prove the existence and linear stability of quasi-periodic solutions for the system with periodic boundary conditions. The method of proof is based on a Nash-Moser iterative scheme in the scale of Sobolev spaces developed by Berti and Bolle, but we have to adapt it substantially to deal with the specific system considered here.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2022)
Article
Engineering, Mechanical
Mingwu Li, George Haller
Summary: In Part I of this paper, the authors constructed reduced-order models for harmonically excited mechanical systems with internal resonances using spectral submanifold theory. By locating the solution branches of equilibria of the corresponding reduced-order model, they were able to extract forced response curves formed by periodic orbits of the full system. In Part II, the authors use bifurcations of the equilibria of the reduced-order model to predict bifurcations of the periodic response of the full system, specifically predicting the existence of two-dimensional and three-dimensional quasi-periodic attractors and repellers in periodically forced mechanical systems of arbitrary dimension.
NONLINEAR DYNAMICS
(2022)
Article
Mathematics, Applied
Melanie Kobras, Valerio Lucarini, Maarten H. P. Ambaum
Summary: In this study, a minimal dynamical system derived from the classical Phillips two-level model is introduced to investigate the interaction between eddies and mean flow. The study finds that the horizontal shape of the eddies can lead to three distinct dynamical regimes, and these regimes undergo transitions depending on the intensity of external baroclinic forcing. Additionally, the study provides insights into the continuous or discontinuous transitions of atmospheric properties between different regimes.
PHYSICA D-NONLINEAR PHENOMENA
(2024)
Article
Mathematics, Applied
Shu-hong Xue, Yun-yun Yang, Biao Feng, Hai-long Yu, Li Wang
Summary: This research focuses on the robustness of multiplex networks and proposes a new index to measure their stability under malicious attacks. The effectiveness of this method is verified in real multiplex networks.
PHYSICA D-NONLINEAR PHENOMENA
(2024)
Article
Mathematics, Applied
Julien Nespoulous, Guillaume Perrin, Christine Funfschilling, Christian Soize
Summary: This paper focuses on optimizing driver commands to limit energy consumption of trains under punctuality and security constraints. A four-step approach is proposed, involving simplified modeling, parameter identification, reformulation of the optimization problem, and using evolutionary algorithms. The challenge lies in integrating uncertainties into the optimization problem.
PHYSICA D-NONLINEAR PHENOMENA
(2024)
Article
Mathematics, Applied
Alain Bourdier, Jean-Claude Diels, Hassen Ghalila, Olivier Delage
Summary: In this article, the influence of a turbulent atmosphere on the growth of modulational instability, which is the cause of multiple filamentation, is studied. It is found that considering the stochastic behavior of the refractive index leads to a decrease in the growth rate of this instability. Good qualitative agreement between analytical and numerical results is obtained.
PHYSICA D-NONLINEAR PHENOMENA
(2024)
Article
Mathematics, Applied
Ling An, Liming Ling, Xiaoen Zhang
Summary: In this paper, an integrable fractional derivative nonlinear Schrodinger equation is proposed and a reconstruction formula of the solution is obtained by constructing an appropriate Riemann-Hilbert problem. The explicit fractional N-soliton solution and the rigorous verification of the fractional one-soliton solution are presented.
PHYSICA D-NONLINEAR PHENOMENA
(2024)
Article
Mathematics, Applied
Marzia Bisi, Nadia Loy
Summary: This paper proposes and investigates general kinetic models with transition probabilities that can describe the simultaneous change of multiple microscopic states of the interacting agents. The mathematical properties of the kinetic model are proved, and the quasi-invariant asymptotic regime is studied and compared with other models. Numerical tests are performed to demonstrate the time evolution of distribution functions and macroscopic fields.
PHYSICA D-NONLINEAR PHENOMENA
(2024)
Article
Mathematics, Applied
Carlos A. Pires, David Docquier, Stephane Vannitsem
Summary: This study presents a general theory for computing information transfers in nonlinear stochastic systems driven by deterministic forcings and additive and/or multiplicative noises. It extends the Liang-Kleeman framework of causality inference to nonlinear cases based on information transfer across system variables. The study introduces an effective method called the 'Causal Sensitivity Method' (CSM) for computing the rates of Shannon entropy transfer between selected causal and consequential variables. The CSM method is robust, cheaper, and less data-demanding than traditional methods, and it opens new perspectives on real-world applications.
PHYSICA D-NONLINEAR PHENOMENA
(2024)
Article
Mathematics, Applied
Feiting Fan, Minzhi Wei
Summary: This paper focuses on the existence of periodic and solitary waves for a quintic Benjamin-Bona-Mahony (BBM) equation with distributed delay and diffused perturbation. By transforming the corresponding traveling wave equation into a three-dimensional dynamical system and applying geometric singular perturbation theory, the existence of periodic and solitary waves are established. The uniqueness of periodic waves and the monotonicity of wave speed are also analyzed.
PHYSICA D-NONLINEAR PHENOMENA
(2024)
Article
Mathematics, Applied
Wangbo Luo, Yanxiang Zhao
Summary: We propose a generalized Ohta-Kawasaki model to study the nonlocal effect on pattern formation in binary systems with long-range interactions. In the 1D case, the model displays similar bubble patterns as the standard model, but Fourier analysis reveals that the optimal number of bubbles for the generalized model may have an upper bound.
PHYSICA D-NONLINEAR PHENOMENA
(2024)
Article
Mathematics, Applied
Corentin Correia, Ana Cristina Moreira Freitas, Jorge Milhazes Freitas
Summary: The emergence of clustering of rare events is due to periodicity, where fast returns to target sets lead to a bulk of high observations. In this research, we explore the potential of a new mechanism to create clustering of rare events by linking observable functions to a finite number of points belonging to the same orbit. We show that with the right choice of system and observable, any given cluster size distribution can be obtained.
PHYSICA D-NONLINEAR PHENOMENA
(2024)
Article
Mathematics, Applied
Enyu Fan, Changpin Li
Summary: This paper numerically studies the Allen-Cahn equations with different kinds of time fractional derivatives and investigates the influences of time derivatives on the solutions of the considered models.
PHYSICA D-NONLINEAR PHENOMENA
(2024)
Article
Mathematics, Applied
Yuhang Zhu, Yinghao Zhao, Chaolin Song, Zeyu Wang
Summary: In this study, a novel approach called Time-Variant Reliability Updating (TVRU) is proposed, which integrates Kriging-based time-dependent reliability with parallel learning. This method enhances risk assessment in complex systems, showcasing exceptional efficiency and accuracy.
PHYSICA D-NONLINEAR PHENOMENA
(2024)
Article
Mathematics, Applied
Chiara Cecilia Maiocchi, Valerio Lucarini, Andrey Gritsun, Yuzuru Sato
Summary: The predictability of weather and climate is influenced by the state-dependent nature of atmospheric systems. The presence of special atmospheric states, such as blockings, is associated with anomalous instability. Chaotic systems, like the attractor of the Lorenz '96 model, exhibit heterogeneity in their dynamical properties, including the number of unstable dimensions. The variability of unstable dimensions is linked to the presence of finite-time Lyapunov exponents that fluctuate around zero. These findings have implications for understanding the structural stability and behavior modeling of high-dimensional chaotic systems.
PHYSICA D-NONLINEAR PHENOMENA
(2024)
Article
Mathematics, Applied
Christian Klein, Goksu Oruc
Summary: A numerical study on the fractional Camassa-Holm equations is conducted to construct smooth solitary waves and investigate their stability. The long-time behavior of solutions for general localized initial data from the Schwartz class of rapidly decreasing functions is also studied. Additionally, the appearance of dispersive shock waves is explored.
PHYSICA D-NONLINEAR PHENOMENA
(2024)
Article
Mathematics, Applied
Vasily E. Tarasov
Summary: This paper extends the standard action principle and the first Noether theorem to consider the general form of nonlocality in time and describes dissipative and non-Lagrangian nonlinear systems. The general fractional calculus is used to handle a wide class of nonlocalities in time compared to the usual fractional calculus. The nonlocality is described by a pair of operator kernels belonging to the Luchko set. The non-holonomic variation equations of the Sedov type are used to describe the motion equations of a wide class of dissipative and non-Lagrangian systems. Additionally, the equations of motion are considered not only with general fractional derivatives but also with general fractional integrals. An application example is presented.
PHYSICA D-NONLINEAR PHENOMENA
(2024)