Article
Mathematics, Applied
Ningbo An, Qishao Wang, Xiaochuan Zhao, Qingyun Wang
Summary: This paper proposes a distributed control method based on the differential flatness property of robot swarms. A swarm differential flatness mapping is established, and a distributed controller is designed based on this mapping. The effectiveness of the method is validated through a numerical simulation.
APPLIED MATHEMATICS AND MECHANICS-ENGLISH EDITION
(2023)
Article
Mathematics, Applied
Jaume Llibre, Claudia Valls
Summary: The paper addresses one of the main problems in the qualitative theory of planar differential equations: the study of their limit cycles. By providing an algebraic curve, the paper characterizes the planar polynomial differential systems of degree greater or equal to n that allow the algebraic curve's ovals to serve as hyperbolic limit cycles.
JOURNAL OF GEOMETRY AND PHYSICS
(2023)
Article
Mathematics, Applied
Joyce A. Casimiro, Jaume Llibre
Summary: Many articles have been published on continuous and discontinuous piecewise differential systems in the plane since the beginning of this century. The increasing number of applications for modeling natural phenomena has led to great interest in these systems. One major challenge in understanding the dynamics of planar differential systems is controlling their limit cycles. Most papers studying continuous piecewise differential systems have used straight lines as the separating boundary. However, this work considers continuous piecewise differential systems separated by a circle and consisting of one linear and one quadratic differential center. The goal is to determine the maximum number of limit cycles such a continuous piecewise differential system can exhibit.
RESULTS IN MATHEMATICS
(2023)
Article
Mathematics, Applied
Nassima Debz, Amel Boulfoul, Abdelhak Berkane
Summary: The paper investigates the maximum number of limit cycles that can emerge from a linear center in perturbed planar polynomial differential systems. By utilizing first and second order averaging theory, the study estimates the maximum limit cycles that this class of systems can exhibit.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
Article
Mathematics, Applied
J. L. Bravo, L. A. Calderon, M. Fernandez
Summary: New criteria have been established to determine upper bounds on the number of limit cycles of periodic Abel differential equations with two periodic invariant curves, one of them bounded. These criteria have been applied to obtain upper bounds of either zero or one limit cycle for planar differential systems.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2021)
Article
Mathematics, Applied
Belen Garcia, Jaume Llibre, Jesus S. Perez del Rio, Set Perez-Gonzalez
Summary: In this paper, the authors investigate the number of algebraic limit cycles in discontinuous piecewise linear differential systems separated by a straight line, with polynomial first integrals on both sides. They assume that at least one of the systems is Hamiltonian and find that under this assumption, piecewise differential systems have no more than one limit cycle. The study provides a characterization of linear differential systems with polynomial first integrals.
Article
Computer Science, Information Systems
Zian Yu, Wangqiang Niu
Summary: A control method combining trajectory planning and backstepping is proposed for solving the antisway problem of underactuated overhead cranes under wind disturbance. The method includes proposing flat outputs to represent the crane system dynamics, providing relevant constraints for the trolley's desired position and swing angle suppression, obtaining planned trajectory through optimal parameters of the flat output, and designing a tracking controller to reduce deviation caused by wind disturbance. Simulation results demonstrate the effectiveness and robustness of the proposed method.
Article
Mathematics
Maria Alberich-Carraminana, Antoni Ferragut, Jaume Llibre
Summary: This paper investigates the effects of planar birational transformations on quadratic planar differential systems, providing a new family of quadratic systems with algebraic limit cycles of degree 5. The paper also classifies known families of quadratic differential systems with algebraic limit cycles by the action of quadratic plane Cremona maps, and provides phase portraits on the Poincare disk for all these families.
ADVANCES IN MATHEMATICS
(2021)
Article
Engineering, Mechanical
Masoumeh Safartoobi, Morteza Dardel, Hamidreza Mohammadi Daniali
Summary: This paper investigates a passive walking robot model with flexible legs, utilizes numerical techniques to find suitable initial conditions for gait cycles, and demonstrates that elastic legs result in non-periodic motion patterns for small slope angles.
MECHANISM AND MACHINE THEORY
(2021)
Article
Mathematics, Applied
M. J. alvarez, A. Gasull, R. Prohens
Summary: This paper proves that any complex differential equation with two monomials of the form z = az(k)z10pxl + bz(m)zn, where k, l, m, n are non-negative integers and a, b ∈ C, has at most one limit cycle, and characterizes the hyperbolic nature of such a limit cycle. Furthermore, we address the center-focus problem and examine the number, position, and type of critical points for equations of this form. We also provide a geometric distribution result regarding the stability of critical points, similar to the Berlinskii theorem.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2023)
Article
Mathematics, Applied
Victoriano Carmona, Fernando Fernandez-Sanchez, Douglas D. Novaes
Summary: In this paper, the family of planar piecewise linear differential systems with two zones separated by a straight line is considered. The maximum number of limit cycles for these systems has been a well-studied problem in the research literature. By using a novel integral characterization of Poincare half-maps, this paper proves that the optimal uniform upper bound for the number of limit cycles is one, without distinguishing the matrices spectra. Additionally, it is proven that the existing limit cycle, if it exists, is hyperbolic and its stability is determined by a simple condition in terms of the system parameters.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2023)
Article
Mathematics, Applied
K. K. D. Adjai, J. Akande, A. V. R. Yehossou, M. D. Monsia
Summary: This article presents some classes of polynomial mixed Lienard-type differential equations that can generate many equations with exact solutions, which serve as counterexamples to the classical existence theorems.
Article
Mathematics, Applied
Yingying Zeng, Tayyab Mahmood, Yinhe Lin, Weinian Zhang
Summary: This paper explores the iteration of single-plateau functions, a class of continuous functions with infinitely many forts, and investigates the changes in number and length of plateaux. The changes are formulated using indices like flatness, plateau limit, and limit length, and computations are done for all nine types of single-plateau functions.
SCIENCE CHINA-MATHEMATICS
(2021)
Article
Multidisciplinary Sciences
Jaume Llibre, Claudia Valls
Summary: This study focuses on the maximum number of limit cycles in planar piecewise differential systems formed by linear Hamiltonian saddles. The findings demonstrate that the number of limit cycles varies depending on the continuity and separation of the systems, with different scenarios leading to different outcomes.
Article
Mathematics, Applied
Gui Lin Ji, Chang Jian Liu, Peng Heng Li
Summary: This paper studies the bifurcation of limit cycles separated by a straight line in planar piecewise smooth systems. A simpler form of Abelian integrals for piecewise smooth systems is proposed. In the application, the existence of 10 limit cycles and 12 small-amplitude limit cycles are respectively proved for piecewise quadratic system.
ACTA MATHEMATICA SINICA-ENGLISH SERIES
(2022)