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Search Results (3)

Search Parameters:
Keywords = unstable manifold of equilibrium

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15 pages, 2697 KiB  
Article
Exploring the Influence of Oblateness on Asymptotic Orbits in the Hill Three-Body Problem
by Vassilis S. Kalantonis
AppliedMath 2025, 5(1), 30; https://doi.org/10.3390/appliedmath5010030 - 17 Mar 2025
Viewed by 889
Abstract
We examine the modified Hill three-body problem by incorporating the oblateness of the primary body and focus on its asymptotic orbits. Specifically, we analyze and characterize homoclinic and heteroclinic connections associated with the collinear equilibrium points. By systematically varying the oblateness parameter, we [...] Read more.
We examine the modified Hill three-body problem by incorporating the oblateness of the primary body and focus on its asymptotic orbits. Specifically, we analyze and characterize homoclinic and heteroclinic connections associated with the collinear equilibrium points. By systematically varying the oblateness parameter, we determine conditions for the existence and location of these orbits. Our results confirm the presence of both homoclinic orbits, where trajectories asymptotically connect an equilibrium point to itself, and heteroclinic orbits, which establish connections between two distinct equilibrium points, via their stable and unstable invariant manifolds, which are computed both analytically and numerically. To achieve precise computations, we employ differential correction techniques and leverage the system’s inherent symmetries. Numerical calculations are carried out for orbit multiplicities up to twelve, ensuring a comprehensive exploration of the dynamical properties. Full article
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16 pages, 313 KiB  
Review
Review of Some Modified Generalized Korteweg–de Vries–Kuramoto–Sivashinsky Equations (Part II)
by Marie-Thérèse Aimar and Abdelkader Intissar
Foundations 2024, 4(4), 630-645; https://doi.org/10.3390/foundations4040039 - 4 Nov 2024
Cited by 1 | Viewed by 933
Abstract
In part I of this work to appear in Foudations-MDPI 2024, some existence and uniqueness results for the solutions of some equations were reviewed, such as the Korteweg–de Vries equation (KdV), the Kuramoto–Sivashinsky equation (KS), the generalized Korteweg–de Vries–Kuramoto–Sivashinsky equation (gKdV-KS), and the [...] Read more.
In part I of this work to appear in Foudations-MDPI 2024, some existence and uniqueness results for the solutions of some equations were reviewed, such as the Korteweg–de Vries equation (KdV), the Kuramoto–Sivashinsky equation (KS), the generalized Korteweg–de Vries–Kuramoto–Sivashinsky equation (gKdV-KS), and the nonhomogeneous boundary value problem for the KdV-KS equation in quarter plane. The main objective of this paper is to review some results of the existence of global attractors for the evolution equations with nonlinearity of the form N(ux), where ux denotes the derivative of u with respect to x, focusing in particular on the Kuramoto–Sivashinsky equation in one and two dimensions. In order to illustrate the general abstract results, we have chosen to discuss in detail the existence of global attractors for the Kuramoto–Sivashinsky (KS) equation in 1D and 2D. Once a global attractor is obtained, the question arises whether it has special regularity properties. Then we give an integrated version of the homogeneous steady state Kuramoto–Sivashinsky equation in Rn. This work ends with a change from rectangular to polar coordinates in the three-dimensional KS equation to give an energy estimate in this case. Full article
(This article belongs to the Section Mathematical Sciences)
15 pages, 2924 KiB  
Article
Homoclinic Orbits in Several Classes of Three-Dimensional Piecewise Affine Systems with Two Switching Planes
by Yanli Chen, Lei Wang and Xiaosong Yang
Mathematics 2021, 9(24), 3285; https://doi.org/10.3390/math9243285 - 17 Dec 2021
Viewed by 2340
Abstract
The existence of homoclinic orbits or heteroclinic cycle plays a crucial role in chaos research. This paper investigates the existence of the homoclinic orbits to a saddle-focus equilibrium point in several classes of three-dimensional piecewise affine systems with two switching planes regardless of [...] Read more.
The existence of homoclinic orbits or heteroclinic cycle plays a crucial role in chaos research. This paper investigates the existence of the homoclinic orbits to a saddle-focus equilibrium point in several classes of three-dimensional piecewise affine systems with two switching planes regardless of the symmetry. An analytic proof is provided using the concrete expression forms of the analytic solution, stable manifold, and unstable manifold. Meanwhile, a sufficient condition for the existence of two homoclinic orbits is also obtained. Furthermore, two concrete piecewise affine asymmetric systems with two homoclinic orbits have been constructed successfully, demonstrating the method’s effectiveness. Full article
(This article belongs to the Topic Dynamical Systems: Theory and Applications)
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