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Classical energy-momentum methods study the existence and stability properties of solutions of t -dependent Hamilton equations on symplectic manifolds whose evolution is given by their Hamiltonian Lie symmetries. The points of such solutions are called relative equilibrium points . This work devises a new cosymplectic energy-momentum method providing a new and more general framework to study t -dependent Hamilton equations. In fact, cosymplectic geometry allows for using more types of distinguished Lie symmetries (given by Hamiltonian, gradient, or evolution vector fields), relative equilibrium points, and reduction methods, than symplectic techniques. To make our work more self-contained and to fill some gaps in the literature, a review of the cosymplectic formalism and the cosymplectic Marsden–Weinstein reduction is included. Known and new types of relative equilibrium points are characterised and studied. Our methods remove technical conditions used in previous energy-momentum methods, like the Ad ∗ -equivariance of momentum maps. Eigenfunctions of t -dependent Schrödinger equations are interpreted in terms of relative equilibrium points in cosymplectic manifolds. A new cosymplectic-to-symplectic reduction is developed and a new associated type of relative equilibrium points, the so-called gradient relative equilibrium points , are introduced and applied to study the Lagrange points and Hill spheres of a restricted circular three-body system by means of a not Hamiltonian Lie symmetry of the system.

Based on the current situation of student cultivation mode and student management in colleges and universities, this paper puts forward six basic assumptions on student cultivation and management, and combines the assumptions to obtain the game payment matrix of student cultivation mode and student management. The dynamic equations of the cultivation and management process are reproduced by using mathematical calculus to derive the game payment matrix. Under the constraints, the game evolution model equilibrium point is tested for stability, the game evolution state is derived, and the game evolution model is completed to optimize the innovation of college student training mode and student management mechanism. Numerical simulation analysis is performed on the evolutionary game of college student cultivation and management based on the parameter settings of the evolutionary game model. The results show that the evolutionary game model will converge to a stable strategy point when the ratio of managers (teachers) and students’ initial participation in cultivation and management activities is set to (0.05, 0.95), (0.05, 0.05), (0.95, 0.05) or (0.95, 0.95). This study has a beneficial impact on cultivating exceptional talents, and it also provides a solid resource for society to deliver top-notch talent.

A pseudo-Newtonian planar circular restricted three-body problem with two Kerr-like primaries is considered. Using numerical methods, we explore the dynamical properties of the points of equilibrium of the system. In particular, we demonstrate how the two main parameters of the system affect the properties (position and type) of the libration points. For all the equilibria, we present their nature by classifying them not only as linearly stable and unstable but also as maxima, index-1, and index-2 saddles. We also reveal the networks of simple symmetric periodic orbits and their linear stability.

The body-schema concept is revisited in the context of embodied cognition, further developing the theory formulated by Marc Jeannerod that the motor system is part of a simulation network related to action, whose function is not only to shape the motor system for preparing an action (either overt or covert) but also to provide the self with information on the feasibility and the meaning of potential actions. The proposed computational formulation is based on a dynamical system approach, which is linked to an extension of the equilibrium-point hypothesis, called Passive Motor Paradigm: this dynamical system generates goal-oriented, spatio-temporal, sensorimotor patterns, integrating a direct and inverse internal model in a multi-referential framework. The purpose of such computational model is to operate at the same time as a general synergy formation machinery for planning whole-body actions in humanoid robots and/or for predicting coordinated sensory-motor patterns in human movements. In order to illustrate the computational approach, the integration of simultaneous, even partially conflicting tasks will be analyzed in some detail with regard to postural-focal dynamics, which can be defined as the fusion of a focal task, namely reaching a target with the whole-body, and a postural task, namely maintaining overall stability.

We numerically explore the Newton-Raphson basins of convergence, related to the libration points (which act as attractors), in the planar circular restricted five-body problem (CR5BP). The evolution of the position and the linear stability of the equilibrium points is determined, as a function of the value of the mass parameter. The attracting regions, on several types of two dimensional planes, are revealed by using the multivariate version of the classical Newton-Raphson iterative method. We perform a systematic investigation in an attempt to understand how the mass parameter affects the geometry as well as the degree of fractality of the basins of attraction. The regions of convergence are also related with the required number of iterations and also with the corresponding probability distributions.

This paper investigates the trends of three different species interacting in the same ecosystem. The study is conducted on the ecosystem which consists of a prey S 1 and two predators S 2 and S 3 in such a way that S 2 is a predator of S 1 and S 3 is a predator of both S 1 and S 2 . In addition, S 1 and S 2 share the same food which is the main food for S 1 and the alternative food for S 2 . A system of three simultaneous nonlinear partial differential equations is used to represent this situation. The local stability of all constant equilibrium points is discussed after showing their feasibility. In addition, a condition for which a coexisting constant equilibrium point is asymptotically globally stable is also investigated. The results showed that the system has eight constant equilibrium points. Five of them are unstable, while the three remaining are asymptotically locally stable under some conditions on the parameters that are involved in the model.

The present paper deals with the restricted four-body problem, when the third primary placed at the triangular libration point of the restricted three-body problem is an oblate body. The third primary m 3 is not influencing the motion of the dominating primaries m 1 and m 2 . We have studied the motion of m 4 , moving under the influence of the three primaries m i , i =1,2,3, but the motion of the primaries is not being influenced by infinitesimal mass m 4 . The aim of this study is to find the locations of equilibrium points and their stability. We obtain three collinear and five non-collinear equilibrium points. The collinear equilibrium points are unstable for all the mass parameter. The non-collinear points are stable for different mass parameter and oblateness factor. Also, we have considered the autonomous coplanar circular restricted four-body problem with the infinitesimal mass as a low-thrust spacecraft. Artificial equilibrium points are created with the use of continuous low-thrust propulsion. The obtained results show that, in absence of thrust there are unstable equilibrium points close to the third primary. Also, using the constant low-thrust, the artificial equilibrium points can be generated, which move from the natural equilibrium points. Further, it is proved that in certain region closed to the third primary, these points are stable. We have drawn the zero velocity surfaces to determine the possible allowed boundary regions. We observed that for increasing values of oblateness coefficient A , the corresponding possible boundary regions increase where the particle can freely move from one side to another side. Further, for different values of Jacobi constant C , we can find the boundary region where the particle can move in possible allowed partitions. The stability regions of the equilibrium points expanded due to presence of oblateness coefficient and various values of C .

The Newton-Raphson basins of attraction, associated with the libration points (attractors), are revealed in the pseudo-Newtonian planar circular restricted three-body problem, where the primaries have equal masses. The parametric variation of the position as well as of the stability of the equilibrium points is determined, when the value of the transition parameter ϵ varies in the interval [ 0 , 1 ] . The multivariate Newton-Raphson iterative scheme is used to determine the attracting domains on several types of two-dimensional planes. A systematic and thorough numerical investigation is performed in order to demonstrate the influence of the transition parameter on the geometry of the basins of convergence. The correlations between the basins of attraction and the corresponding required number of iterations are also illustrated and discussed. Our numerical analysis strongly indicates that the evolution of the attracting regions in this dynamical system is an extremely complicated yet very interesting issue.

This research explores the impact of reflected radiation on the formation and stability of out-of-plane equilibrium points within the circular restricted three-body problem (CR3BP). The study focuses on scenarios where the larger primary body emits radiation and the smaller primary reflects this radiation. Three distinct cases are analyzed based on the parameter λ, representing the ratio of radial distances from the infinitesimal mass to the primary bodies. The research identifies the conditions under which symmetric out-of-plane equilibrium points can form and evaluates their stability, concluding that these points are linearly unstable in all examined cases. The study also delves into the nature of orbits surrounding these equilibrium points, offering important insights into the complex dynamics influenced by albedo effects. Finally, the findings of this study are applied to a realistic model, specifically the Sun-Earth-Satellite system, to validate the theoretical analysis. Upon applying the theoretical framework to the Sun–Earth–Satellite system, we confirm that out-of-plane equilibrium points do not exist in this real planetary setup. This result is consistent with the derived mathematical conditions, showing that the gravitational and dynamical constraints of the system inhibit the formation of such equilibria. Here, the parameter λ plays a key role in classifying the possible configurations. These insights contribute to a more comprehensive understanding of equilibrium structures in celestial mechanics and their implications for planetary motion and space dynamics.

In this paper, a fractional-order prediction-based feedback control scheme (Fo-PbFC) is proposed to stabilize the unstable equilibrium points and to synchronize the fractional-order chaotic systems (FoCS). The design of Fo-PbFC, derived and based on Lyapunov stabilization arguments and matrix measure, is theoretically rigorous and represents a powerful and simple approach to provide a reasonable trade-off between computational overhead, storage space, numerical accuracy and stability analysis in control and synchronization of a class of FoCS. Numerical simulations are also provided to verify the validity and the feasibility of the proposed scheme by considering the fractional-order Newton–Leipnik chaotic and the fractional-order Mathieu–Van Der Pol hyperchaotic systems as illustrative examples.