scholarly journals On the dynamics of the rigid body with a fixed point: periodic orbits and integrability

2013 ◽  
Vol 74 (1-2) ◽  
pp. 327-333 ◽  
Author(s):  
Juan L. G. Guirao ◽  
Jaume Llibre ◽  
Juan A. Vera
Keyword(s):  
Fixed Point ◽  
Rigid Body ◽  
Periodic Orbits

scholarly journals Twist Periodic Orbits for Continuous Maps of the Eight Space

2013 ◽  
Vol 6 ◽  
pp. 4-8
Author(s):  
Iftichar Mudhar Talb Al-Shraa
Keyword(s):  
Fixed Point ◽  
Periodic Orbit ◽  
Periodic Orbits ◽  
Rotation Number ◽  
Continuous Map ◽  
Continuous Maps

Let g be a continuous map from 8 to itself has a fixed point at (0,0), we prove that g has a twist periodic orbit if there is a rational rotation number.

Chaos in a restricted problem of rotation of a rigid body with a fixed point

2008 ◽  
Vol 13 (3) ◽  
pp. 221-233 ◽  
Author(s):  
A. V. Borisov ◽  
A. A. Kilin ◽  
I. S. Mamaev
Keyword(s):  
Fixed Point ◽  
Rigid Body ◽  
Restricted Problem

scholarly journals On the Periodic Structure of Parallel Dynamical Systems on Generalized Independent Boolean Functions

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1088 ◽  
Author(s):  
Juan A. Aledo ◽  
Ali Barzanouni ◽  
Ghazaleh Malekbala ◽  
Leila Sharifan ◽  
Jose C. Valverde
Keyword(s):  
Fixed Point ◽  
Dynamical Systems ◽  
Periodic Orbits ◽  
Periodic Structure ◽  
Boolean Functions ◽  
Directed Graphs ◽  
General Pattern ◽  
General Situation ◽  
Point System ◽  
Local Functions

In this paper, based on previous results on AND-OR parallel dynamical systems over directed graphs, we give a more general pattern of local functions that also provides fixed point systems. Moreover, by considering independent sets, this pattern is also generalized to get systems in which periodic orbits are only fixed points or 2-periodic orbits. The results obtained are also applicable to homogeneous systems. On the other hand, we study the periodic structure of parallel dynamical systems given by the composition of two parallel systems, which are conjugate under an invertible map in which the inverse is equal to the original map. This allows us to prove that the composition of any parallel system on a maxterm (or minterm) Boolean function and its conjugate one by means of the complement map is a fixed point system, when the associated graph is undirected. However, when the associated graph is directed, we demonstrate that the corresponding composition may have points of any period, even if we restrict ourselves to the simplest maxterm OR and the simplest minterm AND. In spite of this general situation, we prove that, when the associated digraph is acyclic, the composition of OR and AND is a fixed point system.

scholarly journals Transitioning Between Underactuated Periodic Orbits: An Optimal Control Approach for Settling Time Reduction

2018 ◽  
Vol 15 (06) ◽  
pp. 1850027 ◽  
Author(s):  
Sotiris Apostolopoulos ◽  
Marion Leibold ◽  
Martin Buss
Keyword(s):  
Optimal Control ◽  
Fixed Point ◽  
Periodic Orbits ◽  
Domain Of Attraction ◽  
Inequality Constraint ◽  
Settling Time ◽  
Walking Robots ◽  
Zero Dynamics ◽  
Control Approach ◽  
Hybrid Zero Dynamics

In underactuated systems, a transition between two periodic orbits is generally characterized by slow convergence. This is due to the fact that the unactuated degree of freedom (DoF) hinders the state of the system to enter the domain of attraction of the target orbit close to the fixed point of the Poincaré Map. In this paper, we introduce an optimal control algorithm to reduce the settling time of transitions between periodic orbits of underactuated walking robots. This is achieved by utilizing the hybrid zero dynamics (HZD) framework to express the feasibility condition of the transition which can be imposed as an inequality constraint in the proposed optimal control problem. In addition, the cost function penalizes deviations from the fixed point of the target periodic orbit in the zero dynamics manifold while at the same time all dynamic and kinematic assumptions are treated as constraints. Furthermore, high magnitude torques are also penalized. The numerical results show that the proposed methodology can indeed improve the settling time compared to the transition methodology usually found in the bibliography and at the same time provide a feasible and smooth motion.

Remarks on the Covering of the Possible Motion Area by Solutions in Rigid Body Systems

2019 ◽  
Vol 20 (3-4) ◽  
pp. 293-302
Author(s):  
Ivan Polekhin
Keyword(s):  
Fixed Point ◽  
Rigid Body ◽  
Integrable Case ◽  
Analytical Results ◽  
Routh Reduction ◽  
Gyroscopic Forces ◽  
Kovalevskaya Case

AbstractThe problem of motion of a rigid body with a fixed point is considered. We study qualitatively the solutions of the system after Routh reduction. For the Lagrange integrable case, we show that the trajectories of solutions starting at the boundary of a possible motion area can both cover and not cover the entire possible motion area. It distinguishes these systems from the systems without gyroscopic forces, where the trajectories always cover the possible motion area. We also present some numerical and analytical results on the same matter for the Kovalevskaya case.

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