Periodic Orbits in a Second-Order Discontinuous System with an Elliptic Boundary

This paper develops the analytical conditions for the onset and disappearance of motion passability and sliding along an elliptic boundary in a second-order discontinuous system. A periodically forced system, described by two different linear subsystems, is considered mainly to demonstrate the methodology. The passable, sliding and grazing conditions of a flow to the elliptic boundary in the discontinuous dynamical system are provided through the analysis of the corresponding vector fields and [Formula: see text]-functions. Moreover, by constructing appropriate generic mappings, periodic orbits in such a discontinuous system are predicted analytically. Finally, three different cases are discussed to illustrate the existence of periodic orbits with passable and/or sliding flows. The results obtained in this paper can be applied to the sliding mode control in discontinuous dynamical systems.

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Analysis of Discontinuous Dynamical Behaviors of a Friction-Induced Oscillator with an Elliptic Control Law

Mathematical Problems in Engineering ◽

10.1155/2018/5301747 ◽

2018 ◽

Vol 2018 ◽

pp. 1-33 ◽

Cited By ~ 3

Author(s):

Jinjun Fan ◽

Ping Liu ◽

Tianyi Liu ◽

Shan Xue ◽

Zhaoxia Yang

Keyword(s):

Dynamical System ◽

Vector Fields ◽

Control Law ◽

Periodic Motions ◽

Separation Boundary ◽

Discontinuous System ◽

Dynamical Behaviors ◽

Discontinuous Dynamical System ◽

Discontinuous Boundary ◽

Mass Spring

This paper develops the passability conditions of flow to the discontinuous boundary and the sticking or sliding and grazing conditions to the separation boundary in the discontinuous dynamical system of a friction-induced oscillator with an elliptic control law and the friction force acting on the mass M through the analysis of the corresponding vector fields and G-functions. The periodic motions of such a discontinuous system are predicted analytically through the mapping structure. Finally, the numerical simulations are given to illustrate the analytical results of motion for a better understanding of physics of motion in the mass-spring-damper oscillator.

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Complex Dynamics of Bouncing Motions at Boundaries and Corners in a Discontinuous Dynamical System

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4036518 ◽

2017 ◽

Vol 12 (6) ◽

Cited By ~ 3

Author(s):

Jianzhe Huang ◽

Albert C. J. Luo

Keyword(s):

Dynamical System ◽

Dynamical Systems ◽

Vector Fields ◽

Complex Dynamics ◽

Singularity Theory ◽

Local Theory ◽

Periodic Motions ◽

Border Collision Bifurcations ◽

Discontinuous Dynamical System ◽

Stability And Bifurcations

In this paper, from the local theory of flow at the corner in discontinuous dynamical systems, obtained are analytical conditions for switching impact-alike chatter at corners. The objective of this investigation is to find the dynamics mechanism of border-collision bifurcations in discontinuous dynamical systems. Multivalued linear vector fields are employed, and generic mappings are defined among boundaries and corners. From mapping structures, periodic motions switching at the boundaries and corners are determined, and the corresponding stability and bifurcations of periodic motions are investigated by eigenvalue analysis. However, the grazing and sliding bifurcations are determined by the local singularity theory of discontinuous dynamical systems. From such analytical conditions, the corresponding parameter map is developed for periodic motions in such a multivalued dynamical system in the single domain with corners. Numerical simulations of periodic motions are presented for illustrations of motions complexity and catastrophe in such a discontinuous dynamical system.

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Integrable two-dimensional dynamical systems and the characteristic Lie algebras

Proceedings of the Mavlyutov Institute of Mechanics ◽

10.21662/uim2007.1.023 ◽

2007 ◽

Vol 5 ◽

pp. 195-200

Author(s):

A.V. Zhiber ◽

O.S. Kostrigina

Keyword(s):

Dynamical System ◽

Dynamical Systems ◽

Lie Algebra ◽

Lie Algebras ◽

Second Order ◽

System Of Equations ◽

Two Dimensional ◽

Finite Dimensional ◽

Dimensional Dynamical System

In the paper it is shown that the two-dimensional dynamical system of equations is Darboux integrable if and only if its characteristic Lie algebra is finite-dimensional. The class of systems having a full set of fist and second order integrals is described.

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Quantization effect on sliding-mode control of a second-order dynamical system

2012 12th International Workshop on Variable Structure Systems ◽

10.1109/vss.2012.6163509 ◽

2012 ◽

Cited By ~ 2

Author(s):

Yan Yan ◽

Xinghuo Yu ◽

Shuanghe Yu

Keyword(s):

Dynamical System ◽

Sliding Mode Control ◽

Sliding Mode ◽

Second Order ◽

Quantization Effect ◽

Mode Control

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Second order and classical sliding mode control of a MAXPID system: A comparison performance

2014 15th International Conference on Sciences and Techniques of Automatic Control and Computer Engineering (STA) ◽

10.1109/sta.2014.7086741 ◽

2014 ◽

Cited By ~ 1

Author(s):

M. Jouini ◽

S. Dhahri ◽

N. Amara ◽

A. Sellami

Keyword(s):

Sliding Mode Control ◽

Sliding Mode ◽

Second Order ◽

Mode Control ◽

System A

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Periodic Motions With Grazing in a Discontinuous Dynamical System With Two Circular Boundaries

Volume 4: Dynamics, Vibration, and Control ◽

10.1115/imece2019-11489 ◽

2019 ◽

Author(s):

Siyu Guo ◽

Albert C. J. Luo

Keyword(s):

Dynamical System ◽

Periodic Motion ◽

Necessary Conditions ◽

Periodic Motions ◽

Discontinuous System ◽

Discontinuous Dynamical System ◽

Sufficient And Necessary Conditions

Abstract In this paper, studied are periodic motions with grazing in a discontinuous dynamical system with two circular boundaries. The grazing motion is for a periodic motion switching to another periodic motions. Thus, the sufficient and necessary conditions of motion switching, grazing and sliding on the boundaries are discussed first. Periodic motions with grazing in the discontinuous system are presented for illustration of motions switching.

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On the Dynamical Systems Stability Control and Applications

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.555.361 ◽

2014 ◽

Vol 555 ◽

pp. 361-368

Author(s):

Marcel Migdalovici ◽

Daniela Baran ◽

Gabriela Vlădeanu

Keyword(s):

Dynamical System ◽

Dynamical Systems ◽

Nonlinear System ◽

Stability Control ◽

Necessary Condition ◽

Linear Dynamical Systems ◽

System A ◽

First Approximation ◽

Free Parameters ◽

The Stability

The stability control analyzed by us, in this show, is based on our results in the domain of dynamical systems that depend of parameters. Any dynamical system can be considered as dynamical system that depends of parameters, without numerical particularization of them. All concrete dynamical systems, meted in the specialized literature, underline the property of separation between the stable and unstable zones, in sense of Liapunov, for two free parameters. This property can be also seen for one or more free parameters. Some mathematical conditions of separation between stable and unstable zones for linear dynamical systems are identified by us. For nonlinear systems, the conditions of separation may be identified using the linear system of first approximation attached to nonlinear system. A necessary condition of separation between stable and unstable zones, identified by us, is the sufficient order of differentiability or conditions of continuity for the functions that define the dynamical system. The property of stability zones separation can be used in defining the strategy of stability assurance and optimizing of the parameters, in the manner developed in the paper. The cases of dynamical systems that assure the separations of the stable and unstable zones, in your evolution, and permit the stability control, are analyzed in the paper.

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A Terminal Sliding Mode Control of Disturbed Nonlinear Second-Order Dynamical Systems

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4032503 ◽

2016 ◽

Vol 11 (5) ◽

Cited By ~ 4

Author(s):

Pawel Skruch

Keyword(s):

Dynamical Systems ◽

Nonlinear Dynamical Systems ◽

Sliding Mode ◽

Second Order ◽

Control Law ◽

Sliding Surface ◽

Terminal Sliding Mode ◽

Nonlinear Dynamical ◽

Second Order Differential Equations ◽

Bounded Disturbance

The paper presents a terminal sliding mode controller for a certain class of disturbed nonlinear dynamical systems. The class of such systems is described by nonlinear second-order differential equations with an unknown and bounded disturbance. A sliding surface is defined by the system state and the desired trajectory. The control law is designed to force the trajectory of the system from any initial condition to the sliding surface within a finite time. The trajectory of the system after reaching the sliding surface remains on it. A computer simulation is included as an example to verify the approach and to demonstrate its effectiveness.

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