SYNCHRONIZATION OF A CLASS OF SECOND-ORDER NONLINEAR SYSTEMS

2008 ◽  
Vol 18 (11) ◽  
pp. 3461-3471 ◽  
Author(s):  
A. P. MIJOLARO ◽  
L. F. C. ABERTO ◽  
N. G. BRETAS
Keyword(s):  
Asymptotic Behavior ◽  
Vector Field ◽  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Second Order ◽  
Coupled Systems ◽  
Nonlinear Dynamical ◽  
Unbounded Solutions ◽  
Synchronization Region ◽  
Coupling Parameters

The asymptotic behavior of a class of coupled second-order nonlinear dynamical systems is studied in this paper. Using very mild assumptions on the vector-field, conditions on the coupling parameters that guarantee synchronization are provided. The proposed result does not require solutions to be ultimately bounded in order to prove synchronization, therefore it can be used to study coupled systems that do not globally synchronize, including synchronization of unbounded solutions. In this case, estimates of the synchronization region are obtained. Synchronization of two-coupled nonlinear pendulums and two-coupled Duffing systems are studied to illustrate the application of the proposed theory.

scholarly journals On diagrammatic technique for nonlinear dynamical systems

2014 ◽  
Vol 29 (35) ◽  
pp. 1430039
Author(s):  
Mykola Semenyakin
Keyword(s):  
Vector Field ◽  
Dynamical Systems ◽  
Fixed Points ◽  
Vector Fields ◽  
Nonlinear Dynamical Systems ◽  
Radius Of Convergence ◽  
Evolution Operators ◽  
Finite Degree ◽  
Nonlinear Dynamical ◽  
Diagrammatic Technique

In this paper, we investigate phase flows over ℂn and ℝn generated by vector fields V = ∑ Pi∂i where Pi are finite degree polynomials. With the convenient diagrammatic technique, we get expressions for evolution operators ev {V|t} : x(0) ↦ x(t) through the series in powers of x(0) and t, represented as sum over all trees of a particular type. Estimates are made for the radius of convergence in some particular cases. The phase flows behavior in the neighborhood of vector field fixed points are examined. Resonance cases are considered separately.

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

2016 ◽  
Vol 11 (5) ◽  
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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