Fundamental Examples of Nonlinear Dynamical Systems and Associated Second-Order Equation

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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Second-order bounds on the domain of attraction and the rate of convergence of nonlinear dynamical systems and neural networks

IEEE Transactions on Neural Networks ◽

10.1109/72.298225 ◽

1994 ◽

Vol 5 (4) ◽

pp. 551-560 ◽

Cited By ~ 7

Author(s):

N. Peterfreund ◽

Y. Baram

Keyword(s):

Neural Networks ◽

Dynamical Systems ◽

Rate Of Convergence ◽

Nonlinear Dynamical Systems ◽

Domain Of Attraction ◽

Second Order ◽

Nonlinear Dynamical ◽

Order Bounds

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Numerical scheme for period-m motion of second-order nonlinear dynamical systems based on generalized harmonic balance method

Nonlinear Dynamics ◽

10.1007/s11071-015-2518-y ◽

2015 ◽

Vol 84 (1) ◽

pp. 323-340 ◽

Cited By ~ 5

Author(s):

Yuefang Wang ◽

Zhiwei Liu

Keyword(s):

Dynamical Systems ◽

Harmonic Balance ◽

Numerical Scheme ◽

Nonlinear Dynamical Systems ◽

Harmonic Balance Method ◽

Second Order ◽

Balance Method ◽

Nonlinear Dynamical ◽

Generalized Harmonic Balance Method

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SYNCHRONIZATION OF A CLASS OF SECOND-ORDER NONLINEAR SYSTEMS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127408022500 ◽

2008 ◽

Vol 18 (11) ◽

pp. 3461-3471 ◽

Cited By ~ 1

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.

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