On the Stability Behavor of Bifurcated Normal Modes in Coupled Nonlinear Systems

1989 ◽  
Vol 56 (1) ◽  
pp. 155-161 ◽  
Author(s):  
C. H. Pak
Keyword(s):  
Nonlinear Systems ◽  
Calculus Of Variations ◽  
Normal Modes ◽  
Nonlinear Oscillators ◽  
Coupled Nonlinear Oscillators ◽  
Floquet’S Theory ◽  
Generic Bifurcation ◽  
The Stability ◽  
Floquet's Theory

The stability of bifurcated normal modes in coupled nonlinear oscillators is investigated, based on Synge’s stability in the kinematico-statical sense, utilizing the calculus of variations and Floquet’s theory. It is found, in general, that in a generic bifurcation, the stabilities of two bifurcated modes are opposite, and in a nongeneric bifurcation, the stability of continuing modes is opposite to that of the existing mode, and the stabilities of the two bifurcated modes are equal but opposite to that of the continuing mode. Some examples are illustrated.

NONLINEAR DYNAMICS OF DIGITAL PHASE-LOCKED LOOPS WITH DELAY

1994 ◽  
Vol 04 (03) ◽  
pp. 715-726 ◽  
Author(s):  
MARIA DE SOUSA VIEIRA ◽  
ALLAN J. LICHTENBERG ◽  
MICHAEL A. LIEBERMAN
Keyword(s):  
Nonlinear Dynamics ◽  
Chaotic Behavior ◽  
Nonlinear Oscillators ◽  
Coupled Systems ◽  
Phase Locked Loops ◽  
Bifurcation Diagrams ◽  
Digital Phase ◽  
Coupled Nonlinear Oscillators ◽  
Digital Phase Locked Loops ◽  
The Stability

We investigate numerically and analytically the nonlinear dynamics of a system consisting of two self-synchronizing pulse-coupled nonlinear oscillators with delay. The particular system considered consists of connected digital phase-locked loops. We find mapping equations that govern the system and determine the synchronization properties. We study the bifurcation diagrams, which show regions of periodic, quasiperiodic and chaotic behavior, with unusual bifurcation diagrams, depending on the delay. We show that depending on the parameter that is varied, the delay will have a synchronizing or desynchronizing effect on the locked state. The stability of the system is studied by determining the Liapunov exponents, indicating marked differences compared to coupled systems without delay.

On the Sensitivity of Non-Generic Bifurcation of Non-Linear Normal Modes

2007 ◽  
Author(s):  
C. H. Pak ◽  
Y. S. Choi
Keyword(s):  
Normal Mode ◽  
Natural Frequencies ◽  
Normal Modes ◽  
New Normal ◽  
Perturbed System ◽  
Saddle Node ◽  
Non Linear ◽  
The Difference ◽  
Generic Bifurcation ◽  
The Stability

It is shown that a non-generic bifurcation of non-linear normal modes may occur if the ratio of linear natural frequencies is near r-to-one, r = 1, 3, 5 ·······. Non-generic bifurcations are explicitly obtained in the systems having certain symmetry, as observed frequently in literatures. It is found that there are two kinds of non-generic bifurcations, super-critical and sub-critical. The normal mode generated by the former kind is extended to large amplitude, but that by the latter kind is limited to small amplitude which depends on the difference between two linear natural frequencies and disappears when two frequencies are equal. Since a non-generic bifurcation is not generic, it is expected generically that if a system having a non-generic bifurcation is perturbed then the non-generic bifurcation disappears and generic bifurcation appear in the perturbed system. Examples are given to verify the change in bifurcations and to obtain the stability behavior of normal modes. It is found that if a system having a super-critical non-generic bifurcation is perturbed, then two new normal modes are generated, one is stable, but the other unstable, implying a saddle-node bifurcation. If the system having a sub-critical non-generic bifurcation is perturbed, then no new normal mode is generated, but there is an interval of instability on a normal mode, implying two saddle-node bifurcations on the mode. Application of this study is discussed.

The Stability of the Normal Modes of Nonlinear Systems With Polynomial Restoring Forces of High Degree

1963 ◽  
Vol 30 (2) ◽  
pp. 193-198 ◽  
Author(s):  
C. P. Atkinson ◽  
S. J. Bhatt ◽  
Tercio Pacitti
Keyword(s):  
Stability Analysis ◽  
Nonlinear Systems ◽  
Exact Solutions ◽  
Normal Modes ◽  
Analog Computer ◽  
Degree Polynomial ◽  
Restoring Forces ◽  
The Stability ◽  
High Degree ◽  
Two Degree Of Freedom

This paper presents methods for determining the exact solutions of certain two-degree-of-freedom nonlinear systems with high-degree polynomial restoring forces. The exact solutions are the normal modes of the systems. A stability analysis of solutions in the normal modes is given. Some systems have normal modes whose stability is a function of amplitude. The stability of complicated systems can be approximated by superposing the stability characteristics of simpler systems. Some experimental confirmation by analog computer is also presented.

The Origin of Stability Indeterminacy in a Symmetric Hamiltonian

1984 ◽  
Vol 51 (2) ◽  
pp. 399-405 ◽  
Author(s):  
M. R. Hyams ◽  
L. A. Month
Keyword(s):  
Perturbation Theory ◽  
Stability Analysis ◽  
Hamiltonian System ◽  
Phase Plane ◽  
Nonlinear Oscillators ◽  
Two Dimensional ◽  
Periodic Motions ◽  
Coupled Nonlinear Oscillators ◽  
The Stability ◽  
Two Degree Of Freedom

The stability and bifurcation of periodic motions in a symmetric two-degree-of-freedom Hamiltonian system is studied by a reduction to a two-dimensional action-angle phase plane, via canonical perturbation theory. The results are used to explain why linear stability analysis will always be indeterminate for the in-phase mode in a class of coupled nonlinear oscillators.

Nonlinear Dynamics of Coupled Nonlinear Oscillators: Using of Complex Variables

1999 ◽  
Author(s):  
L. I. Manevitch
Keyword(s):  
Nonlinear Dynamics ◽  
Strong Coupling ◽  
Degrees Of Freedom ◽  
Normal Modes ◽  
Nonlinear Oscillators ◽  
Complex Variables ◽  
Dynamic Equations ◽  
Asymptotic Approach ◽  
Weakly Coupled ◽  
Coupled Nonlinear Oscillators

Abstract We present an asymptotic approach to the analysis of coupled nonlinear oscillators with asymmetric nonlinearity based on the complex representation of the dynamic equations The ideas of the approach are previously given on the example of the system with two degrees of freedom. The special attention is paid to the study of localized normal modes in the chain of weakly coupled nonlinear oscillators. We discuss also certain peculiarities of the localization of excitations in the case of strong coupling between oscillators.

scholarly journals Observer-based adaptive neural network backstepping sliding mode control for switched fractional order uncertain nonlinear systems with unmeasured states

2021 ◽  
pp. 002029402110211
Author(s):  
Tao Chen ◽  
Damin Cao ◽  
Jiaxin Yuan ◽  
Hui Yang
Keyword(s):  
Neural Network ◽  
Nonlinear Systems ◽  
Sliding Mode Control ◽  
Fractional Order ◽  
Sliding Mode ◽  
Tracking Error ◽  
Adaptive Neural Network ◽  
Dynamic Surface ◽  
Mode Control ◽  
The Stability

This paper proposes an observer-based adaptive neural network backstepping sliding mode controller to ensure the stability of switched fractional order strict-feedback nonlinear systems in the presence of arbitrary switchings and unmeasured states. To avoid “explosion of complexity” and obtain fractional derivatives for virtual control functions continuously, the fractional order dynamic surface control (DSC) technology is introduced into the controller. An observer is used for states estimation of the fractional order systems. The sliding mode control technology is introduced to enhance robustness. The unknown nonlinear functions and uncertain disturbances are approximated by the radial basis function neural networks (RBFNNs). The stability of system is ensured by the constructed Lyapunov functions. The fractional adaptive laws are proposed to update uncertain parameters. The proposed controller can ensure convergence of the tracking error and all the states remain bounded in the closed-loop systems. Lastly, the feasibility of the proposed control method is proved by giving two examples.

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