Global Nonlinear Dynamics Based on the Method of Complete Bifurcation Groups and Rare Attractors

2009 ◽  
Author(s):  
Mikhail V. Zakrzhevsky
Keyword(s):  
Dynamical Systems ◽  
Bifurcation Analysis ◽  
Degrees Of Freedom ◽  
Nonlinear Dynamical Systems ◽  
Piecewise Linear ◽  
Global Bifurcation ◽  
Main Idea ◽  
Degree Of Freedom ◽  
Strongly Nonlinear ◽  
Driven System

The paper is devoted to the global bifurcation analysis of the models of strongly nonlinear forced or autonomous dynamical systems with one or several-degree-of-freedom by direct numerical and/or analytical methods. A new approach for the global bifurcation analysis for strongly nonlinear dynamical systems, based on the ideas of Poincare´, Birkhoff and Andronov, is proposed. The main idea of the approach is a concept of complete bifurcation groups and periodic branch continuation along stable and unstable solutions, named by the author as a method of complete bifurcation groups (MCBG). The article is illustrated using four archetypal forced dynamical systems with one degree-of-freedom. They are Duffing model with positional force f(x) = x + x3, Duffing double-well potential driven system, pendulum driven system and piecewise-linear (bilinear soft impact) driven dynamical system (Eq. 1–4). x+¨bx+˙x+x3=h1coswt(1)x+¨bx−˙x+x3=h1coswt(2)x+¨bx+˙a1sin(πx)=h1coswt(3)x+¨bx+˙f(x)=h1coswt,(4)f(x)=c1xifx≤d1,c2x−(c2−c1)d1ifx>d1 This paper is a continuation of the author’s previous one [53] with new results such as new bifurcation groups, rare attractors (RA) and protuberances. Some new results for dynamical systems with several degrees-of-freedom, based on the method of complete bifurcation groups may be found in [46–52].

HIGH-DIMENSIONAL CHAOS IN DISSIPATIVE AND DRIVEN DYNAMICAL SYSTEMS

2009 ◽  
Vol 19 (09) ◽  
pp. 2823-2869 ◽  
Author(s):  
Z. E. MUSIELAK ◽  
D. E. MUSIELAK
Keyword(s):  
Dynamical Systems ◽  
Degrees Of Freedom ◽  
Nonlinear Dynamical Systems ◽  
High Dimensional ◽  
Van Der Pol ◽  
Nonlinear Dynamical ◽  
Onset Of Chaos ◽  
General Rules ◽  
Van Der Pol Oscillators ◽  
Control And Synchronization

Studies of nonlinear dynamical systems with many degrees of freedom show that the behavior of these systems is significantly different as compared with the behavior of systems with less than two degrees of freedom. These findings motivated us to carry out a survey of research focusing on the behavior of high-dimensional chaos, which include onset of chaos, routes to chaos and the persistence of chaos. This paper reports on various methods of generating and investigating nonlinear, dissipative and driven dynamical systems that exhibit high-dimensional chaos, and reviews recent results in this new field of research. We study high-dimensional Lorenz, Duffing, Rössler and Van der Pol oscillators, modified canonical Chua's circuits, and other dynamical systems and maps, and we formulate general rules of high-dimensional chaos. Basic techniques of chaos control and synchronization developed for high-dimensional dynamical systems are also reviewed.

The Dynamics of a Nonharmonically Excited System Having Rigid Amplitude Constraints

1992 ◽  
Vol 59 (3) ◽  
pp. 693-695 ◽  
Author(s):  
Pi-Cheng Tung
Keyword(s):  
Dynamic Response ◽  
Bifurcation Analysis ◽  
Piecewise Linear ◽  
Coefficient Of Restitution ◽  
Degree Of Freedom ◽  
Linear Oscillator ◽  
Chaotic Motions ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Excited System

We consider the dynamic response of a single-degree-of-freedom system having two-sided amplitude constraints. The model consists of a piecewise-linear oscillator subjected to nonharmonic excitation. A simple impact rule employing a coefficient of restitution is used to characterize the almost instantaneous behavior of impact at the constraints. In this paper periodic and chaotic motions are found. The amplitude and stability of the periodic responses are determined and bifurcation analysis for these motions is carried out. Chaotic motions are found to exist over ranges of forcing periods.

SINGULARITY, SWITCHABILITY AND BIFURCATIONS IN A 2-DOF, PERIODICALLY FORCED, FRICTIONAL OSCILLATOR

2013 ◽  
Vol 23 (03) ◽  
pp. 1330009 ◽  
Author(s):  
ALBERT C. J. LUO ◽  
MOZHDEH S. FARAJI MOSADMAN
Keyword(s):  
Dynamical Systems ◽  
Bifurcation Analysis ◽  
Degrees Of Freedom ◽  
Eigenvalue Analysis ◽  
Analytical Dynamics ◽  
Periodic Motions ◽  
Stability And Bifurcation ◽  
Mapping Structures ◽  
Stability And Bifurcation Analysis ◽  
The Stability

In this paper, the analytical dynamics for singularity, switchability, and bifurcations of a 2-DOF friction-induced oscillator is investigated. The analytical conditions of the domain flow switchability at the boundaries and edges are developed from the theory of discontinuous dynamical systems, and the switchability conditions of boundary flows from domain and edge flows are presented. From the singularity and switchability of flow to the boundary, grazing, sliding and edge bifurcations are obtained. For a better understanding of the motion complexity of such a frictional oscillator, switching sets and mappings are introduced, and mapping structures for periodic motions are adopted. Using an eigenvalue analysis, the stability and bifurcation analysis of periodic motions in the friction-induced system is carried out. Analytical predictions and parameter maps of periodic motions are performed. Illustrations of periodic motions and the analytical conditions are completed. The analytical conditions and methodology can be applied to the multi-degrees-of-freedom frictional oscillators in the same fashion.

scholarly journals The direct method of Lyapunov for nonlinear dynamical systems with fractional damping

2020 ◽  
Vol 102 (4) ◽  
pp. 2017-2037
Author(s):  
Matthias Hinze ◽  
André Schmidt ◽  
Remco I. Leine
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Direct Method ◽  
Infinite Delay ◽  
Functional Differential Equations ◽  
Nonlinear Damping ◽  
Degree Of Freedom ◽  
Lyapunov Functionals ◽  
State Variables ◽  
Fractional Damping

AbstractIn this paper, we introduce a generalization of Lyapunov’s direct method for dynamical systems with fractional damping. Hereto, we embed such systems within the fundamental theory of functional differential equations with infinite delay and use the associated stability concept and known theorems regarding Lyapunov functionals including a generalized invariance principle. The formulation of Lyapunov functionals in the case of fractional damping is derived from a mechanical interpretation of the fractional derivative in infinite state representation. The method is applied on a single degree-of-freedom oscillator first, and the developed Lyapunov functionals are subsequently generalized for the finite-dimensional case. This opens the way to a stability analysis of nonlinear (controlled) systems with fractional damping. An important result of the paper is the solution of a tracking control problem with fractional and nonlinear damping. For this problem, the classical concepts of convergence and incremental stability are generalized to systems with fractional-order derivatives of state variables. The application of the related method is illustrated on a fractionally damped two degree-of-freedom oscillator with regularized Coulomb friction and non-collocated control.

Optimal Control of a Nonlinear Magnetic Bearing System Using the Trigonometric Collocation Method

2008 ◽  
Author(s):  
C. Nataraj ◽  
Steven Marx
Keyword(s):  
Optimal Control ◽  
Collocation Method ◽  
Degrees Of Freedom ◽  
Magnetic Bearing ◽  
Degree Of Freedom ◽  
Lookup Table ◽  
Base Excitation ◽  
Strongly Nonlinear ◽  
Trigonometric Collocation ◽  
Bearing System

Magnetic bearings are non-contacting, with the rotor being suspended between electromagnets, and therefore they can eliminate the need for lube oil and reduce machinery wear. The magnetic bearing is naturally unstable, and very nonlinear. This paper proposes a method designed to suppress the motion of a nonlinear magnetic bearing system rotor due to base excitation. The method combines PD feedback with feedforward optimal control, where a measured base motion is used to select a control signal designed to suppress the rotor response. The signal is generated from a combination of subharmonic frequencies and optimized coefficients stored in a lookup table. The trigonometric collocation method (TCM) is used to generate solutions for the four degree-of-freedom system made up of a shaft suspended at each end by a magnetic bearing. The TCM method uses a trigonometric series to simulate the multiharmonic behavior of each degree-of-freedom of strongly nonlinear systems. The method is easy to use and its advantage over numerical methods is that it demands less computation, particularly with higher numbers of degrees-of-freedom.

Sign in / Sign up

Export Citation Format

Share Document