Lucid Analysis of Periodically Forced Nonlinear Systems via Normal Forms

2019 ◽  
Author(s):  
Peter M. B. Waswa ◽  
Sangram Redkar
Keyword(s):  
Nonlinear Systems ◽  
Ad Hoc ◽  
Normal Forms ◽  
Numerical Results ◽  
Duffing Equation ◽  
System State ◽  
Periodic Forcing ◽  
Periodic Coefficients ◽  
Duffing’S Equation ◽  
State Augmentation

Abstract This paper presents a straightforward methodology to analyze periodically forced nonlinear systems with constant and periodic coefficients via normal forms. We demonstrate how the intuitive system state augmentation facilitates construction of normal forms by avoiding ad-hoc addition of equation variables, book-keeping parameters and detuning parameters. Moreover, this technique directly connects the periodic forcing terms and periodic coefficients of the nonlinearity with the augmented states — making it applicable to all periodically forced nonlinear systems. Accuracy of this approach is successfully verified via fulfilled compliance between analytical and numerical results of forced Duffing’s equation and Mathieu-Duffing equation.

Analytical Study of the Convection Heat Transfer From an Isothermal Wedge Surface to Fluids

2012 ◽  
Vol 134 (6) ◽  
Author(s):  
M. Bachiri ◽  
A. Bouabdallah
Keyword(s):  
Heat Transfer ◽  
Prandtl Number ◽  
Ad Hoc ◽  
Convection Heat Transfer ◽  
Analytical Approximation ◽  
Numerical Results ◽  
Convection Heat ◽  
Governing Equations ◽  
Wedge Surface ◽  
Prandtl Numbers

In this work, we attempt to establish a general analytical approximation of the convection heat transfer from an isothermal wedge surface to fluids for all Prandtl numbers. The flow has been assumed to be laminar and steady state. The governing equations have been written in dimensionless form using a similarity method. A simple ad hoc technique is used to solve analytically the governing equations by proposing a general formula of the velocity profile. This formula verifies the boundary conditions and the equilibrium of the governing equations in the whole spatial region and permits us to obtain analytically the temperature profiles for all Prandtl numbers and for various configurations of the wedge surface. A comparison with the numerical results is given for all spatial regions and in wide Prandtl number values. A new Nusselt number expression is obtained for various configurations of the wedge surface and compared with the numerical results in wide Prandtl number values.

A New Approach for Computing the Normal Forms of High Dimensional Nonlinear Systems

2010 ◽  
Author(s):  
Shuping Chen ◽  
Wei Zhang ◽  
Minghui Yao
Keyword(s):  
Nonlinear Systems ◽  
Normal Form ◽  
Cantilever Beam ◽  
Normal Forms ◽  
Nonlinear Differential Equations ◽  
Real Life ◽  
High Dimensional ◽  
Computation Method ◽  
Normal Form Theory ◽  
Averaged Equations

Normal form theory is very useful for direct bifurcation and stability analysis of nonlinear differential equations modeled in real life. This paper develops a new computation method for obtaining a significant refinement of the normal forms for high dimensional nonlinear systems. The method developed here uses the lower order nonlinear terms in the normal form for the simplifications of higher order terms. In the theoretical model for the nonplanar nonlinear oscillation of a cantilever beam, the computation method is applied to compute the coefficients of the normal forms for the case of two non-semisimple double zero eigenvalues. The normal forms of the averaged equations and their coefficients for non-planar non-linear oscillations of the cantilever beam under combined parametric and forcing excitations are calculated.

Observer normal forms for a class of nonlinear systems by means of coupled auxiliary dynamics

2020 ◽  
Vol 30 (13) ◽  
pp. 4960-4978
Author(s):  
Li‐Fei Wang ◽  
Driss Boutat ◽  
Da‐Yan Liu
Keyword(s):  
Nonlinear Systems ◽  
Normal Forms

Control of chaos in nonlinear systems with time-periodic coefficients

2006 ◽  
Vol 364 (1846) ◽  
pp. 2417-2432 ◽  
Author(s):  
S.C Sinha ◽  
Alexandra Dávid
Keyword(s):  
Nonlinear Systems ◽  
Periodic Orbit ◽  
Nonlinear Control ◽  
State Feedback ◽  
State Feedback Control ◽  
Control Technique ◽  
Periodic Coefficients ◽  
Full State ◽  
Full State Feedback ◽  
Time Periodic

In this study, some techniques for the control of chaotic nonlinear systems with periodic coefficients are presented. First, chaos is eliminated from a given range of the system parameters by driving the system to a desired periodic orbit or to a fixed point using a full-state feedback. One has to deal with the same mathematical problem in the event when an autonomous system exhibiting chaos is desired to be driven to a periodic orbit. This is achieved by employing either a linear or a nonlinear control technique. In the linear method, a linear full-state feedback controller is designed by symbolic computation. The nonlinear technique is based on the idea of feedback linearization. A set of coordinate transformation is introduced, which leads to an equivalent linear system that can be controlled by known methods. Our second idea is to delay the onset of chaos beyond a given parameter range by a purely nonlinear control strategy that employs local bifurcation analysis of time-periodic systems. In this method, nonlinear properties of post-bifurcation dynamics, such as stability or rate of growth of a limit set, are modified by a nonlinear state feedback control. The control strategies are illustrated through examples. All methods are general in the sense that they can be applied to systems with no restrictions on the size of the periodic terms.

Bifurcation in the Duffing equation with independent parameters, II

1978 ◽  
Vol 79 (3-4) ◽  
pp. 317-326 ◽  
Author(s):  
Jack K. Hale ◽  
Hildebrando M. Rodrigues
Keyword(s):  
Parameter Space ◽  
Duffing Equation ◽  
Damping Term ◽  
Duffing's Equation ◽  
Duffing’S Equation ◽  
All Solutions ◽  
Independent Parameters

SynopsisIn a previous paper, the authors gave a complete description of the number of even harmonic solutions of Duffing's equation without damping for the parameters varying in a full neighbourhood of the origin in the parameter space. In this paper, the analysis is extended to the case of an independent small damping term. It is also shown that all solutions of the undamped equation are even functions of time.

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