Control of chaos via extended delay feedback

We study the inducement of passive nonlinear sinks in linear vibrating systems. These are substructures that absorb vibrational energy in a one-way, irreversible fashion. The systems considered are composed of strongly coupled, grounded damped linear oscillators with a strongly nonlinear attachment at the end. Applying a complex averaging technique we derive a set of modulation equations that is directly amenable to physical interpretation, and provides insight into the energy pumping phenomenon. For the case of a two DOF system we show that nonlinear energy pumping occurs when a certain frequency of envelope modulation crosses through zero; then the dynamics of the envelope modulation of the motion resemble the dynamics of a forced rigid body. For the case of an impulsively loaded multi-DOF chain with a nonlinear attachment at the end, we show that after some initial transients the response of the nonlinear attachment sets to a motion dominated by a fast frequency identical to the lower bound of the propagation zone of the linear chain. This feature reduces the study of energy pumping in the chain to a two DOF equivalent problem. The applications of the energy pumping phenomenon to practical engineering problems are discussed.

Inducing Passive Nonlinear Energy Sinks in Vibrating Systems

In this paper, the homotopy–perturbation method proposed by J.-H. He is adopted for solving pure strong nonlinear second-order differential equation. For the oscillatory differential equation the initial approximate solution is assumed in the form of Jacobi elliptic function and the forementioned method is used for obtaining of the approximate analytic solution. Two types of differential equations are considered: with strong cubic and strong quadratic nonlinearity. The obtained solution is compared with exact numerical one. The difference between these solutions is negligible for a long time period. The method is found to work extremely well in the examples, but the theoretical reasons are not yet clear.

Homotopy–perturbation method for pure nonlinear differential equation

An elliptic Lindstedt--Poincare (L--P) method is presented for the steady-state analysis of strongly non-linear oscillators of the form$\ddot x + c_1 x + c_3 x^3 = \varepsilon f(x,\dot x)$ , in which the Jacobian elliptic functions are employed instead of the usual circular functions in the classical L--P perturbation procedure. This method can be viewed as a generalization of the L--P method. As an application of this method, three types of the generalized Van der Pol equation with $f(x,\dot x) = (c_0 - c_2 x^2 )\dot x$ are studied in detail.

A modified Lindstedt-Poincaré method for certain strongly non-linear oscillators☆

An elliptic Lindstedt--Poincare (L--P) method is presented for the steady-state analysis of strongly non-linear oscillators of the form
 
$$\ddot x + c_1 x + c_3 x^3 = \varepsilon f(x,\dot x)$$

 , in which the Jacobian elliptic functions are employed instead of the usual circular functions in the classical L--P perturbation procedure. This method can be viewed as a generalization of the L--P method. As an application of this method, three types of the generalized Van der Pol equation with 
$$f(x,\dot x) = (c_0 - c_2 x^2 )\dot x$$

 are studied in detail.

An Elliptic Lindstedt--Poincaré Method for Certain Strongly Non-Linear Oscillators

A new route is described for eliminating chaos in nonlinear oscillators by changing only the shape of a weak nonlinear periodic perturbation and illustrated with the example of the Duffing-Holmes oscillator forced with the Jacobian elliptic function sn. Two techniques are used in the illustration: applying the Melnikov-Holmes analysis, and studying the behavior of the Lyapunov exponent from a simple recursion relation which models an unstable limit cycle. The connection with related previously described routes is also discussed in a general setting.

Routes to suppressing chaos by weak periodic perturbations.

Construction of approximate analytical solutions to a new class of non-linear oscillator equations

Improvement of a Krylov-Bogoliubov method that uses Jacobi elliptic functions

It is shown that the Krylov–Bogoliubov methods that give the approximate oscillatory solution of the equation in terms of Jacobi elliptic functions are applicable not only when c 1>0 and c 3>0, but also when c 1>0 and c 3 0. In particular, the most precise of these methods, the Christopher-Brocklehurst method, is discussed in detail. Its accuracy has been improved by utilizing the transformation properties of elliptic functions with respect to their parameters.

J. Díaz Bejarano

Papers

Routes to suppressing chaos by weak periodic perturbations.

Construction of approximate analytical solutions to a new class of non-linear oscillator equations

Improvement of a Krylov-Bogoliubov method that uses Jacobi elliptic functions

Extension and improvement to the Krylov–Bogoliubov methods using elliptic functions

A generalization of the method of harmonic balance