Steady-state self-oscillations and chaotic behavior of a controlled electromechanical device by using the first Lyapunov value and the Melnikov theory

Abstract

In this paper regular and chaotic oscillations in a controlled electromechanical transducer are investigated. The nonlinear control laws are defined by an electric tension excitation and an external force applied to the mobile piece of the transducer.The paper shows that an Andronov-Poincaré-Hopf bifurcation appears as long as adequate parameters are chosen for the nonlinear control laws. The stability of the weak focuses associated to such bifurcation is examined according to the sign of the first Lyapunov value, showing that chaotic behavior can arise when the first Lyapunov value is varied harmonically. The appearance of a homoclinic orbit is investigated assuming an approximated model for the device. On the basis of the parametric equations of the homoclinic orbit and the presence of harmonic disturbances on the platform, it is demonstrated that chaotic oscillations can also appear, and they have been examined by means of the Melnikov theory. Chaotic behavior is corroborated by means of the sensitive dependence, Lyapunov exponents and power spectral density, and it is applied to drive the transducer mobile piece to a predefined set point assuming that noise due to the measurement process can appear in the control signals. The steady state error associated to such random noise is eliminated by adding a PI linear controller to the control force. Numerical simulations are used to corroborate the analytical results.

Analysis of the state equations of a real gas at high pressures with the virial coefficients obtained from controlled chaotic oscillations

“…It is well known that the stability of a weak focus depends on the sign of the first Lyapunov value (Guckenheimer et al, 1983;Wiggins, 2000;Pérez_Polo et al, 2014), which is calculated through the normal form of Eqs (24). For this purpose, we obtain the Jordan canonical form of Eq (24) by means of the eigenvector associated to the eigenvalue +iω to obtain the transformation matrix P t , which defines a coordinate transformation as:…”

“…Linearizing the vector field of Eqs (40) along a trajectory and applying the Gram-Smidth orthogonalization method (Pérez-Polo et al, 2014;Lichtenberg et al, 1992;Pérez-Polo et al, 2007), the Lyapunov exponents are obtained as function of time as shown in Fig 6 a). In particular, the whole set of Lyapunov exponents at t = 400 s is given by (Benettin et al, part I-part II 1980):…”

“…The parameter values and the control laws have been chosen to obtain three equilibrium points. One of them is always a saddle and the other two are weak focuses, whose stability is analyzed by calculating the sign of the first Lyapunov value (Guckenheimer et al, 1983;Wiggins, 2000;Pérez-Polo et al, 2014). With the harmonic variation of the first Lyapunov value between negative values (stable weak focus) and positive values (unstable weak focus) the piston position jumps from one weak focus to the other one, thus providing a route to chaotic oscillations.…”