Inverted pendulum under hysteretic control: stability zones and periodic solutions

In this paper, the mathematical model of the stabilization of the inverted pendulum with vertically oscillating suspension under hysteretic control is constructed. In the frame of the presented model, the stability criteria for the linearized equations of motion are found. We have made the numerical construction of the stability zones in the two-dimensional parameter space. Dependencies between initial conditions and driven parameters that provide periodic oscillations of the pendulum are obtained.     

Existence and uniqueness of periodic solutions for fourth-order nonlinear periodic systems

In this paper, we study the existence, uniqueness and stability of the periodic solutions for fourth-order nonlinear nonhomogeneous periodic systems with slowly changing coefficients by using the method of Liapunor Function. We obtain some sufficient conditions which guarantee the existence, uniqueness and asymptotic stability of the periodic solutions of these systems and estimate the extent to which the coefficients are allowed to change.     

Closed-form periodic solutions for piecewise-linear vibrating systems

In this paper, the global robust point dissipativity of an uncertain neural networks model with mixed time-varying delays is investigated, based on Lyapunov theory and inequality techniques. First, the concept of global robust point dissipativity is introduced. Next, some sufficient conditions are given for checking the global robust point dissipativity and the global exponential robust dissipativity of the uncertain neural networks model. Finally, illustrated examples are given to show the effectiveness of our results.     

The relative variance criterion for stability of delay systems

A new approach to the study of delay systems, applicable to physiological control systems and other systems where little information about the time delays is available, is examined. The method is based on the fact that stability information can be deduced from the statistical properties of the probability distribution that encodes the structure of the time delay. The main statistical variables used are the usual expectation parameter,E, and a modified variance, calledrelative variance and denotedR, that is invariant under time scale changes. Recent work of the author has shown that stability often improves asR increases whileE remains fixed. A four-parameter family of delay models is analysed in this paper, and the (E, R) pair is found to be a reliable indicator of stability over the global parameter domain of the family.     

A graphic stability criterion for non-commensurate fractional-order time-delay systems

This study focuses on a graphical approach to determine the stability of non-commensurate fractional-order systems with time-delay. By the iterative decomposition technique, the fractional-order systems described by the transfer functions are represented as a series of closed loop systems. The innermost closed loop system is straightforward to test the stability by its coefficient and the fractional-order. A function with respect to each open loop system is defined, and the stability of a non-commensurate fractional-order system is determined by the unstable poles of the innermost system and the times of the curve depending on the defined function encircling the origin in the clockwise direction. Finally, three illustrative examples are provided to demonstrate the effectiveness of the proposed criterion.     

Non-conservative oscillatory systems with periodic solutions

This note extends the work of previous authors to the closed orbits of non-linear oscillatory systems. The simple phase plane analysis is used.     

Resonant almost periodic oscillations in systems with slow varying parameters

We consider two-dimensional systems with fast rotating phase and slow varying parameters and assume that the right-hand side of a system almost periodically depends on fast and slow times. We investigate the conditions of the existence and stability of resonant almost periodic solutions. As an example, we consider forced oscillations of a mathematical pendulum under the action of a sum of two small forces with closed frequencies.     

An algorithm for constructing periodic solutions of Lyapunov systems

Moscow Aviation Institute. Translated from Prikladnaya Mekhanika, Vol. 27, No. 2, pp. 91–95, February, 1991.     

A general reduction method for periodic solutions in conservative and reversible systems

We introduce a general reduction method for the study of periodic solutions near equilibria in autonomous systems which are either conservative or reversible. We impose no restrictions on the linearization at the equilibrium, allowing higher multiplicities and all kinds of resonances. It is shown that the problem reduces to a similar problem for a reduced system, which is itself conservative or reversible, but also has an additionalS ¹-symmetry. This symmetry allows to immediately write down the bifurcation equations. Moreover, the reduced system can be calculated up to any order by a normal form reduction on the original system. The method of proof combines normal forms with the Liapunov-Schmidt method. A similar approach was already introduced for Hamiltonian systems in [9], and for equivariant systems in [3]; this paper extends the results of these papers to the cases of conservative and reversible systems.The research in this paper was supported by the EEC Science Project on Bifurcation Theory and Its Applications.     

Equivalent linearizations for practical hysteretic systems

Experimental testing of a friction damped base isolation system has indicated a need for a new model of friction damping and for an appropriate equivalent linearization technique. The model for the damping adopted is a combination of viscous damping, constant Coulomb friction and linear Coulomb friction.This model is incorporated into the equation of motion for a single-degree-of-freedom system and the exact solutions are given for free vibrations and for steady-state vibrations excited by a harmonic force. The exact solution is taken as a basis for an equivalent linearization technique that can be used in conjunction with conventional design spectra for a practical design of such a system.     

Lyapunov exponents estimation for hysteretic systems

This article discusses the Lyapunov exponent estimation of non-linear hysteretic systems by adapting the classical algorithm by Wolf and co-workers [Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A., 1985. Determining Lyapunov exponents from a times series. Physica D 16, 285–317.]. This algorithm evaluates the divergence of nearby orbits by monitoring a reference trajectory, evaluated from the equations of motion of the original hysteretic system, and a perturbed trajectory resulting from the integration of the linearized equations of motion. The main issue of using this algorithm for non-linear, rate-independent, hysteretic systems is related to the procedure of linearization of the equations of motion. The present work establishes a procedure of linearization performing a state space split and assuming an equivalent viscous damping in order to represent hysteretic dissipation in the linearized system. The dynamical response of a single-degree of freedom pseudoelastic shape memory alloy (SMA) oscillator is discussed as an application of the proposed algorithm. The restitution force of the oscillator is provided by an SMA element described by a rate-independent, hysteretic, thermomechanical constitutive model. Two different modeling cases are considered for isothermal and non-isothermal heat transfer conditions, and numerical simulations are performed for both cases. The evaluation of the Lyapunov exponents shows that the proposed procedure is capable of quantifying chaos capturing the non-linear dissipation of hysteretic systems.