9R10. Regular and Chaotic Oscillations. Foundations of Engineering Mechanics. - PS Landa (Dept of Phys, Moscow State Univ, Vorobyovy Gory, Moscow, 119899, Russia). Springer-Verlag, Berlin. 2001. 395 pp. ISBN 3-540-41001-5. $79.95.
Reviewed by RA Ibrahim (Dept of Mech Eng, Wayne State Univ, 5050 Anthony Wayne Dr, Rm 2119 Engineering Bldg, Detroit MI 48202).
Professor Polina Landa is a well known physicist for her work in the area of nonlinear oscillations and has significant contributions in both deterministic and stochastic nonlinear dynamical systems. In this book, she has made an excellent attempt to establish the general laws of oscillation theory for a wide spectrum of mechanical and physical problems. This reviewer believes that she has succeeded in achieving that goal.
The book begins with an introduction that classifies dynamical systems and different types of analytical models. Next, there is one chapter summarizing the main analytical techniques commonly used in analyzing nonlinear systems. These methods include the method of slowly time-varying amplitudes (known as the van der Pol method), the asymptotic Krylov-Bogolyubov method, the averaging method, and the Whitham method. The main body of the book is divided in two main parts: oscillations in autonomous systems covered in nine chapters, and oscillations in nonautonomous systems addressed in three chapters. There is also an appendix and a list of 375 references prevailed by Russian writers.
The book is part of a series of research monographs on Foundations of Engineering Mechanics written by former Soviet Union scholars who are recruited by the series editors V I Babitsky and J Wittenburg.
The first part begins with Chapter 3, which deals with the general properties of autonomous systems in terms of phase space and typical singular points. Systems with more degrees of freedom may possess complex structured attractors that can be classified into stochastic and chaotic attractors. Stochastic attractors involve only a finite or an infinite number of saddle cycles. On the other hand, attractors involving both saddle and stable cycles with small attraction basins are referred to as chaotic. The quantitative characteristics of such strange attractors are given in terms of the Hausdorff dimension, correlation dimension, Lyapunov exponents, and Lyapunov dimension. This chapter analyzed several nonlinear single-degree-of-freedom systems including the simple pendulum, a pendulum placed between opposite poles of a magnet, systems described by Duffing equation, oscillations of a bubble in fluid. Oscillations of species and systems with slowly varying natural frequency are considered in Chapter 4. Autonomous systems with many degrees of freedom are treated in Chapter 5. These include the normal oscillations in linear nonlinear conservative systems, and periodically linear and nonlinear chain systems. Chapter 5 also addressed the stochasticity in Hamiltonian systems close to integrable ones.
Chapter 6 introduces another class of autonomous dynamical systems that exhibit self-oscillatory motion such as the van der Pol and Rayleigh oscillators due to negative linear damping and positive nonlinear damping. Such oscillators, such as the model vacuum tube generators, exhibit the most fundamental features of all self-oscillatory systems including limit cycle and relaxation type motion. Other systems including the classical example of friction-induced vibration and the Neimark pendulum experiencing impacts at consecutive instants are treated. Self-oscillatory systems with one and a half degrees of freedom are analyzed in Chapter 7. The source of self-excitation is conditioned by the so-called inertia feedback. Examples involving linear and nonlinear inertia feedback cover a wide spectrum of physical and engineering applications. Chapter 8 is devoted to self-oscillatory two and more degree-of-freedom systems. Depending on the natural frequencies of the system and coupling parameters, the system response reveals exchange of energy or the absence of self-oscillations, or flutter as in the case of aeroelastic structures, or chaotic behavior as in the case of the vocal model, or radiation of sound of a “singing” flame. The analysis of self-oscillatory systems is extended in Chapter 9 to include synchronization and chaotization by imposing an external harmonic excitation. This chapter obviously belongs to non-autonomous systems, but the author preferred to address it in the first part because it is more related to the problem of self-excitation. Chapters 10 and 11 consider systems involving two or more self-oscillatory subsystems that weakly interact with each other. Cases of weak and strong linear coupling are treated. Parametric synchronization of two generators with different frequencies is considered for the case parametric resonance of the difference type, where the parametric excitation frequency equals the difference of the natural frequencies of the system. Chapter 11 considers three different synchronization regimes based on the relationship of their natural frequencies. These include close frequencies, multiple frequencies, and equal frequency differences of neighboring oscillators.
The second part of the book deals with different topics pertaining to nonautonomous systems. Chapter 12 considers forced excitation of nonlinear single- and two-degree-of-freedom systems in the neighborhood of harmonic, subharmonic, and superharmonic resonance conditions. It also treats passage through resonance when the excitation frequency is slowly varying with time. Chaotic regimes are discussed for periodically driven nonlinear systems such as the Duffing oscillator, a gas bubble in liquid under the action of sound field, and the Vallis model. The analysis is extended to include the excitation of two coupled nonlinear oscillators with stiffness nonlinearity in the neighborhood of the main resonance, and combination resonance of summed and difference types. Parametric excitation and simultaneous forced and parametric excitations of nonlinear systems are considered in Chapter 13. Complex response characteristics including chaotic motion and on-off-intermittency are usually manifested in the response for certain values of system and excitation parameters. Chapter 14 addresses the changes in the dynamical behavior and properties of nonlinear systems under high-frequency vibration or noise. The author acknowledges the contribution of Professor II Blekhman in preparing this chapter. The changes in the system dynamical behavior include the appearance and disappearance of attractors and repellers induced by the high-frequency vibration as applied to a pendulum subjected to a support motion that makes an angle with the vertical, and Brownian motion (known as stochastic ratchets) with saw-tooth potential. The case of Brownian motion with saw-tooth potential is treated using the Fokker-Planck equation. For this problem, noise-induced transition was found to occur if fluctuation transitions through each potential barrier are more frequent in one direction than in the other. The effect of the particle mass was then considered for two limiting cases: very small mass and sufficiently large mass. The last topic of this chapter deals with the problem of stochastic and vibrational resonances.
This book, Regular and Chaotic Oscillations, is clear and well written. It is recommended for researchers and students working in the area of nonlinear vibration.