We present the new effect of phase synchronization of weakly coupled self-sustained chaotic oscillators. To characterize this phenomenon, we use the analytic signal approach based on the Hilbert transform and partial Poincar\'e maps. For coupled R\"ossler attractors, in the synchronous regime the phases are locked, while the amplitudes vary chaotically and are practically uncorrelated. Coupling a chaotic oscillator with a hyperchaotic one, we observe another new type of synchronization, where the frequencies are entrained, while the phase difference is unbounded. A relation between the phase synchronization and the properties of the Lyapunov spectrum is studied.

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Phase synchronization of chaotic oscillators

This paper features a survey of some recent developments in asymptotic techniques, which are valid not only for weakly nonlinear equations, but also for strongly ones. Further, the obtained approximate analytical solutions are valid for the whole solution domain. The limitations of traditional perturbation methods are illustrated, various modied perturbation techniques are proposed, and some mathematical tools such as variational theory, homotopy technology, and iteration technique are introduced to overcome the shortcomings. In this paper the following categories of asymptotic methods are emphasized: (1) variational approaches, (2) parameter-expanding methods, (3) parameterized perturbation method, (4) homotopy perturbation method (5) iteration perturbation method, and ancient Chinese methods. The emphasis of this article is put mainly on the developments in this eld in China so the references, therefore, are not exhaustive.

Some asymptotic methods for strongly nonlinear equations

This book, written by two major figures in adaptive control, provides a wealth of material for researchers, practitioners, and students. While some researchers in adaptive control may note the absence of a particular topic, the book‘s scope represents a high-gain instrument. It can be used by designers of control systems to enhance their work through the information on many new theoretical developments, and can be used by mathematical control theory specialists to adapt their research to practical needs. The book is strongly recommended to anyone interested in adaptive control.

Adaptive Control

This book treats nonlinear dynamics in both Hamiltonian and dissipative systems. The emphasis is on the mechanics for generating chaotic motion, methods of calculating the transitions from regular to chaotic motion, and the dynamical and statistical properties of the dynamics when it is chaotic. The book is intended as a self consistent treatment of the subject at the graduate level and as a reference for scientists already working in the field. It emphasizes both methods of calculation and results. It is accessible to physicists and engineers without training in modern mathematics. The new edition brings the subject matter in a rapidly expanding field up to date, and has greatly expanded the treatment of dissipative dynamics to include most important subjects. It can be used as a graduate text for a two semester course covering both Hamiltonian and dissipative dynamics.

Regular and Chaotic Dynamics

Preface Introduction 1. Fluid field equations and modal analysis in rigid containers 2. Linear forced sloshing 3. Viscous damping and sloshing suppression devices 4. Weakly nonlinear lateral sloshing 5. Equivalent mechanical models 6. Parametric sloshing (Faraday's waves) 7. Dynamics of liquid sloshing impact 8. Linear interaction of liquid sloshing with elastic containers 9. Nonlinear interaction under external and parametric excitations 10. Interactions with support structures and tuned sloshing absorbers 11. Dynamics of rotating fluids 12. Microgravity sloshing dynamics Bibliography Index.

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Liquid Sloshing Dynamics

Introduction to Discontinuous ODEs Mathematical Background for Multivalued Formulations Properties of Numerical Schemes Stick-Slip Oscillator with Two Degrees of Freedom Piecewiselinear Approximations Chua Circuit with Discontinuities One DOF Mechanical System with Friction A Mechanical System with 7 DOF Triple Pendulum with Impacts Analytical Prediction of Stick-Slip Chaos and other topics.

Bifurcation and Chaos in Nonsmooth Mechanical Systems

This survey is devoted to a significant role of var ious dry friction laws in engineering sciences. Both advantages and disadvantages of a frict ional process are illustrated and discussed, but excluding the nature of friction. It is shown how the classic friction laws and modern friction theories exists in today pure and a pplied sciences. Static and dynamic friction models are described. An important role of purely theoretical and experimental investigations in developing the appropriate fricti on models is outlined putting an emphasis on new approaches (models proposed by Bay-Wanheim, Dahl, Bliman-Sorine, LundGrenoble, as well as atomic scale and fractal model s, among others). Friction treated as a complex process being in interaction with wear, hea t emission and deformation is also discussed. Then an impact of dry friction models on today dynamical systems theory is reviewed. Finally, an application of friction to mo del a brake mechanism as a mechanical system with two degrees-of-freedom including experimental and numerical analyses is given. The review paper contains 254 references.

https://msvlab.hre.ntou.edu.tw/AMR-Awre-2005.pdf

Analysis of Dynamic Systems With Various Friction Laws

The paper focuses on continuous models derived from a discrete microstructure Various continualization procedures that take into account the nonlocal interaction between variables of the discrete media are analysed

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Improved Continuous Models for Discrete Media

The Principles of the Theory of Stability.- An Introduction to Stability of a Wheeled Vehicle.- Longitudinal Dynamics.- A Transversal Dynamics.- Vertical Dynamics.- Transversal Tilt Dynamics.- Longitudinal Tilt Dynamics.- Road Wheel Rotational Dynamics.- Modeling of a Piston - Connecting Rod - Crankshaft System.- Modeling of a Duo-Servo Brake.

Nonlinear dynamics of a wheeled vehicle

1 Asymptotic Approximations.- 2 Regular Perturbations of Parameters.- 3 Singular Perturbation Problems.- 4 Boundary Value Problems of Isotropic Cylindrical Shells.- 5 Boundary Value Problems - Orthotropic Shells.- 6 Composite Boundary Value Problems - Isotropic Shells.- 7 Composite Boundary Value Problems - Orthotropic Shells.- 8 Averaging.- 9 Continualization.- 10 Homogenization.- 11 Intermediate Asymptotics - Dynamical Edge Effect Method.- 12 Localization.- 13 Improvement of Perturbation Series.- 14 Matching of Limiting Asymptotic Expansions.- 15 Complex Variables in Nonlinear Dynamics.- 16 Other Asymptotical Approaches.- Afterword.- References.

Jan Awrejcewicz

Papers

Bifurcation and Chaos in Nonsmooth Mechanical Systems

Analysis of Dynamic Systems With Various Friction Laws

Improved Continuous Models for Discrete Media

Nonlinear dynamics of a wheeled vehicle

Asymptotical Mechanics of Thin-Walled Structures