Stability Theory of Dynamical Systems

Biography of Nam Parshad Bhatia Born in Lahore, India (now Pakistan) in 1932, Dr. Nam P. Bhatia studied physics and mathematics at Agra University. He then went to Germany and completed a doctorate in applied mathematics in Dresden in 1961. After returning to India briefly, he came to the United States in 1962 at the invitation of Solomon Lefschetz. In the US, Dr. Bhatia held research and teaching positions at the Research Institute of Advanced Studies, Baltimore, MD, Case Western Reserve University, Cleveland, OH, and the University of Maryland Baltimore County (UMBC). He was instrumental in developing the graduate programmes in Applied Mathematics, Computer Science, and Statistics at UMBC. Dr. Bhatia is currently Professor Emeritus at UMBC where he continues to pursue his research interests, which include the general theory of Dynamical and Semi-Dynamical Systems with emphasis on Stability, Instability, Chaos, and Bifurcations. Biography of Giorgio P. Szegö Giorgio Szegö was born in Rebbio, Italy, on July 10, 1934. After his studies at the University of Pavia and at the Technische Hochschule Darmstadt, he joined the Research Institute of Advanced Studies in Baltimore in 1961. From 1964 he held positions at the universities of Milano and Venice as well as several universities and research institutions in France, Spain, UK, and USA. He is currently Professor at the University of Roma "La Sapienza". Szegö's research contributions range from stability theory of ordinary differential equations to optimization theory.

I. Dynamical Systems.- 1. Definition and Related Notation.- 2. Examples of Dynamical Systems.- Notes and References.- II. Elementary Concepts.- 1. Invariant Sets and Trajectories.- 2. Critical Points and Periodic Points.- 3. Trajectory Closures and Limit Sets.- 4. The First Prolongation and the Prolongational Limit Set.- Notes and References.- III. Recursive Concepts.- 1. Definition of Recursiveness.- 2. Poisson Stable and Non-wandering Points.- 3. Minimal Sets and Recurrent Points.- 4. Lagrange Stability and Existence of Minimal Sets.- Notes and References.- IV. Dispersive Concepts.- 1. Unstable and Dispersive Dynamical Systems.- 2. Parallelizable Dynamical Systems.- Notes and References.- V. Stability Theory.- 1. Stability and Attraction for Compact Sets.- 2. Liapunov Functions: Characterization of Asymptotic Stability.- 3. Topological Properties of Regions of Attraction.- 4. Stability and Asymptotic Stability of Closed Sets.- 5. Relative Stability Properties.- 6. Stability of a Motion and Almost Periodic Motions.- >Notes and References.- V. Flow near a Compact Invariant Set.- 1. Description of Flow near a Compact Invariant Set.- 2. Flow near a Compact Invariant Set (Continued).- Notes and References.- VII. Higher Prolongations.- 1. Definition of Higher Prolongations.- 2. Absolute Stability.- 3. Generalized Recurrence.- Notes and References.- VIII. ?1-Liapunov Functions for Ordinary Differential Equations.- 1. Introduction.- 2. Preliminary Definitions and Properties.- 3. Local Theorems.- 4. Extension Theorems.- 5. The Structure of Liapunov Functions.- 6. Theorems Requiring Semidefinite Derivatives.- 7. On the Use of Higher Derivatives of a Liapunov Function.- Notes and References.- IX. Non-continuous Liapunov Functions for Ordinary Differential Equations.- 1.Introduction.- 2. A Characterization of Weak Attractors.- 3. Piecewise Differentiable Liapunov Functions.- 4. Local Results.- 5. Extension Theorems.- 6. Non-continuous Liapunov Functions on the Region of Weak Attraction.- Notes and References.- References.- Author Index.

From the reviews:

"The book presents a systematic treatment of the theory of dynamical systems and their stability written at the graduate and advanced undergraduate level. ... The book is well written and contains a number of examples and exercises." (Alexander Olegovich Ignatyev, Zentralblatt MATH, Vol. 993 (18), 2002)