Averaging Methods in Nonlinear Dynamical Systems

1. Front Matter

   Pages N2-x

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2. Basic Material

   • Jan A. Sanders, Ferdinand Verhulst

   Pages 1-8

3. Asymptotics of Slow-time Processes, First Steps

   • Jan A. Sanders, Ferdinand Verhulst

   Pages 9-32

4. The Theory of Averaging

   • Jan A. Sanders, Ferdinand Verhulst

   Pages 33-66

5. Attraction

   • Jan A. Sanders, Ferdinand Verhulst

   Pages 67-82

6. Averaging over Spatial Variables: Systems with Slowly Varying Frequency and Passage through Resonance

   • Jan A. Sanders, Ferdinand Verhulst

   Pages 83-123

7. Normal Forms

   • Jan A. Sanders, Ferdinand Verhulst

   Pages 124-142

8. Hamiltonian Systems

   • Jan A. Sanders, Ferdinand Verhulst

   Pages 143-179

9. Appendices

   • Jan A. Sanders, Ferdinand Verhulst

   Pages 181-235

10. Back Matter

    Pages 236-249

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In this book we have developed the asymptotic analysis of nonlinear dynamical systems. We have collected a large number of results, scattered throughout the literature and presented them in a way to illustrate both the underlying common theme, as well as the diversity of problems and solutions. While most of the results are known in the literature, we added new material which we hope will also be of interest to the specialists in this field. The basic theory is discussed in chapters two and three. Improved results are obtained in chapter four in the case of stable limit sets. In chapter five we treat averaging over several angles; here the theory is less standardized, and even in our simplified approach we encounter many open problems. Chapter six deals with the definition of normal form. After making the somewhat philosophical point as to what the right definition should look like, we derive the second order normal form in the Hamiltonian case, using the classical method of generating functions. In chapter seven we treat Hamiltonian systems. The resonances in two degrees of freedom are almost completely analyzed, while we give a survey of results obtained for three degrees of freedom systems. The appendices contain a mix of elementary results, expansions on the theory and research problems.

• Hamiltonian
• Systems
• asymptotic analysis
• dynamical system
• dynamical systems
• field
• form
• function
• functions
• hamiltonian system
• online
• sets
• solution

• Department of Mathematics and Computer Science, Free University, Amsterdam, The Netherlands

  Jan A. Sanders

• Mathematical Institute, State University of Utrecht, Utrecht, The Netherlands

  Ferdinand Verhulst

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