Input-Output Stabilization Problem For Some Nonlinear Dynamical Systems With Bifurcation

Covers in a progressive fashion a number of analysis tools and design techniques directly applicable to nonlinear control problems in high performance systems (in aerospace, robotics and automotive areas).

Applied Nonlinear Control

1. Introduction and Simple Properties.- 1.1. Introduction.- 1.2. Chaotic Ordinary Differential Equations.- 1.3. Our Approach to the Lorenz Equations.- 1.4. Simple Properties of the Lorenz Equations.- 2. Homoclinic Explosions: The First Homoclinic Explosion.- 2.1. Existence of a Homoclinic Orbit.- 2.2. The Bifurcation Associated with a Homoclinic Orbit.- 2.3. Summary and Some General Definitions.- 3. Preturbulence, Strange Attractors and Geometric Models.- 3.1. Periodic Orbits for the Hopf Bifurcation.- 3.2. Preturbulence and Return Maps.- 3.3. Strange Attractor and Homoclinic Explosions.- 3.4. Geometric Models of the Lorenz Equations.- 3.5. Summary.- 4. Period Doubling and Stable Orbits.- 4.1. Three Bifurcations Involving Periodic Orbits.- 4.2. 99.524 100.795. The x2y Period Doubling Window.- 4.3. 145 166. The x2y2 Period Doubling Window.- 4.4. Intermittent Chaos.- 4.5. 214.364 ?. The Final xy Period Doubling Window.- 4.6. Noisy Periodicity.- 4.7. Summary.- 5. From Strange Attractor to Period Doubling.- 5.1. Hooked Return Maps.- 5.2. Numerical Experiments.- 5.3. Development of Return Maps as r Increases: Homoclinic Explosions and Period Doubling.- 5.4. Numerical Experiments on Periodic Orbits.- 5.5. Period Doubling and One-Dimensional Maps.- 5.6. Global Approach and Some Conjectures.- 5.7. Summary.- 6. Symbolic Description of Orbits: The Stable Manifolds of C1 and C2.- 6.1. The Maxima-in-z Method.- 6.2. Symbolic Descriptions from the Stable Manifolds of C1 and C2.- 6.3. Summary.- 7. Large r.- 7.1. The Averaged Equations.- 7.2. Analysis and Interpretation of the Averaged Equations.- 7.3. Anomalous Periodic Orbits for Small b and Large r.- 7.4. Summary.- 8. Small b.- 8.1. Twisting Around the z-Axis.- 8.2. Homoclinic Explosions with Extra Twists.- 8.3. Periodic Orbits Without Extra Twisting Around the z-Axis.- 8.4. Heteroclinic Orbits Between C1 and C2.- 8.5. Heteroclinic Bifurcations.- 8.6. General Behaviour When b = 0.25.- 8.7. Summary.- 9. Other Approaches, Other Systems, Summary and Afterword.- 9.1. Summary of Predicted Bifurcations for Varying Parameters ?, b and r.- 9.2. Other Approaches.- 9.3. Extensions of the Lorenz System.- 9.4. Afterword - A Personal View.- Appendix A. Definitions.- Appendix B. Derivation of the Lorenz Equations from the Motion of a Laboratory Water Wheel.- Appendix C. Boundedness of the Lorenz Equations.- Appendix D. Homoclinic Explosions.- Appendix E. Numerical Methods for Studying Return Maps and for Locating Periodic Orbits.- Appendix F. Computational Difficulties Involved in Calculating Trajectories which Pass Close to the Origin.- Appendix G. Geometric Models of the Lorenz Equations.- Appendix H. One-Dimensional Maps from Successive Local Maxima in z.- Appendix I. Numerically Computed Values of k(r) for ? = 10 and b = 8/3.- Appendix J. Sequences of Homoclinic Explosions.- Appendix K. Large r the Formulae.

The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors

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Input-Output Stabilization Problem For Some Nonlinear Dynamical Systems With Bifurcation

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References

Applied Nonlinear Control

The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors

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