Leonov, Gennadij A.; Reitmann, Volker; Smirnova, Vera B.
Non-local methods for pendulum-like feedback systems. (English)
[B] Teubner-Texte zur Mathematik. Stuttgart: Teubner. vii, 242 p. (1992).
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Leonov, Gennadij A.; Reitmann, Volker; Smirnova, Vera B. Non-local methods for pendulum-like feedback systems. (English) [B] Teubner-Texte zur Mathematik. Stuttgart: Teubner. vii, 242 p. (1992). |
| 1. | Systems with Multiple Equilibria |
| 1.1 | Global Properties |
| 1.2 | Feedback Control Equations |
| 1.3 | The Transfer Function |
| 1.4 | The Yakubovich-Kalman Theorem |
| 2. | Pendulum-Like Systems |
| 2.1 | The Two Canonical Forms |
| 2.2 | Second-order Pendulum-Like Systems |
| a) General Properties | |
| b) The Case rS0 | |
| c) The Case r=0. Existence of Separatrix-Loops | |
| d) Non-Local Bifurcations in the Case r>0 | |
| e) Several Remarks about the Case r i (-1/a,0) | |
| 2.3 | The Lyapunov Direct Method in the Standard Form |
| 2.4 | Monostability and Gradient-Like Behavior of Phase-Controlled Systems with Nonlinearities having Mean Value Zero |
| 3. | Invariant Cones |
| 3.1 | Extension of the Circle Criterion |
| 3.2 | Systems with One Nonlinearity and a Bounded Forcing Term |
| 3.3 | Systems with Vector-Valued Nonlinearities |
| 3.4 | Bakaev Stability |
| 4. | The Bakaev-Guzh Technique |
| 4.1 | Basic Tools for Vector Fields on Riemannian Manifolds |
| 4.2 | Lyapunov-Type Results for Boundedness and Convergence |
| 4.3 | The Bakaev-Guzh Technique for Vector Fields |
| 4.4 | Second Order Systems of the Josephson-Type |
| 5. | The Method of Non-Local Reduction |
| 5.1 | A Lyapunov-Type Theorem |
| 5.2 | The Idea of Non-Local Reduction for Autonomous Systems of Indirect Control |
| 5.3 | Gradient-Like Behavior of Pendulum-Like Systems |
| 5.4 | Gradient-Like Behavior of Equations studied in the Theory of Phase-locked loops |
| 5.5 | Non-Local Reduction of Higher dimensional Systems to Autonomous Systems in the Plane |
| 5.6 | Non-Local Reduction in Non-Autonomous Pendulum-Like Systems |
| 5.7 | A Constructive Approach for Determining Lagrange Stability Criteria in the Non-Autonomous Case |
| 6. | Circular Solutions and Cycles |
| 6.1 | Pendulum-Like Systems with a Single Sign-Constant Nonlinearity |
| 6.2 | Frequency-domain Conditions for Existence of Circular Solutions and Cycles of the Second Kind |
| 6.3 | Circular Solutions and Cycles of the Second Kind in Concrete Systems of Phase Synchronization |
| 6.4 | Cycles of the First Kind |
| 7. | Synchronous Machines Equations |
| 7.1 | Special Properties of Synchronous Machines Equations |
| 7.2 | Gradient-Like Behavior of Machine Equations with a Zero Load |
| 7.3 | Application of Non-Local Reduction Method to some Classes of Synchronous Machines |
| 7.4 | Synchronous Machines Equations with a Forcing Term |
| 7.5 | The Equation of a Synchronous Machine with a Speed Governor |
| 7.6 | The Dynamics of Two Coupled Synchronous Machines |
| 8. | Integro-Differential Equations |
| 8.1 | General Setting |
| 8.2 | A priori Integral Estimates |
| 8.3 | Bakaev-Guzh Technique |
| 8.4 | Non-Local Reduction Principle |
| 8.5 | Phase Synchronization Systems |
| 9. | Cycle Slipping in Phase-Controlled Systems |
| 9.1 | Frequency-Domain Conditions for Cycle Slipping in ODE´s |
| 9.2 | Distributed Parameter Systems |
| 10. | Discrete Systems |
| 10.1 | Introduction |
| 10.2 | Boundedness by Positively Invariant Cones |
| 10.3 | Bakaev-Guzh Technique for Discrete Systems |
| 10.4 | The Method of Non-Local Reduction |
