Textbook  Qualitative Methods for Nonlinear Dynamics, Parts I and II by L.Shilnikov, A.Shilnikov, D.Turaev, and L.Chua, World Scientific Publ.

CONTENT
PREFACE
Chapter 1. BASIC CONCEPTS
1.1 Necessary background from the theory of ordinary differential equations
1.2. Dynamical systems. Basic notions.
1.3. Qualitative integration of dynamical systems

Chapter 2. STRUCTURALLY STABLE EQUILIBRIUM STATES OF DYNAMICAL SYSTEMS
2.1 Notion of an equilibrium state. A linearized system.
2.2 Qualitative investigation of 2- and 3-dimensional linear systems.
2.3 High-dimensional linear systems. Invariant phantom
2.4 Behavior of trajectories of a linear system near saddle equilibrium states.
2.5 Topological classification of structurally stable equilibrium states
2.6 Stable equilibrium states. Leading and non-leading manifolds
2.7 Saddle equilibrium states. Invariant manifolds
2.8 Solution near a saddle. The boundary-value problem
2.9 Problem of smooth linearization. Resonances

Chapter 3. STRUCTURALLY STABLE PERIODIC TRAJECTORIES OF DYNAMICAL SYSTEMS
3.1 A Poincare map. A fixed point. Multipliers
3.2 Non-degenerate linear one- and two-dimensional maps
3.3 Fixed points of high-dimensional linear maps
3.4 Topological classification of fixed points
3.5 Properties of nonlinear maps near a stable fixed point
3.6 Saddle fixed points. Invariant manifolds
3.7 The boundary-value problem near a saddle fixed point
3.8 Behavior of linear maps near saddle fixed points Examples
3.9 Geometrical properties of nonlinear saddle maps
3.10 Normal coordinates in a neighborhood of a periodic trajectory
3.11 The variational equations
3.12 Stability of periodic trajectories. Saddle periodic trajectories
3.13 Smooth equivalence and resonances
3.14 Autonomous normal forms
3.15 The principle of contraction mappings. Saddle maps

Chapter 4. INVARIANT TORI
4.1. Non-autonomous systems
4.2. Theorem on the existence of an invariant torus. The annulus principle
4.3. Theorem on persistence of an invariant torus
4.4. Basics of the theory of circle diffeomorphisms. Synchronization problems

Chapter 5. CENTER MANIFOLD. LOCAL CASE
5.1. Reduction to the center manifold
5.2. A boundary-value problem
5.3. Theorem on invariant foliation
5.4. Proof of theorems on center manifolds

Chapter 6. CENTER MANIFOLD. NON-LOCAL CASE
6.1. Center manifold theorem for a homoclinic loop
6.2. The Poincare map near a homoclinic loop
6.3. Proof of the center manifold theorem near a homoclinic loop
6.4. Center manifold theorem for heteroclinic cycles

Appendix A. SPECIAL FORM OF SYSTEMS NEAR A SADDLE EQUILIBRIUM STATE
Appendix B. FIRST ORDER ASYMPTOTIC FOR THE TRAJECTORIES NEAR A SADDLE FIXED POINT

INTRODUCTION TO PARTII
Chapter 7. Structurally Stable Systems
7.1: Rough systems on a plane. Andronov-Pontryagin theorem
7.2: The set of center motions
7.3: General classification of center motions
7.4: Remarks on roughness of high-order dynamical systems
7.5: Morse-Smale systems
7.6: Some properties of Morse-Smale systems

Chapter 8. BIFURCATIONS OF DYNAMICAL SYSTEMS
8.1. Systems of first degree of non-roughness
8.2. Remarks on bifurcations of multi-dimensional systems
8.3. Structurally unstable homoclinic and heteroclinic orbits. Moduli of topological equivalence
8.4. Bifurcations in finite-parameter families of systems. Andronov’s setup.

Chapter 9. THE BEHAVIOR OF DYNAMICAL SYSTEMS ON STABILITY BOUNDARIES OF EQUILIBRIUM STATES
9.1. The reduction theorems. The Lyapunov functions
9.2. The first critical case
9.3. The second critical case

Chapter 10. THE BEHAVIOR OF DYNAMICAL SYSTEMS ON STABILITY BOUNDARIES OF PERIODIC TRAJECTORIES
10.1. The reduction of the Poincar´e map. Lyapunov functions
10.2. The first critical case
10.3. The second critical case
10.4. The third critical case. Weak resonances
10.5. Strong resonances
10.6. Passage through strong resonance on stability boundary
10.7. Additional remarks on resonances

Chapter 11. LOCAL BIFURCATIONS ON THE ROUTE OVER STABILITY BOUNDARIES
11.1. Bifurcation surface and transverse families
11.2. Bifurcation of an equilibrium state with one zero exponent
11.3. Bifurcation of periodic orbits with multiplier +1
11.4. Bifurcation of periodic orbits with multiplier −1
11.5. Andronov–Hopf bifurcation
11.6. Birth of invariant torus
11.7. Bifurcations of resonant periodic orbits accompanying the birth of invariant torus

Chapter 12. GLOBAL BIFURCATIONS AT THE DISAPPEARANCE OF SADDLE-NODE EQUILIBRIUM STATES AND PERIODIC ORBITS
12.1. Bifurcations of a homoclinic loop to a saddle-node equilibrium state
12.2. Creation of an invariant torus
12.3. The formation of a Klein bottle
12.4. The blue sky catastrophe
12.5. On embedding into the flow

Chapter 13. BIFURCATIONS OF HOMOCLINIC LOOPS OF SADDLE EQUILIBRIUM STATES
13.1. Stability of a separatrix loop on the plane
13.2. Bifurcation of a limit cycle from a separatrix loop of a saddle with non-zero saddle value
13.3. Bifurcations of a separatrix loop with zero saddle value
13.4. Birth of periodic orbits from a homoclinic loop (the case dim Wu = 1)
13.5. Behavior of trajectories near a homoclinic loop in the case dim Wu > 1
13.6. Codimension-two bifurcations of homoclinic loops
13.7. Bifurcations of the homoclinic-8 and heteroclinic cycles
13.8. Estimates of the behavior of trajectories near a saddle equilibrium state

Chapter 14. SAFE AND DANGEROUS BOUNDARIES
14.1. Main stability boundaries of equilibrium states and periodic orbits
14.2. Classification of codimension-one boundaries of stability regions
14.3. Dynamically definite and indefinite boundaries of stability regions

APPENDIX C: Examples, Problems & Exercises