Canard Cycles: From Birth to Transition (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, 73)

Preface
	Part I
	Part II
	Part III
	Guide for the Reader
Acknowledgements
Contents
Part I Basic Notions
	1 Basic Definitions and Notions
		1.1 Slow–Fast Families of Vector Fields
		1.2 Examples
		1.3 Examples Where More Than One Admissible Expression Is Needed
		1.4 Normally Hyperbolic Versus Contact Points
	2 Local Invariants and Normal Forms
		2.1 Normal Forms Near Contact Points
		2.2 Invariants at Contact Points
		2.3 Example
		2.4 Remarks About Contact Points
		2.5 Invariants at Normally Hyperbolic Points
	3 The Slow Vector Field
		3.1 Definition
		3.2 Differential 1-Forms Along the Critical Curve
		3.3 Calculating Slow Vector Fields
		3.4 The Slow Vector Field Near Contact Points
		3.5 Slow Singularities
	4 Slow–Fast Cycles
		4.1 Definitions
		4.2 Elementary Slow–Fast Segments
		4.3 Regular Common Cycles
		4.4 Canard Cycles
		4.5 Ordinary Canard Cycles
		4.6 Transitory Cycles
			4.6.1 Singular Points in the Slow Vector Field: Transition to Singular Homoclinic
			4.6.2 Loss of Hyperbolicity on One of the Two Branches in a Layer
			4.6.3 Birth of Canard Cycles
	5 The Slow Divergence Integral
		5.1 Preliminaries
		5.2 Definition and Intrinsic Nature of Slow Divergence Integral
		5.3 Invariance of the Slow Divergence Integral Under Equivalences
		5.4 Slow Divergence Integral Near Singularities of the Slow Vector Field
		5.5 Slow Divergence Integral Near Contact Points
		5.6 Slow Divergence Integral of a Slow–Fast Cycle
		5.7 Examples
			5.7.1 Van der Pol
			5.7.2 Some Canards in Quartic Liénard Systems
			5.7.3 Zeros in the Slow Divergence Integral
	6 Breaking Mechanisms
		6.1 Normal Forms for Generic Jump Points and Generic Turning Points
		6.2 Generic Jump Breaking Mechanism
		6.3 Generic Hopf Breaking Mechanism
		6.4 Formal Power Series Methods for the Generic Hopf Breaking Mechanism
		6.5 Generic Breaking Mechanisms
		6.6 Other Breaking Mechanisms
		6.7 Examples
	7 Overview of Known Results
		7.1 Periodic Orbits Near Common Cycles
			7.1.1 Existence of Periodic Orbits Near Common Cycles
			7.1.2 Multiple Periodic Orbits Near Common Cycles
		7.2 Unicity of Periodic Orbits Near Unbalanced Canard Cycles
		7.3 Existence of Periodic Orbits Near Ordinary Canard Cycles
		7.4 Entry–Exit Relations
		7.5 Multiple Periodic Orbits in Layers
		7.6 Contact Points of Higher Singularity Order Or Contact Order
		7.7 Canard Cycles with Singularities in the Slow Vector Field
		7.8 Multi-Layer Canard Cycles
			7.8.1 Two-Layer Canard Cycles and Their Transitory Boundaries
			7.8.2 More Than Two Layers
		7.9 Birth of Canard Cycles
			7.9.1 Birth of Canards in Liénard Systems
			7.9.2 The Conjecture
			7.9.3 The Infinite Codimension Case
			7.9.4 Birth of Canard Cycles for the Slow–Fast Bogdanov–Takens Singularity
			7.9.5 Birth of Canard Cycles for More Degenerate Contact Points
Part II Technical Tools
	8 Blow-up of Contact Points
		8.1 Blow-up Procedure
		8.2 Blow-up of a Generic Jump Point
		8.3 Blow-up of Regular Contact Points
			8.3.1 The Saddle s-
			8.3.2 The Saddle s+
			8.3.3 The Semi-Hyperbolic Points
		8.4 Blow-up of a Generic Turning Point
			8.4.1 Blow-up of the Turning Point for |a|ε
			8.4.2 Asymptotic Expansions in the Blow-up of the Hopf Point
		8.5 Global Aspects in the Blow-up of Contact Points
			8.5.1 Closed Form Expressions for the Orbits on the Blow-Up Locus of the Generic Jump Point
			8.5.2 Passage Time in the Blow-up of Jump Points
			8.5.3 Separatrices on the Blow-up Locus of the Generic Jump Point
			8.5.4 Regular Splitting of Separatrices in the Blow-up of a Generic Turning Point
			8.5.5 Hopf Breaking Mechanism Revisited
		8.6 From the Hopf Bifurcation to the Polycycle and the Birth of Canards
	9 Center Manifolds
		9.1 Ck-Invariant Manifolds for Diffeomorphisms
		9.2 Ck-Invariant Manifolds for Vector Fields
		9.3 Smooth Invariant Manifolds in Slow–Fast Systems
			9.3.1 The Case of a Closed Critical Curve
			9.3.2 The Case of a Closed Critical Interval
			9.3.3 The Case of a Critical Semi-Hyperbolic Point
			9.3.4 The Case of Singularities of the Slow Vector Field
	10 Normal Forms
		10.1 Preliminaries
			10.1.1 The Path Method
			10.1.2 First Order Differential Equation
				10.1.2.1 Generalities
				10.1.2.2 Affine First Order Equations
				10.1.2.3 Solution for Hyperbolically Attracting Affine Equations
		10.2 Regular Points of the Critical Curve
			10.2.1 The Formal Solution
			10.2.2 The Semi-Formal Solution
			10.2.3 The Final Step
		10.3 Semi-Hyperbolic Points in the Blow-up Locus
			10.3.1 The Formal Solution
				10.3.1.1 Solution Along a Line of Zeros
				10.3.1.2 The Case αβ= 0
			10.3.2 The Semi-Formal Solution
			10.3.3 The Final Step
		10.4 Construction of Center Manifolds
			10.4.1 The Case of a Regular Interval
			10.4.2 The Case of a Semi-Hyperbolic Point
			10.4.3 Intervals Ending at a Semi-Hyperbolic Point
		10.5 Hyperbolic Saddle Points in the Blow-up Locus
			10.5.1 Resonant Monomial Vectors uαvβyγ∂y
			10.5.2 Formal Normal Form
			10.5.3 Reducing to a Differential Equation on Flat Functions
			10.5.4 Solving the Differential Equation on Flat Functions
	11 Smooth Functions on Admissible Monomials and More
		11.1 Admissible Monomials and Functions in Admissible Monomials
		11.2 Derivation
		11.3 Counting the Number of Roots
		11.4 Asymptotically Smooth Functions in Admissible Monomials
		11.5 Functions of Exponentially Flat Type
			11.5.1 Some Basic Properties of the Exponential Term
			11.5.2 Coherence of Definition 11.9
			11.5.3 Composition of Families of Diffeomorphisms of Exponentially Flat Type
Part III Results and Open Problems
	12 Local Transition Maps
		12.1 Transition Along an Arc of Regular Points of the Slow Dynamics
			12.1.1 Transition in a Normal Form Chart
				12.1.1.1 Changing the Exit Section
				12.1.1.2 Restricting to a Starting Section Transverse to the Critical Curve
			12.1.2 General Expressions for Regular Transitions
				12.1.2.1 Transition Between Two Sections Transverse to the Critical Curve
				12.1.2.2 Transition from An Exterior Section
				12.1.2.3 Remark Concerning the Choice of the Coordinate z
			12.1.3 Properties of Transitions Along Regular Arcs
		12.2 Transition Near Semi-Hyperbolic Points
			12.2.1 Equation for the Transition Component Z̃
			12.2.2 A Simple Case
			12.2.3 Preparing the Function G
			12.2.4 Estimates for the Integral I in (12.8)
			12.2.5 Theorems for the Transition Map
			12.2.6 Transitions Near Particular Semi-Hyperbolic Points
				12.2.6.1 Transition Near the Semi-Hyperbolic Point q1 (for λ)
				12.2.6.2 Transition Near the Semi-Hyperbolic Point q2 (for -λ)
				12.2.6.3 Transition Near the Semi-Hyperbolic Point s3 (for X,μ)
		12.3 Transition Near Hyperbolic Saddle Points
			12.3.1 The Transition Map in the Case p=1
			12.3.2 Transition in the General Case (for pN)
			12.3.3 The Saddle Points of Chap.8
		12.4 Transition at a Jump Point
		12.5 Transition Along an Attracting Sequence
		12.6 Transition Along a Hopf Attracting Sequence
	13 Ordinary Canard Cycles
		13.1 Introduction
		13.2 Basic Settings
			13.2.1 Difference Functions
				13.2.1.1 The Case of a Jump Mechanism
				13.2.1.2 The Case of a Hopf Mechanism
				13.2.1.3 A Common Expression for the Two Difference Functions
			13.2.2 Tubular Neighborhood of the Canard Cycle
		13.3 Results of Bifurcation
			13.3.1 A Mild Preparation for Eq.(13.11)
			13.3.2 The Canard Phenomenon
			13.3.3 Formal Power Series Expansion of the Canard Surface for Generic Hopf Breaking Mechanisms
			13.3.4 Canard Explosion, Flying Canard, and Sitting Canards
			13.3.5 Counting the Limit Cycles over a Whole Layer Strip
				13.3.5.1 The Non-orientable Case
				13.3.5.2 The Orientable Case
			13.3.6 Limit Cycles and Bifurcations in a Rescaled Layer
				13.3.6.1 Equivalence of Families of Functions
				13.3.6.2 Asymptotic Expression for the Difference Function ε,()
				13.3.6.3 Bifurcation Results in a Rescaled Layer
			13.3.7 Limit Cycles Outside the Rescaled Layer
				13.3.7.1 Uniform Result for a Simple Zero of the Slow Divergence Integral
	14 Transitory Canard Cycles With Slow–fast Passage Through a Jump Point
		14.1 Statement of the Results
		14.2 Behavior of the Slow Divergence Integral
			14.2.1 The Slow Divergence Integrals J, K, and L
			14.2.2 Slow Divergence Integrals of Slow–fast Cycles
		14.3 Local Study Near the Jump Point q
			14.3.1 Blowing Up the Jump Point
			14.3.2 Transition at the Saddle Point s
			14.3.3 Transition at the Semi-Hyperbolic Point q1
			14.3.4 Transition at the Semi-Hyperbolic Point q2
		14.4 Transition Maps Outside the Jump Point q
		14.5 Cyclicity of the Transitory Canard Cycle 0
			14.5.1 The Displacement Function η
			14.5.2 Structure of the Transition Maps Toward T
			14.5.3 Covering of the Section C
			14.5.4 Unique Maximum Properties
			14.5.5 Proof of Theorem 14.1
		14.6 Limit Cycles and their Unfoldings
			14.6.1 Saddle-Node Bifurcation of Limit Cycles in Case I
			14.6.2 Proof of Theorem 14.5
	15 Transitory Canard Cycles with Fast–fast Passage Through a Jump Point
		15.1 Introduction
		15.2 Blow-up of the Jump Point
		15.3 Transitions Near the Singular Points of λ
			15.3.1 Transition at the Saddle Points s
			15.3.2 Transition at the Semi-Hyperbolic Point q
		15.4 Regular Transitions for λ Along the Blow-up Locus
			15.4.1 Regular Transition Near the Interior of the Blow-up Locus
			15.4.2 Regular Transition Near the Boundary of the Blow-up Locus
		15.5 Cyclicity of the Canard Cycle
			15.5.1 The Displacement Function η
			15.5.2 Normal Form for Transitions Toward T
			15.5.3 From Global to Local Displacement Functions
			15.5.4 Proof of Theorem 15.2 for ηm
			15.5.5 Proof of Theorem 15.3 for ηd
			15.5.6 Proof of Theorem 15.3 for ηu
		15.6 Proof of the Main Theorem
	16 Outlook and Open Problems
		16.1 Introduction
		16.2 Codimension
			16.2.1 Codimension of Contact Points
			16.2.2 Codimension of Jumps Between Contact Points
			16.2.3 Codimension of Singularities of the Slow Vector Field
			16.2.4 Codimension of a Slow–fast Unfolding
			16.2.5 Codimension of a Canard Cycle
		16.3 Desingularization of Unfoldings
			16.3.1 Generic Unfoldings
			16.3.2 Existence of Versal Unfoldings
			16.3.3 Blowing Up of Versal Unfoldings
		16.4 Analytic Slow–fast Unfoldings of Infinite Codimension
		16.5 The Question of Finite Cyclicity for Canard Cycles
		16.6 Disorienting Canard Cycles
		16.7 Recapitulation of Open Problems and Questions
			16.7.1 Questions About Codimension
			16.7.2 Questions About Versal Unfoldings and their Desingularization
			16.7.3 Questions About Asymptotic Properties
			16.7.4 Questions About Analytic Unfoldings and Canard Cycles
			16.7.5 Questions About the Finite Cyclicity Conjecture
			16.7.6 Questions About Disorienting Canard Cycles
References
Index