Certi ed Programs and Proofs, 21 Jan 2020

The Poincare-Bendixson Theorem inIsabelle/HOL

Fabian Immler and Yong Kiam Tan

Carnegie Mellon University

Certified Programs and Proofs, 21 Jan 2020

Outline

Background: The Poincare-Bendixson Theorem

Formalization Challenges

The Monotonicity Lemma

Conclusion and Outlook

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Outline





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Recap: Ordinary Differential Equations (ODEs)

Ordinary differential equations (ODEs) provide mathematicalmodels of real world phenomena.

ODE model:x = v , v = −g

ODE solution:

x(t) = x0 + v0t −g

2t2

v(t) = v0 − gt

Properties of the ball’s falling motioncan be deduced from these solutions.

A model of glycolysis:

x = −x + ay + x2y

y = b − ay − x2y

ODE solution: ???

How can we deduce properties withoutknowing the closed-form solution?

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(Planar) ODEs and Dynamical Systems

x

y

0 1 2 x

0

1

2

y


x = −x + ay + x2y

y = b − ay − x2y

Fundamental Idea (Poincare):

ODEs describe dynamical systemswhose qualitative properties can bededuced directly from the equations.

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(Planar) ODEs and Dynamical Systems

0 1 2 x

0

1

2

y


x = −x + ay + x2y

y = b − ay − x2y

Fundamental Idea (Poincare):

ODEs describe dynamical systemswhose qualitative properties can bededuced directly from the equations.

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The Poincare-Bendixson Theorem

0 1 2 x

0

1

2

y

Theorem (Poincare-Bendixson)

(Under mild assumptions) trajectories of planar dynamical systemsare either periodic or tend towards a periodic trajectory.

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The Poincare-Bendixson Theorem

In our paper/formalization:



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Outline





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Formalization Challenges (the “easy” ones)

Our Claim: the Poincare-Bendixson theorem is an interestingbenchmark for formalized mathematics.

1. Has substantial prerequisite formalized mathematics, e.g.:the Jordan curve theorem, (real) analysis, ODEs.

XIsabelle/HOL and the Archive of Formal Proofs (AFP) meetthese prerequisites.

2. Needs formalization of key dynamical systems concepts, e.g.:limit sets of trajectories, periodic orbits.

XMostly involves formalizing of (real) analysis-type argumentsfollowing standard presentations in textbooks.

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Formalization Challenges (this talk)


3. Textbook proofs rely heavily on sketches, especially for a keylemma that is fundamental to the plane.

This Talk: our answers to these formalization challenges, focusingon the key monotonicity lemma used in the proof.

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Chicone:


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Hartman:


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Palis & de Melo:


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Perko:


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Wiggins:


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Teschl:


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? How can we formalize these sketches in a proof assistant?


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? How can we formalize these sketches in a proof assistant?

4. Textbook proofs argue by symmetry and present only one ofthe (several) cases required.

? How can we effectively formalize these symmetries in orderto minimize duplicated proof effort?


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Outline





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The Monotonicity Lemma (Textbook Proof)

Lemma (Monotonicity)

Successive intersections of a trajectory with a transversal segmentare ordered increasingly (or decreasingly) along the segment.

Definition (Transversal Segment)

A transversal segment is a (closed) 2D line segment where theRHS of the ODE is nowhere zero along the segment.

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Proof.

Suppose trajectory from x1 on the transversal touches thetransversal again at x2:

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Proof.

Construct the Jordan curve J formed by the trajectory and thesegment between x1, x2:

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Proof.

By the Jordan curve theorem, J separates the plane into an insideI and outside O:

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Proof.

Any further intersection at x3 must happen inside by construction,so the intersections are ordered x1 ≤ x2 ≤ x3:

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Proof.

Not quite done! There are several other cases, but the argumentfor them is symmetric:

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The Monotonicity Lemma (Formal Proof)

(Recall) Formalization Challenge:


Subtle Claim: these are the only possibilities that can occur for J.

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Three pieces of information are needed for the (Left) case:

Symmetrically for (Right) case, e.g., flow always crosses frominside to outside between x1 to x2.

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Obvious Problem: Naive case splitting yields 8 cases.

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Obvious Problem: Naive case splitting yields 8 cases.

Not-so-obvious Problem: It is unclear how to finish the prooffrom the impossible cases.

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Key Idea: Construct flow regions r1, r2 that must lie on oppositesides and case split to deduce the three pieces of information.

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Key Idea: reverse flows to obtain other cases by symmetry.

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Formalized using Isabelle/HOL’s (sub-)locales:

ODE f : Rn → Rn

Flow: φ, Thm: P(φ(x0, t)) (≈760 line proof for monotonicity)

rev.ODE −f : Rn → Rn

Flow: φ−f , rev.Thm: P(φ−f (x0, t)) (automatically generated)

⇓export and rewrite φ−f (x0, t) = φ(x0,−t) (≈30 line boilerplate)⇓Flow: φ, Thm: P(φ(x0,−t)) (symmetric lemma proved!)

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ODE f : Rn → Rn




⇓export and rewrite φ−f (x0, t) = φ(x0,−t) (≈30 line boilerplate)⇓

Flow: φ, Thm: P(φ(x0,−t)) (symmetric lemma proved!)

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ODE f : Rn → Rn




⇓export and rewrite φ−f (x0, t) = φ(x0,−t) (≈30 line boilerplate)⇓Flow: φ, Thm: P(φ(x0,−t)) (symmetric lemma proved!)

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Outline





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XWe give the first (as far as we know*) fully rigorousargument for this step, that is amenable to formalization.


XUse Isabelle/HOL’s locale system to formally reverse flows.

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* While preparing these slides, we came across a proof in Croninbased on indexes but we have not attempted to formalize it.

Cronin (on the Poincare-Bendixson theorem):

. . . part of the difficulty in proving thetheorem lies in the fact that all theconsiderations take place in the Euclideanplane and distinguishing between intuitiveand rigorous arguments in the plane issometimes difficult.

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? But our proof is rather different from the textbook sketches.Is this unavoidable? Are there cleaner or more abstract proofs?


XUse Isabelle/HOL’s locale system to formally reverse flows.

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Conclusion



0 1 2 x

0

1

2

y

Future Directions:

1. Formalize other tools for the analysis of planar dynamicalsystems, e.g., Lienard’s theorem, Dulac’s criterion.

2. Generalize beyond planar ODEs, e.g., Poincare-Bendixsontheorem for ODEs defined on the sphere or 2-manifolds.

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The (Left) case of our flow region construction, is sketched here:

Key Idea: Construct flow regions r1, r2 that must lie on oppositesides and case split to deduce the three pieces of information.

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Glycolysis Example Proof

0 1 2 x

0

1

2

y

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Certi ed Programs and Proofs, 21 Jan 2020

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