Approximating Limit Cycles of an Autonomous Delay Differential - PowerPoint PPT Presentation

Approximating Limit Cycles of an Autonomous Delay Differential Equation David E. Gilsinn Mathematical and Computational Sciences Division National Institute of Standards and Technology Gaithersburg, MD 20899-8910 dgilsinn@nist.gov Acknowledgements: Dianne O’Leary, Computer Science Dept., Univ. of MD Tim Burns, MCSD, NIST

OUTLINE • Introduction • Main Existence Theorem • Principal Parameter Estimation Algorithms • Application to a Van der Pol Equation with Delay • Final Observations

INTRODUCTION • Measurement and control of machine tool errors has led to delay differential equation models. (Hanna, Tobias [4]) • Machine tool chatter has been established as a Hopf bifurcation of limit cycles from stable machining. (Gilsinn [1], Nayfeh, et al. [5]) • Chatter is self sustained limit cycles caused by the cutting tool interacting with undulations from a previous cut. • Wish to approximate limit cycles with an analytic form and develop a computable error bound.

SOME BASIC FACTS • An autonomous delay differential equation (DDE) with a fixed delay will be written ( ) = − x t X x t x t h � ( ) ( ( ), ( )) 1 φ with initial condition from the space of continuous functions on [- h ,0] . • If satisfies a Lipschitz condition with respect to x , independent X x y ( , ) of y , there exists a unique solution of (1) for on [- h ,0] . φ • A linear DDE with fixed delay takes the form ( ) = + − x t A t x t B t x t h � ( ) ( ) ( ) ( ) ( ) 2 with initial condition on [- h ,0] , a column vector. (Forward Int.) φ ∈ n x R • The formal adjoint equation to (2) takes the form ( ) = − − + + y t y t A t y t h B t h � ( ) ( ) ( ) ( ) ( ) 3 ψ ψ ∈ n + R t t h for on , some initial interval, , a row vector. (Backward Int.) [ , ] 0 0 Z t s • A solution of the linear DDE (2) is called a fundamental solution ( , ) Z s s = I Z t s = t < s if ( , ) , ( , ) 0, .

OBJECTIVE • Find periodic solution of autonomous delay differential equation (DDE) ( ) = − x t X x t x t h � ( ) ( ( ), ( )) 4 φ with initial condition from the space of continuous − h [ ,0] functions on = π ω T • Period is also unknown. 2 / t ω • Introduce for t to get / ( ) ω = − ω x t X x t x t h � ( ) ( ( ), ( )) 5 2 π • Look for periodic solutions of fixed period 2 π • Desired Result: Given an approximate -periodic ω x t solution and frequency, , of (2), wish to show that ( , ( )) if they satisfy a certain noncriticality condition then 2 π there exists an exact frequency and -periodic ω x t solution, , in a computable neighborhood of ( *, *( )) ω x t ( , ( ))

NOTATION = − ω x x t x t h ( ( ), ( ) ω φ = φ + φ φ = φ φ dX x X x X x ( ; ) ( ) ( ) , ( , ) ω ω ω 1 1 2 2 1 2 ω 2 π x • If are an approximate frequency and - periodic solution then ( , ) ω � = + + π = x t X x k t k t k t ( ) ( ) ( ), ( 2 ) ( ) (6) ω • The variational equation with respect to the approximate solution is ω = z t dX x z � ( ) ( ; ) (7) ω ω Let A(t) = , B(t) = X x ω X x ω 1 ( ) 2 ( ) ρ • is a characteristic multiplier of the linear system (2) if there exists a non-trivial solution of (2) such that + π = ρ x t x t ( 2 ) ( ) (8)

NONCRITCAL APPROXIMATE SOLUTION ω = The pair is noncritical with respect to if the x X x ω ω x � ( ) ( , ) • variational equation about the approximate solution (7) has a ρ characteristic multiplier of multiplicity one with the remaining = v t v 2 π ( ), 1 multipliers unequal to one. If is the periodic solution 2 ρ of the adjoint corresponding to then π 2 � � ω ≠ v t J x t dt ( ) ( , )( ) 0 (13) 0 where � ω = � + ω � J x t x hX x x ( , )( ) ( ) (14) t ω t ω 2 , (Hale [3], Stokes [7])

IMPORTANT LEMMAS LEMMA (Halanay [2]): When the linear DDE coefficients A(t), B(t) are periodic the linear and adjoint systems have the same finite number of independent solutions. ρ LEMMA (Hale [3], Halanay [2]): If is a simple characteristic 2 π multiplier of the linear DDE (2), p(t) a nontrivial -periodic solution 2 π of the linear DDE (2), q(t) a nontrivial - periodic solution of the adjoint (3) and ω = + ω − ω J p t p t hB t p t h ( , )( ) ( ) ( ) ( ) (9) then π ( ) 2 � ω ≠ q t J p t dt ( ) ( , )( ) 0 10 0 ω x π f ( , ) LEMMA (Halanay [2]): If is noncritical, -periodic, such that , 2 π 2 � = v t f t dt ( ) ( ) 0 (11) 0 2 π then there exists a unique -periodic solution of ω = + z t dX x z f t � ( ) ( ; ) ( ) (12) ω ω ≤ z M f f M > which satisfies for some , independent of 0 2

FUNDAMENTAL THEOREM ω x ( , ) THEOREM (Stokes [7]): Let satisfy ω � = + x t X x k t ( ) ( ) ( ) (15) ω ψ ψ ≤ , k r K K and let . Suppose there exist and such that for 1 2 1 φ ≤ φ dX x K ( ; ) (16) ω ω 1 dX x + ψ φ − dX x + ψ φ ≤ K ψ − ψ φ ( ; ) ( ; ) ω ω ω ω 1 2 1 2 v ω x ( , ) Assume is noncritical (in the delay sense) and let be the v = appropriate solution of the adjoint to the variational equation, . 1 − Let 1 � � π 2 1 � α = � � ω v t J x t dt � ( ) ( , )( ) (17) ω π � 2 � � � 0 If M is the constant from the previous lemma, let λ = + α � ω M J x (1 ( , ) ) (18) 1 λ � � λ = + MK (1 ) � 1 � 2 1 M � � Finally, if there is a function C of computable parameters such that α λ λ � �� ω < rC K K h x x ( , , , , , , , , ) 1 (19) 1 1 2 x * 2 π then there exists an exact -periodic solution and an ω * exact frequency so that − ≤ λ x x r * 4 1 ω − ω ≤ α * r 2 (20)

PROOF (OUTLINE) β π z t , ( ) 2 Goal: Find -periodic so that ω ω = ω + β = + x t x t z t , ( ) ( ) ( ) (21) ω + β is and exact solution of ω = − ω x t X x t x t h � ( ) ( ( ), ( )) (22) Substituting (21) into (22) ω = + β − β � ω − z t dX x z R z J x t k t � ( ) ( ; ) ( , ) ( , )( ) ( ) (23) ω ω where ω � = + x X x k t ( ) ( ) (24) ω R z β and is a function of computable parameters ( , ) z β and ( , ). = β g R z g g Strategy: Wish to find a fixed point of a map ( ( ), ( )) such that the perturbation term on the right of (23) is orthogonal to the solution of the adjoint in the noncriticality definition. The Lemma can then be used z t To solve for a ( )

PROOF (CONTINUED) { } = ∈ + π = Define the sets P g g C g t g t 0 | , ( 2 ) ( ) ⊂ ∩ P P C 1 1 { } = ∈ ≤ δ N g P g | = S L T S Construct map as a composition ( ) Define: → × L N R P : 1 β > ∈ g N 0 Given solve for unique so that − β � ω − ⊥ ν g J x t k t t ( , )( ) ( ) ( ) ν ω t x ( ) solution of adjoint for noncritical. ( , ) ∈ z P By Lemma there exists a unique such that 1 � ω = + − β ω − z t dX x z g t J x t k t � ( ) ( ; ) ( ) ( , )( ) ( ) ω ω � ≤ − β ω − z M g J x k ( , ) 2 = β L g g z g ( ) ( ( ), ( )) Now define × → T R P N Define: : 1 β > β ∈ × z R P Given 0, ( , ) , 1 β = β T z R z ( , ) ( , ) → = β Define: S N N S g R z g g : , ( ) ( ( ), ( ))

PROOF (CONTINUED) δ = For in the definition of N there exists a bounded function r α λ λ � ω C K K h x ( , , , , , , , ) 1 1 2 rC < → S N N Such that , which implies is a contraction. 1 : = β g R z g g * * * ( ( ), ( )) Therefore there exists a fixed point The exact solution is then given by ω = ω + β g * * ( ) ω = + x t x t z g t * * ( ) ( ) ( )( ) ω + β g * ( ) Finally, we can show that − ≤ λ x x r * 4 1 ω − ω ≤ α r * 2 NOTE: This provides only O(r) estimates. These may not be optimal bounds but they are computable.

APPLICATION STEPS • Compute the approximation pair ω x ( , ) • Verify that the pair is noncritical α • Compute M and A quote from Stokes [7] “The computational difficulties here are considerably greater than in the case of ordinary differential equations,…, but they are not insurmountable” He never produced an example. This talk describes the first application.