Four-dimensional Silnikov-type dynamics in x ( t ) = x ( t d - PDF document

Four-dimensional ˇ Sil’nikov-type dynamics in x ′ ( t ) = − α · x ( t − d ( x t )) (Joint work with Hans-Otto Walther; in progress) Bernhard Lani-Wayda Southern Ontario Dynamics Day, Toronto 2013

Result of H.-O. Walther: Existence of solution homoclinic to 0 for x ′ ( t ) = − α · x ( t − d ( x t )) , if the delay function d is chosen appropriately. Spectrum at zero: ( d = 1 , α ≈ 5 π/ 2) ρ 2 > | ρ 1 | , 0 > ρ 1 > ρ. ρ 1 + iω 1 Im ✻ ❘ ρ 2 + iω 2 ✴ ✲ Re C C S ρ

Aim of joint work: Show existence of symbolic dynamics for a return map of the above equation. Sil’nikov in R 4 (1967).) (Famous precursor: Result of ˇ We describe the essential framework without reference to an equation: 1) ( X, || || ) Banach space, decomposition X = S × C × C 2) C 0 − semigroup T : R + 0 → L c ( X, X ), T ( t )( x s , z 1 , z 2 ) = ( T S ( t ) x s , e ( ρ 1 + iω 1 ) t z 1 , e ( ρ 2 + iω 2 ) t z 2 ) where || T S ( t ) || ≤ Ke ρt for some K > 0, and ρ < ρ 1 < 0 < ρ 2 , ρ 2 > | ρ 1 | . 3) Consider the sets � � � � || x S || < r 1 /K, | z 1 | = r 1 , 0 < | z 2 | < r 2 S r 1 ,r 2 := ( x S , z 1 , z 2 ) ∈ X , � � � � || x S || < r 1 /K, | z 1 | < r 1 , | z 2 | = r 2 Σ r 1 ,r 2 := ( x S , z 1 , z 2 ) ∈ X . For x = ( x S , z 1 , z 2 ) ∈ S r 1 ,r 2 there exists a unique time τ ( x ) > 0 such that T ( τ ( x )) x ∈ Σ r 1 ,r 2 , namely τ ( x ) := 1 log( r 2 | z 2 | ) . ρ 2

The local map. P 0 : S r 1 ,r 2 → Σ r 1 ,r 2 , P 0 ( x ) := T ( τ ( x )) x. Explicitly: For x = ( x S , z 1 , z 2 ) ∈ S r 1 ,r 2 , z 2 = r 2 e iθ 2 , z 1 = r 1 e iθ 1 , � r 2 � ρ 1 /ρ 2 · e i ( ω 1 τ ( x )+ θ 1 ) ) , r 2 e i ( ω 2 τ ( x )+ θ 2 ) P 0 ( x ) = ( y S , r 1 ) | z 2 | � �� � =: w 2 � �� � =: w 1 where || y S || ≤ || x S || Ke ρτ ( x ) < r 1 e ρτ ( x ) . Note: | w 1 | ∼ | z 2 | − ρ 1 /ρ 2 , 0 < exponent < 1. (Thus, 1 >> | w 1 | >> | z 2 | .) × Σ r 1 ,r 2 S × ✻ P 0 θ 2 θ 1 r 2 ☛ ☛ r 1 × × S r 1 ,r 2 S

2 ∈ [0 , 2 π ) and a C 1 map The global map. Assume there exists θ ∗ 1 , θ ∗ P 1 , with values in S r 1 ,r 2 and defined on the set � � � � max {|| y S || , | w 1 | , | θ 2 − θ ∗ Σ ∗ y = ( y S , w 1 , w 2 = r 2 e iθ 2 ) ∈ Σ r 1 ,r 2 r 1 ,r 2 := 2 |} < δ 2 such that with y ∗ := (0 , 0 , r 2 e iθ ∗ 2 ) ∈ Σ r 1 ,r 2 and x ∗ = ( x ∗ S , r 1 e iθ ∗ 1 , 0) ∈ S r 1 ,r 2 one has P 1 ( y ∗ ) = x ∗ . θ ∗ 2 y ∗ Σ r 1 ,r 2 P 1 ❄ θ ∗ 1 S r 1 ,r 2 x ∗

Domain of P 1 : y 1 ✻ y ∗ × S .............................................. ✲ θ 2 ❘ x 1 ✛ ✲ 2 δ 2

The composition. 2 + δ 2 ] . If ϑ ∗∗ > ϑ ∗ > 0 are large enough and δ 1 ∈ Set I 2 := [ θ ∗ 2 − δ 2 , θ ∗ (0 , π/ 2), the set � � ( x S , z 1 = r 1 e iθ 1 , z 2 = r 2 e iθ 2 ) ∈ S r 1 ,r 2 D ϑ ∗ ,ϑ ∗∗ := � | θ 1 − θ ∗ 1 | < δ 1 , − ϑ ∗∗ < θ 2 ≤ − ϑ ∗ , | z 2 | ∈ r 2 · exp[ − ρ 2 � ( I 2 − θ 2 )] ω 2 satisfies P 0 ( D ϑ ∗ ,ϑ ∗∗ ) ⊂ Σ ∗ r 1 ,r 2 , and hence one can define the composition P := P 1 ◦ P 0 : D ϑ ∗ ,ϑ ∗∗ → S r 1 ,r 2 . A typical domain D ϑ ∗ ,ϑ ∗∗ : ✯ θ 1 y 2 ✻ x ∗ x 2 + iy 2 = z 2 = | z 2 | e iθ 2 × S ✲ x 2

Explicit formulas. x S , r 1 e i ˜ Describe P 1 in the form y = ( y S , x 1 + iy 1 , r 2 e iθ 2 ) �→ (˜ θ 1 , ˜ z 2 ), with C 1 functions ˜ x S , ˜ � y ∗ ˜ ∂ ∂ θ 1 , ˜ z 2 , and partial derivatives � y ∗ ˜ z 2 , θ 1 , etc. � � ∂θ 2 ∂x 1 For x = ( x S , r 1 e iθ 1 , z 2 ) ∈ D ϑ ∗ , set τ := τ ( x ) (as above) , r ′ 1 := r 1 ( r 2 / | z 2 | ) ρ 1 /ρ 2 , x 1 := r ′ 1 cos( ω 1 τ + θ 1 ) , y 1 := r ′ 1 sin( ω 1 τ + θ 1 ) , y S := T ( τ ) x S , || y S || ≤ r 1 e ρτ ∼ | z 2 | | ρ/ρ 2 | � 1 + < ∇ 3 ˜ 0 , r 1 exp { i [ θ ∗ � y ∗ , ( x 1 , y 1 , θ 2 − θ ∗ Then P ( x ) = θ 1 � 2 ) > + E 1 ] } , � � y ∗ , ( x 1 , y 1 , θ 2 − θ ∗ < ∇ 3 ˜ z 2 � 2 ) > + E 2 + E 3 + E 4 , where E 1 , E 2 = o ( r ′ 1 + r 2 ( ω 2 τ + θ 2 − θ ∗ 2 )) , E 3 = O ( || y S || ) , E 4 = (˜ x S , 0 , 0), x S || = O ( r ′ and || ˜ 1 + δ 2 r 2 ) . (Briefly: Taylor expansion of first order w.r.t. 3d-Variables, but only to zero order w.r. to S .)

Set Y 3 := span( ∂ ∂ , ∂ X 3 := span( ∂ , ∂ , ∂ , ) � � y ∗ , ) � � x ∗ , ∂θ 2 ∂x 1 ∂y 1 ∂θ 1 ∂x 2 ∂y 2 then T y ∗ Σ r 1 ,r 2 = S ⊕ Y 3 , T x ∗ S r 1 ,r 2 = S ⊕ X 3 with a corresponding projection pr 3 to X 3 . Transversality conditions: 1) pr 3 ◦ DP 1 ( y ∗ ) is invertible on Y 3 ; 2) ζ 2 := ∂ ˜ z 2 � y ∗ � = 0, or equivalently: DP 1 ( y ∗ ) ∂ � y ∗ �∈ R ∂ � x ∗ . � � � ∂θ 2 ∂θ 1 ∂θ 2 (Geometric meaning: The image of D ϑ ∗ under P is not coaxial with D ϑ ∗ ,ϑ ∗∗ .) Consequences: a) With U 1 := pr 3 DP 1 ( y ∗ ) span( ∂ ∂ ∂x 1 , ∂y 1 ) � � y ∗ , one has X 3 = U 1 ⊕ R · ζ 2 . ∂ b) Let H ⊂ X 3 be a plane containing ζ 2 and such that ∂θ 1 �∈ H ; then pr x 2 ,y 2 is an isomorphism on H . (particularly convenient choice possible).

Choice of N 0 , N 1 . With suitably chosen numbers ϑ 0 , ϑ 00 , ϑ 1 , ϑ 11 and ε 1 > 0, the sets N 0 := D ϑ 0 ,ϑ 00 , N 1 := D ϑ 1 ,ϑ 11 have the properties below: a) (their images lie on different sides of the plane x ∗ + H ). b) For fixed ¯ θ 1 and j ∈ { 1 , 2 } , the map N j ∋ (0 , r 1 e i ¯ θ 1 , z 2 ) �→ pr x 2 ,y 2 pr X 3 P ((0 , r 1 e i ¯ θ 1 , z 2 )) is homeomorphic. (Easier to see for pr H ; then use that pr x 2 ,y 2 is isomorphic on H .)

Main Theorem. ∀ ( ...s − 2 s − 1 s 0 s 1 s 2 ... ) ∈ { 0 , 1 } Z ∃ trajectory ( x j ) j ∈ Z of P with x j ∈ N s j for all j ∈ Z . Proof (ideas): 1) For a finite, periodic symbol sequence α = ( s 0 , s 1 , ..., s k = s 0 ) ∈ { 0 , 1 } k +1 and a map f defined on N 0 ∪ N 1 , define N α,f := N s 0 ∩ f − 1 ( N s 1 ) ∩ ... ∩ f − k ( N s k ) . Lemma (Zgliczy´ nski). If f, g are homotopic maps and the invariant set is disjoint to ∂N 0 ∪ ∂N 1 throughout the homotopy, then ind( f k , N α,f ) = ind( g k , N α,g ) . 2) Three homotopies as in the lemma: a) P ∼ P 3 := pr X 3 ◦ P ; (eliminate S − component from image of P ) b) P 3 ∼ ˜ P 3 ; (eliminate θ 1 − dependence) c) ˜ P 3 ∼ P 2 := pr x 2 ,y 2 ◦ ˜ P 3 (project values to x 2 , y 2 -space). 3) With the Lemma and the reduction property of fixed point index: ind( P k , N α ) = ind( P k 2 , N α ) = ind( P k 2 , N α ∩ ( x 2 , y 2 ) − space) .

4) ( N 0 ∪ N 1 ) ∩ ( x 2 , y 2 ) − space consists of two sets homeomorphic to a ball in R 2 , mapped by P 2 homeomorphically to a larger ball containing both. 5) Lemma. For a map f as in the situation of 4), ind( f k , N α ) = ± 1. 6) Corollary. There is a periodic orbit of P obeying α . 7) The main theorem now follows with a standard compactness argument, using that P is compact and that periodic symbol sequences are dense in the space of all symbol sequences (with the product topology). Thank you for your attention!

References [1] L.P. ˇ Sil’nikov, The existence of a denumerable set of periodic motions in four- dimensional space in an extended neighborhood of a saddle-focus, Dokl. Akad. Nauk. SSR, Tom. 172 , No. 1, 1967. Translation: Soviet Math. Dokl. Vol. 8 , No. 1 (1967). [2] H. Steinlein, Nichtlineare Funktionalanalysis, Lecture at Ludwig-Maximilians- Universit¨ at M¨ unchen, 1986/87. [3] H.-O. Walther, A homoclinic loop generated by variable delay, preprint, submit- ted 2012. [4] S. Wiggins, Global bifurcations and chaos, Springer-Verlag, New York, 1988, pp. 267-275. [5] E. Zeidler, Nonlinear Functional Analysis and its Applications I (Fixed Point Theorems), Springer-Verlag, New York, 1986 (Second Printing, 1992). (In particular, Section 13.7, pp. 574 - 578.) [6] P. Zgliczy´ nski, Fixed point for iterations of maps, topological horseshoe and chaos , Topological Methods in Nonlinear Analysis (Journal of the Juliusz Schauder Center) Vol. 8 (1996), 169-177.