LECTURE 8: Topics in Chaos Ricker Equation. Period doubling bifurcation. Period doubling cascade. A Quadratic Equation Ricker Equation 1.0. x x 4 0.

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1 LECTURE 8: Topcs Chaos Rcker Equato (t ) = (t ) ep( (t )) Perod doulg urcato Perod doulg cascade A Quadratc Equato Rcker Equato (t ) = (t ) ( (t ) ). (t ) = (t ) ep( (t ))

2 The perod doulg cascade perod Perod doulg cascade or * * * < <. What are the dyamcs or. < <? For >? We ll look rst at Case : = The Case : > ad ally (& oly rely) Case :. < < ( ) = ( ) oto :,, Case : = t ( ) = t ( ) t ( ) ( ) () < or > ort teds to DEFINITION : Let X e a metrc space. A map : X X s (topologcally) trastve or all pars o ope sets UV X U V ( k ), there s a k Z such that ( ).. A cowe aalyss shows orts ragg over the whole terval

3 U ( U) X te & trastve => X = ort o a perodc ort Assume X s te () ( U) () ( U) p X s a perodc pot ts ort s a perodc cycle () ( U) DEFINITION ( k ) ( U) V A map : X X(a metrc space) s chaotc () s trastve () the set o perod pots s dese X THEOREM : X X = metrc space () I there ests a dese ort, the s trastve. cots : I I = terval R (ot ecessarly te) m () I X R s closed ad ouded ad s cotuous, THEOREM (Vellekoop & Berglud, 99) the s trastve there ests a dese ort X. s trastve the set o perodc pots s dese I. Proo o (): Let a X have a dese ort UV, X e artrary ope sets ( m) ( ) tegers m< such that ( a) U, ( a) V U ( m ) ( a ) ( ) ( a ) m steps V ( m ) ( U) V COROLLARY : s trastve o I chaotc o I Theorem => COROLLARY : I = closed & ouded. The s chaotc o I has a dese ort Not ecessarly true o o-tervals, hgher dmesoal Eucldea spaces, or other metrc spaces

4 EXAMPLE Irratoal rotato o the crcle θ ( θ πα ( ) e ), e = α =rratoal ( t) Show { e θ πα } s dese Theorem => trastve ( t) However, e θ πα s ot a perodc,.e., there are o perodc orts ( ) e θ πα e θ DEFINITION: : X tal codtos (SIC) : X (a metrc space) has sestvty to δ > such that gve X ad ay ope U X, U ( m) ( m) ( ( ), ( )) y U or whch d y > δ or some teger m U y δ ( m ) ( ) ( m ) ( y) THEOREM (Baks et al., 99) cots I : X X = metrc space s chaotc, the has SIC I geeral, coverse s alse EXAMPLE: The Tet Map T : I =, I o,. T( ) = ( ) o., Eample: lear repellor t t () = a(), a> t t () yt () = a () y() ustale equlrum.

5 Pck a teger. The sutervals Clam : k k,, k =,, partto the ut terval T ( ) k k [,] oto :,, Proo y ducto o Case = s ovous rom graph o T() Iducto hypothess : Wat to show T ( m) k k m m ( m ) T m m oto :,, k k oto :,, k k T k k ( m), m m, m m T oto [,] Cosder sutervals rom the let hal sde o : [,] deto (y ducto hypothess) Smlar argumets or sutervals rom the rght hal sde o the ut terval Gve ope sets UV,,. For some teger m U cotas a suterval ( m ) oto T : U, k k,, m m ( ) T ( m ) T U V s trastve Thereore, the tet map s chaotc (Corollary ) has a dese ort (Theorem ) has a dese set o perodc orts (Theorem ) possesses SIC (Theorem ) EXAMPLE: The Sht Map { { } : = } Σ = = = or y d(, y) = s a metrc o Σ = Note : = y or =,, d(, y) Note : d(, y) < = y or =,,

6 Sht map σ : Σ Σ σ { } { } () σ s cotuous Gve ε > choose such that < ε Let δ = d(, y) < δ = y or =,, (Note ) σ ( ), σ ( y) agree or =,, d( σ( ), σ( y)) < ε (Note ) () Perodc pots are dese Perodc pots are sequeces wth repeatg locks: { p p p p p p } = { p p} { } Gve ay = dee y y { } { } = = { } y = Note d(, y) lm y = () There ests a dese ort We costruct the tal pot a dese ort as ollows Lst all locks o legth :, z z = { } Add all locks o legth :,,,,, z = { } Cotue y addg all locks o legth z = { } Add all locks o legth :,,, z = { } 6

7 To prove the ort startg at z s dese, suppose we are gve ay = { } ad ay ε >. Pck so that < ε The lock appears z. I t appears at posto m the { } m σ ( ) ( z) = Note d( σ ( z), ) < ε ( m) Theorem mples σ s trastve. IN SUMMARY We ve show the sht map s trastve ad has a dese set o perodc pots By deto, the sht map s chaotc Theorem mples the sht map has SIC EXAMPLE: Aother Sht Map {(. ) : } = = or d(, y) y = s a metrc o = Sht map σ : Σ σ Σ (. ) (. ) Smlar argumets show σ s chaotc XY=, metrc spaces DEFINITION: h: X Y s a homeomorphsm t s oe-oe, oto, ad -cotuous. : X X, g: Y Y Let g h: X Y deote the composte o h ad g. DEFINITION: s (topologcally) cojugate to g there ests a homeomorphsm h: X Y h = g h such that