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Introduktion till dynamiska system Period 3, 2012 Introduction to Dynamical Systems

Hemuppgifter till fredagen den 27 januari Exercises for Friday, January 27

H¨anvisningarna ¨ar till Devaney: An Introduction to Chaotic Dynamical Systems

References and exercises are taken from the text book Devaney: An Introduction to Chaotic Dynamical Systems

1. Det dynamiska systemet definierat av funktionen sin x konvergerar mot 0. Bevisa detta!

L˚at allts˚a xn+1= sin xn, n= 0, 1, 2 . . .. Verifiera numeriskt Niclas Carlssons formel:

xn ≈r 3 n f¨or stora n och s˚a gott som oberoende av x0>0.

Efter ungef¨ar hur m˚anga steg ¨ar avst˚andet till origo mindre ¨an 10−5 ? Samma fr˚aga f¨or 12sin x.

The dynamical system defined by the function sin x converges to 0. Prove this fact!

Let xn+1= sin xn, n= 0, 1, 2 . . .. Verify numerically Niclas Carlsson’s formula:

xn≈r 3 n, for large n, practically independently of the starting point x0>0.

About how many iterations are needed in order for xn to be closer than 10−5to the origin? Same questions for 12sin x.

2. (cf. Exerc. 4, p. 43)

L˚at Σ best˚a av alla f¨oljder i Σ2 som uppfyller: om sj = 0 s˚a sj+1 = 1. Med andra ord best˚ar Σ av de f¨oljder i Σ2 som aldrig har tv˚a nollor efter varann.

(a) Visa att σ avbildar Σ p˚a sig sj¨alv och att Σ ¨ar en sluten delm¨angd av Σ2. (b) Visa att de periodiska punkterna f¨or σ ligger t¨att i Σ.

(c) Visa att Σ inte inneh˚aller n˚agon ¨oppen m¨angd.

Let Σ consist of all sequences in Σ2 satisfying: if sj= 0 then sj+1= 1. In other words, Σ consists of only those sequences in Σ2 which never have two consecutive zeros.

(a) Show that σ preserves Σ and that Σ is a closed subset of Σ2. (b) Show that periodic points of σ are dense in Σ.

(c) Show that Σ contains no open set.

Var god v¨and! - Please turn over!

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3.

Komplexa affina avbildningar

L˚at f vara en komplex affin avbildning: f (z) = az + b, z komplext, d¨ar a och b ocks˚a antas vara komplexa och |a| < 1.

Unders¨ok om systemet har fixpunkter och periodiska punkter samt n¨ar dessa ¨ar attraktiva eller repellerande.

Visa speciellt att banan {fn(z0)}asymptotiskt i huvudsak beror av a och b och inte n¨amnv¨art av begynnel- sev¨ardet z0. Beskriv banan f¨or n˚agra speciella v¨arden p˚a a.

Complex Affine Maps

Let f be a complex affine map: f (z) = az + b, z complex, where a and b are assumed to be complex and

|a| < 1.

Does the system have fixed points and periodic points? If so, determine whether they are attracting or repelling. Show in particular that the asymptotics of the forward orbit {fn(z0)} principally depends on a and b and very little on the initial value z0. Describe the orbit for some special values of a.

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References

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