By Prof. Dr. H. G. Schuster, Lecturer Wolfram Just(auth.)
A brand new variation of this well-established monograph, this quantity offers a complete evaluation over the nonetheless attention-grabbing box of chaos examine. The authors contain fresh advancements equivalent to platforms with constrained levels of freedom yet positioned additionally a powerful emphasis at the mathematical foundations. in part illustrated in colour, this fourth variation gains new sections from utilized nonlinear technological know-how, like regulate of chaos, synchronisation of nonlinear structures, and turbulence, in addition to contemporary theoretical techniques like unusual nonchaotic attractors, on-off intermittency and spatio-temporal chaotic movement.
Chapter 1 advent (pages 1–5):
Chapter 2 Experiments and straightforward types (pages 7–18):
Chapter three Piecewise Linear Maps and Deterministic Chaos (pages 19–31):
Chapter four common habit of Quadratic Maps (pages 33–67):
Chapter five The Intermittency path to Chaos (pages 69–88):
Chapter 6 unusual Attractors in Dissipative Dynamical platforms (pages 89–125):
Chapter 7 The Transition from Quasiperiodicity to Chaos (pages 127–159):
Chapter eight average and abnormal movement in Conservative structures (pages 161–181):
Chapter nine Chaos in Quantum structures? (pages 183–192):
Chapter 10 Controlling Chaos (pages 193–205):
Chapter eleven Synchronization of Chaotic structures (pages 207–215):
Chapter 12 Spatiotemporal Chaos (pages 217–229):
Chapter thirteen Outlook (pages 231–232):
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Additional info for Deterministic Chaos: An Introduction, Fourth Edition
F n (x0 ). . can be characterized a) by a Liapunov exponent, which tells us how adjacent points become separated under the action of f ; b) by the invariant density, which serves as a measure of how the iterates become distributed over the unit interval; and c) by the correlation function C(m), which measures the correlation between iterates that are m steps apart. For the triangular map, the Liapunov exponent is λ = log 2r, which changes its sign at r = 1/2. It therefore serves as an order parameter for the onset of chaos.
23) makes sense only if ρ(x) is independent of time n, that is, ρ(x) is an eigenfunction of the Frobenius–Perron operator with eigenvalue 1: ρ(x) = Z 1 0 dy δ[x − f (y)] ρ(y) . 2 Characterization of Chaotic Motion 27 Formally, this equation has many solutions (e. , δ(x − x∗ ) where x∗ = f (x∗ ) is an unstable fixed point). But fortunately, only one of the solutions is physically relevant, namely that one which is, for example, obtained by solving eq. 30) on a computer. In the presence of weak random noise (which is caused by rounding errors in the computer or physical fluctuations in real systems), the probability to hit an unstable repelling fixed point x∗ is zero, and therefore such spurious solutions are automatically eliminated (Eckmann and Ruelle, 1985).
1) which is shown in Fig 9. If we start with a value x0 the map generates a sequence of iterates x0 , x1 = σ(x0 ), x2 = σ(x1 ) = σ(σ(x0 )). . In order to investigate the properties of this sequence we write x0 in binary representation: x0 = ∞ ∑ aν 2−ν =ˆ (0, a1 a2 a3 . 2) ν=1 where aν has the values zero or unity. For x0 < 1/2, we have a1 = 0, and x0 > 1/2 implies a1 = 1. Therefore, the first iterate σ(x0 ) can be written as σ(x0 ) = 2x0 2x0 − 1 for for a1 = 0 → = (0, a2 a3 a4 . ) a1 = 0 → Figure 9: The transformation σ(x) = 2x mod l.