From chaos to quasi-periodicity

Кузнецов, А. П.; Migunova, Natalia A.; Сатаев, И. Р.; Sedova, Yuliya V.; Turukina, Ludmila V.

doi:10.1134/s1560354715020070

Cited by 19 publications

(4 citation statements)

References 36 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Remarkably, a practically identical mechanism for the appearance of hyperchaos was recently reported for a Van der Pol oscillator with feedback loops [32]. A similar mechanism was also found for a system of two coupled Rößler oscillators [34].…”

Section: Characteristics Of the Chaotic Regimesupporting

confidence: 76%

Chaotic Antiferromagnetic Nano-Oscillator driven by Spin-Torque

Wolba,

Gomonay,

Kravchuk

2021

Preprint

View full text Add to dashboard Cite

Section: Characteristics Of the Chaotic Regimesupporting

confidence: 76%

Chaotic Antiferromagnetic Nano-Oscillator driven by Spin-Torque

Wolba,

Gomonay,

Kravchuk

2021

Preprint

View full text Add to dashboard Cite

“…Other authors have considered related numerical methods (see section 3.9), in particular [8][9][10], which we will compare to our approach when we introduce our averaging method in section 3. See also [11][12][13][14][15][16][17][18][19][20]. We announced some of the results presented here in [21].…”

Section: Main Convergence Resultsmentioning

confidence: 72%

“…Other authors have considered related numerical methods before, in particular [4,5], which we will compare to when we introduce our method. See also [6,7,8,9,10,11,12,13,14,15].…”

Section: Introductionmentioning

confidence: 99%

Quantitative quasiperiodicity

et al. 2017

View full text Add to dashboard Cite

The Birkhoff Ergodic Theorem concludes that time averages, i.e., Birkhoff averages, B N (f ) ∶= Σ N −1 n=0 f (x n ) N of a function f along a length N ergodic trajectory (x n ) of a function T converge to the space average ∫ f dµ, where µ is the unique invariant probability measure. Convergence of the time average to the space average is slow. We use a modified average of f (x n ) by giving very small weights to the "end" terms when n is near 0 or N − 1. When (x n ) is a trajectory on a quasiperiodic torus and f and T are C ∞ , our Weighted Birkhoff average (denoted WB N (f )) converges "super" fast to ∫ f dµ with respect to the number of iterates N , i.e. with error decaying faster than N −m for every integer m. Our goal is to show that our Weighted Birkhoff average is a powerful computational tool, and this paper illustrates its use for several examples where the quasiperiodic set is one or two dimensional. In particular, we compute rotation numbers and conjugacies (i.e. changes of variables) and their Fourier series, often with 30-digit accuracy. d j=1 a j ρ j = 0, then every a j = 0. We then say such a ρ is irrational.Let T be a C ∞ quasiperiodic map. The quasiperiodicity persists for most small perturbations by the Kolmogorov-Arnold-Moser theory. We believe that quasiperiodicity is one of only three types of invariant sets with a dense trajectory that can occur in typical smooth maps. The other two types are periodic sets and chaotic sets. See [1] for the statement of our formal conjecture of this triumvirate. For example, quasiperiodicity occurs in a system of weakly coupled oscillators, in which there is an invariant smooth *

show abstract

“…These quasiperiodic orbits occur in both Hamiltonian and more general systems [3][4][5][6][7][8][9][10][11][12][13][14]. Luque and Villanueva [3] have published an effective method for computing rotation numbers, see their Figure 11.…”

mentioning

confidence: 99%

Measuring quasiperiodicity

Das¹,

Dock²,

Saiki³

et al. 2016

EPL

View full text Add to dashboard Cite

The Birkhoff Ergodic Theorem asserts under mild conditions that Birkhoff averages (i.e. time averages computed along a trajectory) converge to the space average. For sufficiently smooth systems, our small modification of numerical Birkhoff averages significantly speeds the convergence rate for quasiperiodic trajectories -by a factor of 10 25 for 30-digit precision arithmetic, making it a useful computational tool for autonomous dynamical systems. Many dynamical systems and especially Hamiltonian systems are a complex mix of chaotic and quasiperiodic behaviors, and chaotic trajectories near quasiperiodic points can have long near-quasiperiodic transients. Our method can help determine which initial points are in a quasiperiodic set and which are chaotic. We use our weighted Birkhoff average to study quasiperiodic systems, to distinguishing between chaos and quasiperiodicity, and for computing rotation numbers for selfintersecting curves in the plane. Furthermore we introduce the Embedding Continuation Method which is a significantly simpler, general method for computing rotation numbers.

show abstract

From chaos to quasi-periodicity

Cited by 19 publications

References 36 publications

Chaotic Antiferromagnetic Nano-Oscillator driven by Spin-Torque

Chaotic Antiferromagnetic Nano-Oscillator driven by Spin-Torque

Quantitative quasiperiodicity

Measuring quasiperiodicity

Contact Info

Product

Resources

About