Chimera States for Coupled Oscillators

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Predict R(x) and Θ(x) theoreticallyExpanding G with a trigonometric identity and using Fourier, 12 gives18Re iΘ = e iβ 〈he iΘ 〉 + Ae iβ 〈he iΘ cos(x ′ )〉 cos(x) (14)= c + a cos(x) (15)Where the angular brackets denotes the spatial average:〈f〉 = 12π∫ π−πf(x ′ )dx ′ (16)The sinus term is dropped because R(x ′ ) = R(−x ′ ) , Θ(x ′ ) = Θ(−x ′ )/w
Predict R(x) and Θ(x) theoreticallyThe assumed evenness implies formulas for R(x) and Θ(x) that possess symmetryso R satisfies19To solve for a and c we first rewriteR 2 = (Re iΘ )(Re −iΘ ) (17)= (c + a cos(x))(c ∗ + a ∗ cos(x)) (18)= |c 2 | + 2Re(ca ∗ ) cos(x) + |a| 2 cos 2 (x) (19)he iΘ = (Re iΘ ) h R = ∆ − √ ∆ 2 − R 2 (x)c + a ∗ cos(x)(20)Making use of 15 we obtain/wc = e iβ 〈 ∆ − √ ∆ 2 − |c + a cos(x)| 2c + a ∗ cos(x)a = Ae iβ 〈 ∆ − √ ∆ 2 − |c + a cos(x)| 2c + a ∗ cos(x)〉 (21)cos(x) 〉 (22)
Page 1 and 2: Where innovation startsChimera Stat
Page 3 and 4: References3• Daniel M. Abrams, et
Page 5 and 6: Introduction5ChimeraA mathematical
Page 7 and 8: Simulation7Discretized EquationSimu
Page 9 and 10: Mean field theory9Relative phase θ
Page 11 and 12: Mean field theory11Time independenc
Page 13 and 14: Predict R(x) and Θ(x) theoreticall
Page 15 and 16: Predict R(x) and Θ(x) theoreticall
Page 17: Predict R(x) and Θ(x) theoreticall
Page 21 and 22: Conclusion21Conclusion• The meanf

Chimera States for Coupled Oscillators

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