南京师大学报自然科学版第 35 卷第 3 期 0 年因有两个第一这是一个经典简并系统 KAM 定理的条件不满足文献上非 KAM 系统研究的很少第二由于该系统数值处理上较困难本征值本征态能谱统计等方面的研究结果还没有看到过报导系统的经典动力学一维周期受击简谐振子系统的哈密顿量

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1 JOURNA OF NANJING NORMA UNIVERITYNatural cience Edition Vol 35 No 3 ept κ Poisson κ = 30 κ = < 0 > 0 κ = 0 3 Σ γ γ κ 043 A Classical Dynamics and Quasi-Energy pectral tatistics of a Periodically Kicked Harmonic Oscillator Yang huangbo Wei Dong chool of Physics and TechnologyNanjing Normal UniversityNanjing 0046China AbstractThis paper studies the classical dynamics and quasienergy spectral statistics for a periodically kicked Harmonic oscillator system under the nonresonance condition It is found that as we increase the kicking strength κand the phase space structure starts from tori for integrable system to completely chaotic for nonintegrable system the nearest neighbor spacing distribution for the quasienergy spectral keeps the Poissonian distributionand this is similar to that of the periodically kicked free rotor The result of spectral rigidities shows that except the case of κ = 30the rigidities for κ = bunched increase linearly with for < 0 and spread increase nonlinearly with and the rigidity for κ = 0 3 tends to saturation for > 0 The number variance Σ skewness γ excess γ are not sensitive to the change of κ Key wordschaosquasienergynearest neighbor spacing distributionspectral rigidityhigher moment 3 3 t -9 Casati D Wintgen H Marxer Wigner GOE D Kilbane 3 k Poisson Wigner RMT Poisson Wigner GOE Eric Heller 4 scar yangshuangbo@ njnu edu cn 37

2 南京师大学报自然科学版第 35 卷第 3 期 0 年因有两个第一这是一个经典简并系统 KAM 定理的条件不满足文献上非 KAM 系统研究的很少第二由于该系统数值处理上较困难本征值本征态能谱统计等方面的研究结果还没有看到过报导系统的经典动力学一维周期受击简谐振子系统的哈密顿量被写为 p H = + μωc x + κcosx δ t nt = H0 + κcosx δ t nt μ n = n = 它的正则运动方程为 p H H = p = = μωc x + κsinx δ t nt p μ x n = 对运动方程在一个周期 nt 0 n + T 0 内积分我们得到如下的映射方程 x = ( x x n + = ( x n + = ) κ sinx n cosω c T ω c x n sinω c T μ sinω c T κ + x n cosω c T x n + sinx n ωc μ + n 3 ) 4 ωc πq q 为简单有理数时为谐振 ω 打击此时系统称为谐振系统它的经典动力学已在文献 5 中讨论过了在这篇文章中我们取 q 为无理这里 κ 是打击强度令 ω 为打击频率 T 为周期则当 ω c T = π 数即非谐振系统仍然采用自然单位即取 μ = ω c = = H0 = 0 KAM 定理的条件不被满足相空间不存在 KAM 不变环非谐 I 振下相空间结构随打击强度 κ 变化当 κ 很小时同心圆或者环充满相空间随着 κ 增大越来越多的环从简谐振子系统是经典简并系统外面高能量处逐渐被破坏直到圆心能量最小处其破坏机制并非清楚图 a f 显示不同打击强 5 可以看出当 κ = 0 3 时相空间充满了环度下一个非谐振系统的相空间结构其频率比为 q = 槡 b κ=08 d κ=0 c κ=5 e κ=6 图 Fig 38 f κ=300 a κ=03 5 不同打击强度 κ 下相空间的结构 q = 槡 5 Phase space structure at different kicking strength κ q = 槡

3 κ κ = 0 8 κ κ = 5 8 κ = 6 Floquet U^ x T= e - i T H^ dt 0 = e - i H^ 0 T e - i κcosx 5 H^ 0 = ^p μ + μω c x ψ 0 Floquet U^ x T ψ T ψ T = U^ x Tψ 0 6 U^ x T U^ ψ f = λ f ψ f 7 λ f U^ x T n U^ x T 5 Floquet λ f λ f = e iε r +iε i = e -ε i e iε r ε r 3 U^ κ U^ λ f 3 E i ε i = fe i f ε i i = n f ε i = NE i NE i E i ε i ε i = E i dn /de E i ΔN /ΔE ΔN E i ΔE E i - ΔE / E i + ΔE / ΔE E i ΔE = 0 0 x n x n+ s s + ds psdsps 0 psds = ps N s ds s s + ds ΔNps= ΔN /NΔs Poisson Ps= exp- s Wigner GOE Ps = 39 ε r

4 π ( ) sexp - πs [ 4 ] RMT Poisson a - f κ = Poisson 6 Poisson Wigner GOE (a) κ=03 (b) κ=07 (c) κ=6 (d) κ=0 (e) κ=6 (f) κ=30 Fig a - f κ q = 槡 5 - a - fthe nearest neighbor spacing distribution for the nonresonant system at different kicking strength κ q = 槡 5-3 E i E E + Δ 3 E = min A B E + E NE'- AE' - B de' 8 NE' i A B AE' + B E < E' < E + Δ 3 E E Δ珔 3 Ps Δ珔 3 Poisson E = j Δ珔 3 = / Wigner GOE Δ珔 3 Poisson E i < E + < E i + 40 Δ 3 E = min A B E + E NE'- AE' - B de' =

5 min NE'- AE' - B E + [ de' + B NE'- AE' - B de' ] = A Ei E ji j = j E Ej+ j - ae' - b de' + E j E + E i j i - ae' - b de' 9 E i ab E Δ珔 3 = Δ 3 E E 3a - f κ κ = 0 3 Δ珔 3 Poisson γ = 7 8 a a 3a κ = 6 c Poisson c 3b c 3b κ = 0 8 d Poisson d 3c κ = 6 κ = 30 e f Poisson e f 3d e κ 3f κ = 30 Poisson κ = 30 3e f Fig 3 (a) κ=03 (d) κ=6 3 (b) κ=6 (e) κ=30 (c) κ=0 (f) κ=30 κ=0 κ=6 κ=6 κ=03 a - f κ q = 槡 5 - a - fpectral rigidities for nonresonant systems with different kicking strengthand its comparisonq = 槡 E NE E E 珚 N= NE E Σ = NE - 珚 N = NE - 珚 N 0 4

6 E 珚 N NE Δ珔 3 Σ 3 4 skewnessγ excessγ γ = M 3 M 3 γ = M 4-3 M M k NE 珚 N k M k M 3 = NE - 珚 N k = NE - 珚 N 3 = 3 NE 3-3 NE 珚 N+ 珚 N 3 4 M 4 = NE - 珚 N 4 = NE 4-4 NE 3 珚 N+ 6 NE 珚 N - 3 珚 N 4 5 珚 N NE NE 3 NE N(E,) 4 κ=6 珚 N= NE κ = 6q = 槡 5 - Fig 4 The average value 珚 N= NE versus for κ = 6q = 槡 5-4 κ = 6 珚 N 4 珚 N κ = Σ γ γ 珚 N 0 40 (a) κ=03 (c) κ=6 (e) κ=6 撞撞撞酌, 酌 (b) κ=03 酌酌酌, 酌 (d) κ=6 酌酌酌, 酌酌酌 (f) κ=6 Fig 5 5 κ Σ γ γ q = 槡 5 - Number variance Σ skewness γ and excess γ for nonresonance system with different kicking strength κ q = 槡 5 - ( 下转第 47 页 ) 4

7 Bruce C EGolde R H The lightning discharge J J Inst Elect Pt Rakov V A Uman M A Review and evaluation of lightning return stroke models including some aspects of their application J IEEE Trans Electromag Compat Baba YRakov V A On the transmission lines model for lightning return stroke representantion J Geophydical Recearch etters doi0 09 /003G Marcos Rubinstein An approximate formula for the calculation of the horizontal electric field from lightning at close intermediateand long range J IEEE Trans Electromag Compat Moini RKordi BRaif G Zet al A new lightning return stroke model based on antenna theory J J Geophys Res D ( 上接第 4 页 ) κ < Casati GChirikov B VFord Jet al tochastic behavior of a quantum pendulum under periodic perturbation J ect Notes Phys Wintgen DMarxer H evel statistics of a quantized cantori system J Phys Rev ett Kilbane DCummings AO ullivan Get al Quantum statistics of a kicked particle in an infinite potential well J Chaosolitons and Fractals Heller E JO Connor P WGehlen J The eigenfunctions of classical chaostic systems J Physica cripta J Izrailev F M imple models of quantum chaosspectrum and eigenfunctions J Phys Rep Casti GChirikov B VGuarneri I Energy-level statistics of integrable quantum systems J Phys Rev ett Honig AWintgen D pectral properties of strongly perturbed Coulomb systemsfluctuation properties J Phys Rev A J

35 3 0 z TM Maxwell r - E r z - H z + E z r = ε E r t = μ H t rh = ε E z r t + σe z 3 + σe z 4 Fig ΔrΔz Δt Maxwell H n+0 5 i + 0 5 j + 0 5= H n-0 5 +

35 3 0 z TM Maxwell r - E r z - H z + E z r = ε E r t = μ H t rh = ε E z r t + σe z 3 + σe z 4 Fig ΔrΔz Δt Maxwell H n+0 5 i + 0 5 j + 0 5= H n-0 5 + 35 3 0 9 JOURNAL OF NANJING NORMAL UNIVERSITYNatural Science Edition Vol 35 No 3 Sept 0 0003 FDTD TL LEMP TM863 A 00-466003-0043-05 Comparison of Two Lightning Return Stroke Models Yin Jie College of ScienceNanjing