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第92行： − + − − Bifurcation theory has been applied to connect quantum systems to the dynamics of their classical analogues in atomic systems,<ref>{{Cite journal |title=Quantum manifestations of bifurcations of closed orbits in the photoabsorption spectra of atoms in electric fields |first=J. |last=Gao |first2=J. B. |last2=Delos |journal=Phys. Rev. A |volume=56 |issue=1 |pages=356–364 |year=1997 |doi=10.1103/PhysRevA.56.356 |bibcode = 1997PhRvA..56..356G }}</ref><ref>{{Cite journal |title=Quantum Manifestations of Bifurcations of Classical Orbits: An Exactly Solvable Model |first=A. D. |last=Peters |first2=C. |last2=Jaffé |first3=J. B. |last3=Delos |journal=Phys. Rev. Lett. |volume=73 |issue=21 |pages=2825–2828 |year=1994 |pmid=10057205 |doi=10.1103/PhysRevLett.73.2825 |bibcode=1994PhRvL..73.2825P}}</ref><ref>{{Cite journal |title=Closed Orbit Bifurcations in Continuum Stark Spectra | last1 = Courtney | first1 = Michael | last2 = Jiao | first2 = Hong | last3 = Spellmeyer | first3 = Neal | last4 = Kleppner | first4 = Daniel | last5 = Gao | first5 = J. | last6 = Delos | first6 = J. B. |journal=Phys. Rev. Lett. |volume=74 |issue=9 |pages=1538–1541 |year=1995 |pmid=10059054 |doi=10.1103/PhysRevLett.74.1538 |bibcode=1995PhRvL..74.1538C|display-authors=etal}}</ref> molecular systems,<ref>{{Cite journal |title=Bifurcation diagrams of periodic orbits for unbound molecular systems: FH2 |first=M. |last=Founargiotakis |first2=S. C. |last2=Farantos |first3=Ch. |last3=Skokos |first4=G. |last4=Contopoulos |journal=Chemical Physics Letters |volume=277 |issue=5–6 |year=1997 |pages=456–464 |doi=10.1016/S0009-2614(97)00931-7 |bibcode=1997CPL...277..456F}}</ref> and [[resonant tunneling diode]]s.<ref>{{Cite journal |title=Quantum Wells in Tilted Fields:Semiclassical Amplitudes and Phase Coherence Times |first=T. S. |last=Monteiro |lastauthoramp=yes |first2=D. S. |last2=Saraga |journal=Foundations of Physics |volume=31 |issue=2 |year=2001 |pages=355–370 |doi=10.1023/A:1017546721313 }}</ref> Bifurcation theory has also been applied to the study of [[laser dynamics]]<ref>{{Cite journal |title=The dynamical complexity of optically injected semiconductor lasers |first=S. |last=Wieczorek |first2=B. |last2=Krauskopf |first3=T. B. |last3=Simpson |lastauthoramp=yes |first4=D. |last4=Lenstra |journal=Physics Reports |volume=416 |issue=1–2 |year=2005 |pages=1–128 |doi=10.1016/j.physrep.2005.06.003 |bibcode = 2005PhR...416....1W }}</ref> and a number of theoretical examples which are difficult to access experimentally such as the kicked top<ref>{{Cite journal |title=Quantum entanglement dependence on bifurcations and scars in non-autonomous systems. The case of quantum kicked top |first=G. |last=Stamatiou |lastauthoramp=yes |first2=D. P. K. |last2=Ghikas |journal=Physics Letters A |volume=368 |issue=3–4 |year=2007 |pages=206–214 |doi=10.1016/j.physleta.2007.04.003 |arxiv = quant-ph/0702172 |bibcode = 2007PhLA..368..206S }}</ref> and coupled quantum wells.<ref>{{Cite journal |title=Chaos in a Mean Field Model of Coupled Quantum Wells; Bifurcations of Periodic Orbits in a Symmetric Hamiltonian System |first=J. |last=Galan |first2=E. |last2=Freire |journal=Reports on Mathematical Physics |volume=44 |issue=1–2 |year=1999 |pages=87–94 |doi=10.1016/S0034-4877(99)80148-7 |bibcode=1999RpMP...44...87G}}</ref> The dominant reason for the link between quantum systems and bifurcations in the classical equations of motion is that at bifurcations, the signature of classical orbits becomes large, as [[Martin Gutzwiller]] points out in his classic<ref>{{Cite journal |title=Beyond quantum mechanics: Insights from the work of Martin Gutzwiller |first=D. |last=Kleppner |first2=J. B. |last2=Delos |journal=Foundations of Physics |volume=31 |issue=4 |year=2001 |pages=593–612 |doi=10.1023/A:1017512925106 }}</ref> work on [[quantum chaos]].<ref>{{Cite book |first=Martin C. |last=Gutzwiller |title=Chaos in Classical and Quantum Mechanics |year=1990 |publisher=Springer-Verlag |location=New York |isbn=978-0-387-97173-5 }}</ref> Many kinds of bifurcations have been studied with regard to links between classical and quantum dynamics including saddle node bifurcations, Hopf bifurcations, umbilic bifurcations, period doubling bifurcations, reconnection bifurcations, tangent bifurcations, and cusp bifurcations. − − Bifurcation theory has been applied to connect quantum systems to the dynamics of their classical analogues in atomic systems, molecular systems, and resonant tunneling diodes. Bifurcation theory has also been applied to the study of laser dynamics and a number of theoretical examples which are difficult to access experimentally such as the kicked top and coupled quantum wells. The dominant reason for the link between quantum systems and bifurcations in the classical equations of motion is that at bifurcations, the signature of classical orbits becomes large, as Martin Gutzwiller points out in his classic work on quantum chaos. Many kinds of bifurcations have been studied with regard to links between classical and quantum dynamics including saddle node bifurcations, Hopf bifurcations, umbilic bifurcations, period doubling bifurcations, reconnection bifurcations, tangent bifurcations, and cusp bifurcations. − − 分岔理论已经应用于连接量子系统的动力学他们的经典类似物在原子系统，分子系统和共振隧穿二极管。分岔理论也被应用于激光动力学的研究，以及一些理论上难以通过实验获得的例子，如踢陀螺和耦合量子阱。正如 Martin Gutzwiller 在他关于量子混沌的经典著作中指出的那样，量子系统和经典运动方程之间存在联系的主要原因是在分岔时，经典轨道的特征变得很大。研究了经典动力学与量子动力学之间的联系，包括鞍点分岔、 Hopf 分岔、脐点分岔、周期倍增分岔、重联分岔、切线分岔和尖点分岔。 − + +
→Applications in semiclassical and quantum physics
[[Bogdanov-Takens分岔]]是研究余维数为2的分岔的一个很好的例子。
[[Bogdanov-Takens分岔]]是研究余维数为2的分岔的一个很好的例子。
==Applications in semiclassical and quantum physics==
==在半经典与量子物理中的应用==
分岔理论已经应用于将量子系统与原子系统中经典类似的动力学联系起来，<ref>{{Cite journal |title=Quantum manifestations of bifurcations of closed orbits in the photoabsorption spectra of atoms in electric fields |first=J. |last=Gao |first2=J. B. |last2=Delos |journal=Phys. Rev. A |volume=56 |issue=1 |pages=356–364 |year=1997 |doi=10.1103/PhysRevA.56.356 |bibcode = 1997PhRvA..56..356G }}</ref><ref>{{Cite journal |title=Quantum Manifestations of Bifurcations of Classical Orbits: An Exactly Solvable Model |first=A. D. |last=Peters |first2=C. |last2=Jaffé |first3=J. B. |last3=Delos |journal=Phys. Rev. Lett. |volume=73 |issue=21 |pages=2825–2828 |year=1994 |pmid=10057205 |doi=10.1103/PhysRevLett.73.2825 |bibcode=1994PhRvL..73.2825P}}</ref><ref>{{Cite journal |title=Closed Orbit Bifurcations in Continuum Stark Spectra | last1 = Courtney | first1 = Michael | last2 = Jiao | first2 = Hong | last3 = Spellmeyer | first3 = Neal | last4 = Kleppner | first4 = Daniel | last5 = Gao | first5 = J. | last6 = Delos | first6 = J. B. |journal=Phys. Rev. Lett. |volume=74 |issue=9 |pages=1538–1541 |year=1995 |pmid=10059054 |doi=10.1103/PhysRevLett.74.1538 |bibcode=1995PhRvL..74.1538C|display-authors=etal}}</ref>也用在分子系统<ref>{{Cite journal |title=Bifurcation diagrams of periodic orbits for unbound molecular systems: FH2 |first=M. |last=Founargiotakis |first2=S. C. |last2=Farantos |first3=Ch. |last3=Skokos |first4=G. |last4=Contopoulos |journal=Chemical Physics Letters |volume=277 |issue=5–6 |year=1997 |pages=456–464 |doi=10.1016/S0009-2614(97)00931-7 |bibcode=1997CPL...277..456F}}</ref>和[[共振隧穿二极管]]中。<ref>{{Cite journal |title=Quantum Wells in Tilted Fields:Semiclassical Amplitudes and Phase Coherence Times |first=T. S. |last=Monteiro |lastauthoramp=yes |first2=D. S. |last2=Saraga |journal=Foundations of Physics |volume=31 |issue=2 |year=2001 |pages=355–370 |doi=10.1023/A:1017546721313 }}</ref>分岔理论也被应用于[[激光动力学]]
<ref>{{Cite journal |title=The dynamical complexity of optically injected semiconductor lasers |first=S. |last=Wieczorek |first2=B. |last2=Krauskopf |first3=T. B. |last3=Simpson |lastauthoramp=yes |first4=D. |last4=Lenstra |journal=Physics Reports |volume=416 |issue=1–2 |year=2005 |pages=1–128 |doi=10.1016/j.physrep.2005.06.003 |bibcode = 2005PhR...416....1W }}</ref>以及一些理论上难以通过实验获得的例子中，如踢陀螺<ref>{{Cite journal |title=Quantum entanglement dependence on bifurcations and scars in non-autonomous systems. The case of quantum kicked top |first=G. |last=Stamatiou |lastauthoramp=yes |first2=D. P. K. |last2=Ghikas |journal=Physics Letters A |volume=368 |issue=3–4 |year=2007 |pages=206–214 |doi=10.1016/j.physleta.2007.04.003 |arxiv = quant-ph/0702172 |bibcode = 2007PhLA..368..206S }}</ref> 和耦合量子阱。<ref>{{Cite journal |title=Chaos in a Mean Field Model of Coupled Quantum Wells; Bifurcations of Periodic Orbits in a Symmetric Hamiltonian System |first=J. |last=Galan |first2=E. |last2=Freire |journal=Reports on Mathematical Physics |volume=44 |issue=1–2 |year=1999 |pages=87–94 |doi=10.1016/S0034-4877(99)80148-7 |bibcode=1999RpMP...44...87G}}</ref>正如[[Martin Gutzwiller]]在他关于量子混沌的经典著作中指出的那样，量子系统和经典运动方程之间存在联系的主要原因是在分岔时，经典轨道的特征变得很大。<ref>{{Cite journal |title=Beyond quantum mechanics: Insights from the work of Martin Gutzwiller |first=D. |last=Kleppner |first2=J. B. |last2=Delos |journal=Foundations of Physics |volume=31 |issue=4 |year=2001 |pages=593–612 |doi=10.1023/A:1017512925106 }}</ref><ref>{{Cite book |first=Martin C. |last=Gutzwiller |title=Chaos in Classical and Quantum Mechanics |year=1990 |publisher=Springer-Verlag |location=New York |isbn=978-0-387-97173-5 }}</ref>关于经典动力学和量子动力学之间的联系，人们研究了许多分岔，包括鞍结分岔、霍普夫分岔、脐点分岔、周期倍增分岔、重联分岔、切线分岔和尖点分岔。
==See also==
==See also==