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Chaos in classical string dynamics in $$\Hat{\gamma}$$ deformed $$\mathrm{AdS}_5 \times T^{1,1}$$. (English) Zbl 1368.81131
Summary: We consider a circular string in $$\Hat{\gamma}$$ deformed $$\mathrm{AdS}_5 \times T^{1, 1}$$ which is localized in the center of $$\mathrm{AdS}_5$$ and winds around the two circles of deformed $$T^{1, 1}$$. We observe chaos in the phase space of the circular string implying non-integrability of string dynamics. The chaotic behaviour in phase space is controlled by energy as well as the deforming parameter $$\Hat{\gamma}$$. We further show that the point like object exhibits non-chaotic behaviour. Finally we calculate the Lyapunov exponent for both extended and point like object in support of our first result.

##### MSC:
 81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory 34D08 Characteristic and Lyapunov exponents of ordinary differential equations 70K55 Transition to stochasticity (chaotic behavior) for nonlinear problems in mechanics 34D45 Attractors of solutions to ordinary differential equations 70H08 Nearly integrable Hamiltonian systems, KAM theory 37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion 83E30 String and superstring theories in gravitational theory
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