标量场模型非线性行为研究

So far, there has been convincing observational evidence that the universe is currently in a phase of accelerated expansion. Scalar field which acts as a candidate for dark energy provides the evidence for the accelerated expansion of the universe, and it becomes a hot topic now. It depends on two necessary conditions to reflect accurately nonlinear behavior of the scalar field dark energy model. One is idealized model, the other is reliable method which is used to detect chaos. In view of the two aspects, we plan to study a conformally coupled scalar field model and an anisotropic inflation model. Firstly, we compare manifold corrections with symplectic integrators by numerical experiments, and discuss some chaotic indicators which are widely used in flat spacetime. Then, as for the most sensitive indicator, it will be used in curve spacetime after being corrected. At last, we will study deeply nonlinear behavior of the scalar field model, discuss dynamic and chaos caused by parameter's change. We hope our research will provide more theoretical interpretation for the accelerated expansion.

观测表明，当前宇宙正在加速膨胀。标量场作为暗能量的候选提供了宇宙加速膨胀的依据，已经成为热门研究领域。准确反映标量场暗能量模型的非线性演化过程取决于两个必要条件，一是较为理想的模型，二是可靠的混沌识别方法。基于以上两个条件的考虑，课题组拟以共形耦合标量场模型和各向异性的暴涨模型为基本研究模型，首先通过数值实验对比研究流形改正方案和辛算法，讨论在平直时空广泛应用的混沌识别指标。然后对灵敏度较高的算法改造后应用到弯曲时空中去。最后深入讨论标量场模型非线性行为，讨论参数的变化所产生的混沌动力学特征。为宇宙加速膨胀提供更多理论依据。

标量场作为暗能量的一个候选，提供了宇宙加速膨胀的机制，研究标量场模型的动力学演化可以为宇宙加速膨胀提供更多理论依据。我们主要从以下四个方面开展了相关研究工作：（1）讨论弦在GB黑洞背景下的混沌行为。一般而言，在Kerr-Newman背景中质点运动是可积的，然而在Schwarzschild和anti-de Sitter-Schwarzschild黑洞背景下弦运动却出现了混沌现象。考虑到引力场理论中高阶曲率作用，我们的研究模型是Ads-GB黑洞。数值上给出GB参数的临界值,我们发现在GB临界值附近弦运动具有不同的动力学行为，低于临界值时有序，而高于临界值时混沌。而且混沌行为随着GB参数的增加而愈发剧烈。这个结论与anti-de Sitter-Schwarzschild黑洞背景下弦的运动是弱混沌的明显不同。（2）Massive gravity框架下讨论标量场模型的可积性问题。在Massive gravity框架下加入标量场，然后用Poincare sections、Lyapunov exponment和 Fast Lyapunov indicator讨论该模型的非线性演化问题，从理论角度说明增加的massive gravity部分其实等价于一个有效的宇宙学常数。（3）关于保守系统和耗散系统的混沌识别算法研究。数值模拟表明Lyapunov exponent (LE)、fast Lyapunov indicator (FLI)、relative finite-time Lyapunov indicator (RLI)、smaller alignment index (SALI)和generalized alignment index (GALI)在短期积分内基本上都能反映系统的规则或混沌特征。但是，除了计算全空间的LE指标谱外，其余四个指标均不能反映系统从混沌通向超混沌的转变过程。而且，SALI还不能区分拟周期轨道和规则轨道。（4）关注混沌指标在其它系统中的应用问题。上述研究成果接收并发表在Phys. Rev. D、Celest Mech Dyn Astr、Int J Bifurcat Chaos等期刊上。