One of the greatest challenges in contemporary physics is to understand the various complex systems in nature. Nonlinear interaction between each element is a major cause of the system's complexity, and therefore the theory of nonlinear dynamics plays an important role in the approach to such issues. The main interests in our laboratory are chaotic dynamics in classical Hamiltonian systems and quantum manifestations of classical chaos (quantum chaos). In particular, we are focusing on quantum tunneling in the presence of chaos; complex WKB methods in chaotic systems; slow relaxation in Hamiltonian systems with internal degrees of freedom; chaotic open quantum systems; and formalisms on adiabatic phenomena.

#### Quantum chaos

Quantum chaos is a name of the research subject exploring the quantum manifestation of classical chaos. As is well known, chaos appears only in classical dynamics and does not exist in quantum mechanics in its strict sense. However, chaos in classical mechanics significantly affects the nature of quantum mechanics, and gives rise to various new phenomena absent in the system without chaos. Our group has been focused on purely quantum mechanical phenomena such as quantum tunneling or localization of wave functions. These have wave mechanical origins and have apparently no link to classical mechanics; nevertheless, classical chaos leaves clear fingerprints, even in such purely quantum phenomena. We take the semiclassical approach to understand how classical chaos enriches the corresponding quantum mechanics.

#### Classical Hamiltonian Dynamics

The phase space of generic Hamiltonian systems is composed of regular and chaotic regions, and various types of invariant structures coexist in a highly complicated manner. The orbits in chaotic components are strongly influenced by those in regular regions, and the nature of long-time correlation significantly differs from that expected in the uniformly hyperbolic system. Our motivation to study classical dynamics in Hamiltonian systems is to seek the origin of slow motions in nature from the microscopic dynamical levels. Our investigation targets systems with a few degrees of freedom to many dimensional systems. For the latter cases, there exist situations in which typical scenarios applied to systems with only a few degrees of freedom do not work.

#### Complex WKB analysis

The semiclassical method is a widely technique used to obtain asymptotic solutions of differential equations or to evaluate the integral with a small (or large) parameter. However, semiclassical expansions are asymptotic at best and divergent in general; therefore, ambiguities remain in its practical use. In particular, since controlling the exponentially small terms is usually beyond the treatment of asymptotic expansions, the so-called Stokes phenomenon has been handled only in a heuristic manner. We are applying recently developed mathematics, called the exact WKB method, to several physical problems such as quantum tunneling in the presence of chaos, multilevel nonadiabatic transition problems, and so on.

#### Exotic quantum holonomy

Adiabatic cycles may induce nontrivial changes in quantum systems. A famous example is Berry's phase, which is also called as a phase holonomy. Besides, (quasi-)eigenenergies and eigenspaces of stationary states exhibit nontrivial change, which is referred to as exotic quantum holonomy. We are investigating this phenomenon in various physical systems and seeking its topological structure.