高坂 研究室

Discontinuity

If a function is not continuous at a point in its domain, one says that it has a discontinuity there. The set of all points of discontinuity of a function may be a discrete set, a dense set, or even the entire domain of the function.

Controlling chaos

Small manipulations of a chaotic system can control chaos, e.g., stabilize an unstable periodic orbit, direct chaotic trajectories to desired locations, or achieve other useful goals.

Bifurcation

A bifurcation of a dynamical system is a qualitative change in its dynamics produced by varying parameters.

Synchronization

In a classical context, synchronization means adjustment of rhythms of self-sustained periodic oscillators due to their weak interaction; this adjustment can be described in terms of phase locking and frequency entrainment.

Piecewise smooth dynamical system

A piecewise-smooth dynamical system (or PWS) is a discrete- or continuous-time dynamical system whose phase space is partitioned in different regions, each associated to a different functional form of the system vector field.

Numerical analysis

Numerical analysis is the area of mathematics and computer science that creates, analyzes, and implements algorithms for solving numerically the problems of continuous mathematics.

Crises

Considering a dynamical system with a chaotic attractor, bifurcations of such attractors can occur as a system parameter is varied. These changes occur due to the collision of the chaotic attractor with an unstable invariant set.

Periodic orbit

A periodic orbit corresponds to a special type of solution for a dynamical system, namely one which repeats itself in time. A dynamical system exhibiting a stable periodic orbit is often called an oscillator.

Pulse coupled oscillators

Pulse coupled oscillators are limit cycle oscillators that are coupled in a pulsatile rather than smooth manner.

Chaos

Chaos describes a system that is predictable in principle but unpredictable in practice. In other words, although the system follows deterministic rules, its time evolution appears random.

Stability

The stability of an orbit of a dynamical system characterizes whether nearby (i.e., perturbed) orbits will remain in a neighborhood of that orbit or be repelled away from it.

Normal forms

A normal form of a mathematical object, broadly speaking, is a simplified form of the object obtained by applying a transformation (often a change of coordinates) that is considered to preserve the essential features of the object.