Circadian and Cell Cycle Dynamics: Coupling and Synchronization

This lecture explores the synchronization and coupling of circadian and cell cycles in mammalian cells, focusing on the impact of cell divisions on the circadian period and the absence of gating in 3T3 cells. The instructor discusses the dynamics of two coupled oscillators, the reconstruction of noisy phase dynamics, and the prediction of stable attractors. The lecture also covers the connection between theory and data, including generative models and optimization. Moreover, it delves into the identification of dynamics from data and the influence of cell proliferation on organ-specific time. The presentation concludes by addressing the potential for identifying brain/neuronal states dynamics from data.

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BIO-341: Dynamical systems in biology

Life is non-linear. This course introduces dynamical systems as a technique for modelling simple biological processes. The emphasis is on the qualitative and numerical analysis of non-linear dynamical

Attractor

In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain close even if slightly disturbed. In finite-dimensional systems, the evolving variable may be represented algebraically as an n-dimensional vector. The attractor is a region in n-dimensional space.

Stable manifold

In mathematics, and in particular the study of dynamical systems, the idea of stable and unstable sets or stable and unstable manifolds give a formal mathematical definition to the general notions embodied in the idea of an attractor or repellor. In the case of hyperbolic dynamics, the corresponding notion is that of the hyperbolic set. The gravitational tidal forces acting on the rings of Saturn provide an easy-to-visualize physical example. The tidal forces flatten the ring into the equatorial plane, even as they stretch it out in the radial direction.

Hidden attractor

In the bifurcation theory, a bounded oscillation that is born without loss of stability of stationary set is called a hidden oscillation. In nonlinear control theory, the birth of a hidden oscillation in a time-invariant control system with bounded states means crossing a boundary, in the domain of the parameters, where local stability of the stationary states implies global stability (see, e.g. Kalman's conjecture).

Rössler attractor

The Rössler attractor ˈrɒslər is the attractor for the Rössler system, a system of three non-linear ordinary differential equations originally studied by Otto Rössler in the 1970s. These differential equations define a continuous-time dynamical system that exhibits chaotic dynamics associated with the fractal properties of the attractor. Rössler interpreted it as a formalization of a taffy-pulling machine. Some properties of the Rössler system can be deduced via linear methods such as eigenvectors, but the main features of the system require non-linear methods such as Poincaré maps and bifurcation diagrams.

Limit cycle

In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity or as time approaches negative infinity. Such behavior is exhibited in some nonlinear systems. Limit cycles have been used to model the behavior of many real-world oscillatory systems. The study of limit cycles was initiated by Henri Poincaré (1854–1912).

Limit Cycles and Excitable Systems BIO-341: Dynamical systems in biology

Explores limit cycles, excitable systems, positive feedback, and phase locking in human circadian rhythms.

Shell Components in Meromorphic Parameter Spaces

Explores shell components in transcendental parameter planes and attracting cycles.

Small Scale Stability: Gradient Systems COM-502: Dynamical system theory for engineers

Covers small scale stability in gradient systems, focusing on trajectory properties and equilibrium point attraction.

Nonlinear Dynamics and Chaos PHYS-460: Nonlinear dynamics, chaos and complex systems

Covers bifurcations, long-term dynamics, logistic map, universality, and correspondence between different maps.

Bifurcation and fractals in complex dynamical systems

Explores bifurcation, fractals, Julia sets, Mandelbrot set, and self-similarity in complex dynamical systems.