Deposition of Submicron Particles by Chaotic Mixing in the Pulmonary Acinus

In this review, the authors outline the evidence that emerged some 30 years ago that the mechanisms thought responsible for the deposition of submicron particles in the respiratory region of the lung were inadequate to explain the measured rate of deposition. They then discuss the background and theory of what is believed to be the missing mechanism, namely chaotic mixing. Specifically, they outline how that the recirculating flow in the alveoli has a range of frequencies of oscillation and some of these resonate with the breathing frequency. If the system is perturbed, the resonating frequencies break into chaos, and they discuss a number of practical ways in which the system can be disturbed. The perturbation of fluid particle trajectories results in Hamiltonian chaos, which produces qualitative changes in those trajectories. They end the review with a discussion of the effects of chaotic mixing on the deposition of inhaled particles in the respiratory region of the lung.

Study on the finite element method of Hamiltonian system with chaos

By comparing with symplectic different methods, the quadratic element is an approximately symplectic method which can keep high accuracy approximate of symplectic structure for Hamiltonian chaos, and it is also energy conservative when there have chaos phenomenon. We use the quadratic finite element method to solve the H[Formula: see text]non–Heiles system, and this method was never used before. Combining with Poincar[Formula: see text] section, when we increase the energy of the systems, KAM tori are broken and the motion from regular to chaotic. Without chaos, three kinds of methods to calculate the Poincar[Formula: see text] section point numbers are the same, and the numbers are different with chaos. In long-term calculation, the finite element method can better keep dynamic characteristics of conservative system with chaotic motion.

Hamiltonian Chaos

Nondissipative or Hamiltonian systems are also capable of chaos as phase space volume is twisted and folded in area-preserving maps like the Standard Map. When nonintegrable terms are added to a potential function, Hamiltonian chaos emerges. The Standard Map (also known as the Chirikov map) for a periodically kicked rigid rotator provides a simple model with which to explore the emergence of Hamiltonian chaos as well as the KAM theory of islands of stability. A periodically kicked harmonic oscillator displays extended chaos in the web map. Hamiltonian classical chaos makes a direct connection to quantum chaos, which is illustrated using the chaotic stadium, for which quantum scars are associated with periodic classical orbits in the stadium.

From Butterflies to Hurricanes

Half a century after Poincaré first glimpsed chaos in the three-body problem, the great Russian mathematician Andrey Kolmogorov presented a sketch of a theorem that could prove that orbits are stable. In the hands of Vladimir Arnold and Jürgen Moser, this became the Kolmo–Arnol–Mos (KAM) theory of Hamiltonian chaos. This chapter shows how KAM theory fed into topology in the hands of Stephen Smale and helped launch the new field of chaos theory. Edward Lorenz discovered chaos in numerical models of atmospheric weather and discovered the eponymous strange attractor. Mathematical aspects of chaos were further developed by Mitchell Feigenbaum studying bifurcations in the logistic map that describes population dynamics.