Chapter 5
Arnol’d Diffusion
Abstract All conservative nonintegrable classical systems, with three or more
degrees of freedom (DoF), can undergo a global transition to chaos due to a
dense web of resonances (the Arnol’d web) that permeates the phase space. Each
resonance line in the Arnol’d web is surrounded by a resonance region and a
stochastic web. The system can diffuse throughout each energy surface, both along
and across the web in a universal diffusion process called Arnol’d diffusion. The
rate of diffusion depends on parameters of the system. Because Arnol’d diffusion
occurs in systems with three or more degrees of freedom, it is difficult to visualize
and difficult to analyze.
The solar system is one of the most studied classical systems that shows the
effects of Arnol’d diffusion. The motion of planets, and other objects in the solar
system, is affected by the Arnol’d web that permeates the solar system phase space.
It also plays an important role in dynamic astronomy (Cincotta, 2002). Another type
of system that shows Arnol’d diffusion, and is amenable to experiment, is a timeperiodically driven optical lattice. A classical periodically modulated optical lattice
with two space dimensions does contain a complete Arnold web and will show a
global transition to chaos. When the dynamics is governed by quantum mechanics,
it will also show the manifestations of Arnol’d diffusion.
Keywords Arnol’d diffusion · Arnol’d web · Nekhoroshev time · Nonlinear
resonance · Driven optical lattice · Solar system stability · Time-periodic optical
lattice · Particle accelerators
5.1 Introduction
All conservative nonintegrable classical systems, with three or more degrees of
freedom (DoF), can undergo a global transition to chaos due to a dense web of
resonances that permeates their phase space Arnol’d [1963, 1964]. Arnol’d showed
that each energy surface of the classical phase space is covered by this web of
resonance lines (the Arnol’d web). The system can diffuse throughout each energy
surface, both along and across the web. This universal diffusion process is called
© Springer Nature Switzerland AG 2021
L. Reichl, The Transition to Chaos, Fundamental Theories of Physics 200,
https://doi.org/10.1007/978-3-030-63534-3_5
133
Arnol’d Diffusion
Abstract All conservative nonintegrable classical systems, with three or more
degrees of freedom (DoF), can undergo a global transition to chaos due to a
dense web of resonances (the Arnol’d web) that permeates the phase space. Each
resonance line in the Arnol’d web is surrounded by a resonance region and a
stochastic web. The system can diffuse throughout each energy surface, both along
and across the web in a universal diffusion process called Arnol’d diffusion. The
rate of diffusion depends on parameters of the system. Because Arnol’d diffusion
occurs in systems with three or more degrees of freedom, it is difficult to visualize
and difficult to analyze.
The solar system is one of the most studied classical systems that shows the
effects of Arnol’d diffusion. The motion of planets, and other objects in the solar
system, is affected by the Arnol’d web that permeates the solar system phase space.
It also plays an important role in dynamic astronomy (Cincotta, 2002). Another type
of system that shows Arnol’d diffusion, and is amenable to experiment, is a timeperiodically driven optical lattice. A classical periodically modulated optical lattice
with two space dimensions does contain a complete Arnold web and will show a
global transition to chaos. When the dynamics is governed by quantum mechanics,
it will also show the manifestations of Arnol’d diffusion.
Keywords Arnol’d diffusion · Arnol’d web · Nekhoroshev time · Nonlinear
resonance · Driven optical lattice · Solar system stability · Time-periodic optical
lattice · Particle accelerators
5.1 Introduction
All conservative nonintegrable classical systems, with three or more degrees of
freedom (DoF), can undergo a global transition to chaos due to a dense web of
resonances that permeates their phase space Arnol’d [1963, 1964]. Arnol’d showed
that each energy surface of the classical phase space is covered by this web of
resonance lines (the Arnol’d web). The system can diffuse throughout each energy
surface, both along and across the web. This universal diffusion process is called
© Springer Nature Switzerland AG 2021
L. Reichl, The Transition to Chaos, Fundamental Theories of Physics 200,
https://doi.org/10.1007/978-3-030-63534-3_5
133