Chapter 2
Fundamental Concepts
Abstract The dynamical behavior of nonlinear conservative systems is determined by global and hidden symmetries that constrain dynamical flow to lowerdimensional surfaces in the phase space. When the number of global symmetries
equals the number of degrees of freedom, the dynamical system is integrable. This
rarely happens,
Symmetry-breaking terms added to a Hamiltonian cause nonlinear resonances
to occur on all scales in the phase space and give rise to a fractal structuring of
the phase space. Chaos appears in the neighborhood of the nonlinear resonances.
The Poincaré surfaces of section provide a numerical tool for testing integrability of
conservative dynamical systems. Non-linear resonances may appear or disappear as
mparameters of the system are varied and the overlap of nonlinear resonances leads
to the onset of chaos.
Kolmogorov, Arnol’d, and Moser (collectively called KAM) developed a rapidly
converging perturbation theory that describes non-resonant regions of the phase
space. In chaotic regions of the phase space, neighboring orbits move apart
exponentially in any direction. The rate of exponential divergence of pairs of orbits
is given by Lyapounov exponents. Systems with positive Lyapounov exponents also
have positive KS metric entropy.
Keywords Noether’s theorem · Integrability · Global symmetries · Hidden
symmetries · KAM tori · Poincaré surface of section · Nonlinear resonance ·
Definition of chaos · Lyapounov exponents · Baker’s transformation
2.1 Introduction
The dynamical behavior of nonlinear conservative systems is determined by the
nature of the symmetries that govern their behavior. These dynamical symmetries
can be categorized as global symmetries or hidden symmetries. Both types of
symmetry constrain the dynamical flow of the system to lower-dimensional surfaces
in the phase space. Global symmetries are related to the space-time symmetries
of the system. The other symmetries do not have an obvious source and were
© 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_2
11
Fundamental Concepts
Abstract The dynamical behavior of nonlinear conservative systems is determined by global and hidden symmetries that constrain dynamical flow to lowerdimensional surfaces in the phase space. When the number of global symmetries
equals the number of degrees of freedom, the dynamical system is integrable. This
rarely happens,
Symmetry-breaking terms added to a Hamiltonian cause nonlinear resonances
to occur on all scales in the phase space and give rise to a fractal structuring of
the phase space. Chaos appears in the neighborhood of the nonlinear resonances.
The Poincaré surfaces of section provide a numerical tool for testing integrability of
conservative dynamical systems. Non-linear resonances may appear or disappear as
mparameters of the system are varied and the overlap of nonlinear resonances leads
to the onset of chaos.
Kolmogorov, Arnol’d, and Moser (collectively called KAM) developed a rapidly
converging perturbation theory that describes non-resonant regions of the phase
space. In chaotic regions of the phase space, neighboring orbits move apart
exponentially in any direction. The rate of exponential divergence of pairs of orbits
is given by Lyapounov exponents. Systems with positive Lyapounov exponents also
have positive KS metric entropy.
Keywords Noether’s theorem · Integrability · Global symmetries · Hidden
symmetries · KAM tori · Poincaré surface of section · Nonlinear resonance ·
Definition of chaos · Lyapounov exponents · Baker’s transformation
2.1 Introduction
The dynamical behavior of nonlinear conservative systems is determined by the
nature of the symmetries that govern their behavior. These dynamical symmetries
can be categorized as global symmetries or hidden symmetries. Both types of
symmetry constrain the dynamical flow of the system to lower-dimensional surfaces
in the phase space. Global symmetries are related to the space-time symmetries
of the system. The other symmetries do not have an obvious source and were
© 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_2
11