Basins of attraction are rarely computed for systems having more than two state
variables. This is because basins of dimension three and higher are very hard to
interpret, should one have the computational power (and patience) to compute
them. Therefore we are usually forced to circumvent the obstacle by employing
‘typical’ initial conditions, well aware that the system might behave otherwise for
other sets of initial conditions. Or to postulate, rather loosely, that the system is
chaotic for ‘almost all’ initial conditions.
6.3.8 Summary on Detection Tools
For characterizing, visualizing and quantifying chaotic phenomena a rather well
developed set of tools is at our disposal: phase plane plots, frequency spectra,
Poincaré maps, Lyapunov exponents, topological dimension and time-horizon
estimates. A positive largest Lyapunov exponent is the strongest indicator of chaos,
perhaps supported by a fractal Poincaré map and a broadband spectrum.
6.4 Universal Routes to Chaos
Chaos is typically observed only when system parameters are within certain ranges;
Otherwise the behavior is regular. Transitions between regular and chaotic motions
occur through bifurcations. Usually these bifurcations are global, that is, they
cannot be deduced from local information on the flow of orbits near singular points
and limit cycles.
A sequence of bifurcations taking a system from a regular state to a chaotic state
is termed a route to chaos. Certain routes to chaos have been observed so often, and
across so many systems and disciplines, that they seem to be somehow universal in
character. This concerns the period-doubling route, the quasiperiodic route, the
transient route and the intermittency route.
For applications the value of knowing these universal routes comes in when a
single step of one of them is observed. One is then warned (or excited by the
chance) that slight changes in system parameters may push the system into a chaotic
regime. This is especially valuable if there is no or very limited control on system
parameters – as for a traffic bridge, once it has been built – or if parameters cannot
be changed just to satisfy the curiosity of the investigator – as with nuclear reactors
and cardiac patients.
6.4.1 The Period-Doubling Route
This route involves a cascade of bifurcations, each characterized by a doubling of
the period of motion. The process accumulates at a critical value of a system
parameter beyond which the motion turns chaotic.
6.3 Tools for Detecting Chaotic Vibrations
341
variables. This is because basins of dimension three and higher are very hard to
interpret, should one have the computational power (and patience) to compute
them. Therefore we are usually forced to circumvent the obstacle by employing
‘typical’ initial conditions, well aware that the system might behave otherwise for
other sets of initial conditions. Or to postulate, rather loosely, that the system is
chaotic for ‘almost all’ initial conditions.
6.3.8 Summary on Detection Tools
For characterizing, visualizing and quantifying chaotic phenomena a rather well
developed set of tools is at our disposal: phase plane plots, frequency spectra,
Poincaré maps, Lyapunov exponents, topological dimension and time-horizon
estimates. A positive largest Lyapunov exponent is the strongest indicator of chaos,
perhaps supported by a fractal Poincaré map and a broadband spectrum.
6.4 Universal Routes to Chaos
Chaos is typically observed only when system parameters are within certain ranges;
Otherwise the behavior is regular. Transitions between regular and chaotic motions
occur through bifurcations. Usually these bifurcations are global, that is, they
cannot be deduced from local information on the flow of orbits near singular points
and limit cycles.
A sequence of bifurcations taking a system from a regular state to a chaotic state
is termed a route to chaos. Certain routes to chaos have been observed so often, and
across so many systems and disciplines, that they seem to be somehow universal in
character. This concerns the period-doubling route, the quasiperiodic route, the
transient route and the intermittency route.
For applications the value of knowing these universal routes comes in when a
single step of one of them is observed. One is then warned (or excited by the
chance) that slight changes in system parameters may push the system into a chaotic
regime. This is especially valuable if there is no or very limited control on system
parameters – as for a traffic bridge, once it has been built – or if parameters cannot
be changed just to satisfy the curiosity of the investigator – as with nuclear reactors
and cardiac patients.
6.4.1 The Period-Doubling Route
This route involves a cascade of bifurcations, each characterized by a doubling of
the period of motion. The process accumulates at a critical value of a system
parameter beyond which the motion turns chaotic.
6.3 Tools for Detecting Chaotic Vibrations
341