256 research outputs found

### Connecting period-doubling cascades to chaos

The appearance of infinitely-many period-doubling cascades is one of the most
prominent features observed in the study of maps depending on a parameter. They
are associated with chaotic behavior, since bifurcation diagrams of a map with
a parameter often reveal a complicated intermingling of period-doubling
cascades and chaos. Period doubling can be studied at three levels of
complexity. The first is an individual period-doubling bifurcation. The second
is an infinite collection of period doublings that are connected together by
periodic orbits in a pattern called a cascade. It was first described by
Myrberg and later in more detail by Feigenbaum. The third involves infinitely
many cascades and a parameter value $\mu_2$ of the map at which there is chaos.
We show that often virtually all (i.e., all but finitely many) ``regular''
periodic orbits at $\mu_2$ are each connected to exactly one cascade by a path
of regular periodic orbits; and virtually all cascades are either paired --
connected to exactly one other cascade, or solitary -- connected to exactly one
regular periodic orbit at $\mu_2$. The solitary cascades are robust to large
perturbations. Hence the investigation of infinitely many cascades is
essentially reduced to studying the regular periodic orbits of $F(\mu_2,
\cdot)$. Examples discussed include the forced-damped pendulum and the
double-well Duffing equation.Comment: 29 pages, 13 figure

### Robust Chaos

It has been proposed to make practical use of chaos in communication, in
enhancing mixing in chemical processes and in spreading the spectrum of
switch-mode power suppies to avoid electromagnetic interference. It is however
known that for most smooth chaotic systems, there is a dense set of periodic
windows for any range of parameter values. Therefore in practical systems
working in chaotic mode, slight inadvertent fluctuation of a parameter may take
the system out of chaos. We say a chaotic attractor is robust if, for its
parameter values there exists a neighborhood in the parameter space with no
periodic attractor and the chaotic attractor is unique in that neighborhood. In
this paper we show that robust chaos can occur in piecewise smooth systems and
obtain the conditions of its occurrence. We illustrate this phenomenon with a
practical example from electrical engineering.Comment: 4 pages, Latex, 4 postscript figures, To appear in Phys. Rev. Let

### Piecewise-linear maps with heterogeneous chaos

Chaotic dynamics can be quite heterogeneous in the sense that in some regions
the dynamics are unstable in more directions than in other regions. When
trajectories wander between these regions, the dynamics is complicated. We say
a chaotic invariant set is heterogeneous when arbitrarily close to each point
of the set there are different periodic points with different numbers of
unstable dimensions. We call such dynamics heterogeneous chaos (or
hetero-chaos), While we believe it is common for physical systems to be
hetero-chaotic, few explicit examples have been proved to be hetero-chaotic.
Here we present two more explicit dynamical systems that are particularly
simple and tractable with computer. It will give more intuition as to how
complex even simple systems can be. Our maps have one dense set of periodic
points whose orbits are 1D unstable and another dense set of periodic points
whose orbits are 2D unstable. Moreover, they are ergodic relative to the
Lebesgue measure.Comment: 16 pages, 9 figure

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