49 research outputs found
Intermingled basins in coupled Lorenz systems
We consider a system of two identical linearly coupled Lorenz oscillators,
presenting synchro- nization of chaotic motion for a specified range of the
coupling strength. We verify the existence of global synchronization and
antisynchronization attractors with intermingled basins of attraction, such
that the basin of one attractor is riddled with holes belonging to the basin of
the other attractor and vice versa. We investigated this phenomenon by
verifying the fulfillment of the mathematical requirements for intermingled
basins, and also obtained scaling laws that characterize quantitatively the
riddling of both basins for this system
Viewing the efficiency of chaos control
This paper aims to cast some new light on controlling chaos using the OGY-
and the Zero-Spectral-Radius methods. In deriving those methods we use a
generalized procedure differing from the usual ones. This procedure allows us
to conveniently treat maps to be controlled bringing the orbit to both various
saddles and to sources with both real and complex eigenvalues. We demonstrate
the procedure and the subsequent control on a variety of maps. We evaluate the
control by examining the basins of attraction of the relevant controlled
systems graphically and in some cases analytically
Lagrangian Reachabililty
We introduce LRT, a new Lagrangian-based ReachTube computation algorithm that
conservatively approximates the set of reachable states of a nonlinear
dynamical system. LRT makes use of the Cauchy-Green stretching factor (SF),
which is derived from an over-approximation of the gradient of the solution
flows. The SF measures the discrepancy between two states propagated by the
system solution from two initial states lying in a well-defined region, thereby
allowing LRT to compute a reachtube with a ball-overestimate in a metric where
the computed enclosure is as tight as possible. To evaluate its performance, we
implemented a prototype of LRT in C++/Matlab, and ran it on a set of
well-established benchmarks. Our results show that LRT compares very favorably
with respect to the CAPD and Flow* tools.Comment: Accepted to CAV 201
No elliptic islands for the universal area-preserving map
A renormalization approach has been used in \cite{EKW1} and \cite{EKW2} to
prove the existence of a \textit{universal area-preserving map}, a map with
hyperbolic orbits of all binary periods. The existence of a horseshoe, with
positive Hausdorff dimension, in its domain was demonstrated in \cite{GJ1}. In
this paper the coexistence problem is studied, and a computer-aided proof is
given that no elliptic islands with period less than 20 exist in the domain. It
is also shown that less than 1.5% of the measure of the domain consists of
elliptic islands. This is proven by showing that the measure of initial
conditions that escape to infinity is at least 98.5% of the measure of the
domain, and we conjecture that the escaping set has full measure. This is
highly unexpected, since generically it is believed that for conservative
systems hyperbolicity and ellipticity coexist
The Conley Index and Rigorous Numerics for Attracting Periodic Orbits
Despite the enormous number of papers devoted to the problem of the exis-tence of periodic trajectories of differential equations, the theory is still far from being satisfactory, especially when concrete differential equations are concerned, because the necessary conditions formulated in many theoretica
Dynamics of the Universal Area-Preserving Map Associated with Period Doubling: Hyperbolic Sets
It is known that the famous Feigenbaum-Coullet-Tresser period doubling
universality has a counterpart for area-preserving maps of {\fR}^2. A
renormalization approach has been used in \cite{EKW1} and \cite{EKW2} in a
computer-assisted proof of existence of a "universal" area-preserving map
-- a map with orbits of all binary periods 2^k, k \in \fN. In this paper, we
consider maps in some neighbourhood of and study their dynamics.
We first demonstrate that the map admits a "bi-infinite heteroclinic
tangle": a sequence of periodic points , k \in \fZ, |z_k|
\converge{{k \to \infty}} 0, \quad |z_k| \converge{{k \to -\infty}} \infty,
whose stable and unstable manifolds intersect transversally; and, for any N
\in \fN, a compact invariant set on which is homeomorphic to a
topological Markov chain on the space of all two-sided sequences composed of
symbols. A corollary of these results is the existence of {\it unbounded}
and {\it oscillating} orbits.
We also show that the third iterate for all maps close to admits a
horseshoe. We use distortion tools to provide rigorous bounds on the Hausdorff
dimension of the associated locally maximal invariant hyperbolic set: 0.7673
\ge {\rm dim}_H(\cC_F) \ge \varepsilon \approx 0.00044 e^{-1797}.$
Study of periodic solutions in discretized two-dimensional sliding-mode control systems
The existence of periodic solutions in discretized 2-D equivalent control-based sliding-mode control systems is studied. Admissibility conditions for the existence of periodic solutions with specific symbol sequences are derived, and admissibility regions for short periodic sequences are found. It is shown that, for certain parameter values, there exist arbitrarily long periodic orbits for arbitrarily small discretization steps. Theoretical results are illustrated with simulation examples
The Dangers of Rounding Errors for Simulations and Analysis of Nonlinear Circuits and Systems?and How to Avoid Them
On Euler's discretization of sliding mode control systems with relative degree restriction
Not availabl
Tikhonov regularization and constrained quadratic programming for magnetic coil design problems
In this work, the problem of coil design is studied. It is assumed that the structure of the coil is known (i.e., the positions of simple circular coils are fixed) and the problem is to find current distribution to obtain the required magnetic field in a given region. The unconstrained version of the problem (arbitrary currents are allowed) can be formulated as a Least-SQuares (LSQ) problem. However, the results obtained by solving the LSQ problem are usually useless from the application point of view. Moreover, for higher dimensions the problem is ill-conditioned. To overcome these difficulties, a regularization term is sometimes added to the cost function, in order to make the solution smoother. The regularization technique, however, produces suboptimal solutions. In this work, we propose to solve the problem under study using the constrained Quadratic Programming (QP) method. The methods are compared in terms of the quality of the magnetic field obtained, and the power of the designed coil. Several 1D and 2D examples are considered. It is shown that for the same value of the maximum current the QP method provides solutions with a higher quality magnetic field than the regularization method