The Rössler system is a classical low-dimensional dynamical system generating different types of attractors. The question whether the Rössler attractor observed for classical parameter values is periodic or chaotic remains an open problem. In this work, we study the problem whether periodic windows are dense in the parameter region close to the classical case. Symbolic representation of trajectories is defined and the ordering of symbol sequences is constructed with a property that for periodic symbol sequences their order agrees with the layout of corresponding periodic windows. Using symbolic descriptions of periodic window, the continuation technique and the bisection method we find sinks existing for parameter values at a distance smaller than 10−15from the classical case. Convergence properties of periodic attractors are studied numerically.

Installing protective devices enhances reliability of power distribution systems by means of failure separation. Finding optimal positions of protective devices to minimize a given objective function can be achieved using various single-objective optimization methods. Solutions obtained for different objective functions may differ significantly. Multiobjective optimization algorithms may be used to solve the optimization problem taking into account more than one objective function. In this work, a new branch and cut algorithm is proposed to solve the multiobjective optimization problem of protective device allocation in radial distribution systems with a single feeder. It is shown that the proposed algorithm can successfully handle very large power distribution systems. The performance of the algorithm is compared with the performance of three other approaches: 1) the exhaustive search; 2) an evolutionary algorithm; and 3) a reinforcement learning algorithm. It is shown that the proposed algorithm outperforms other methods both in terms of the computation time and the quality of the results.

Nonlinear maps are usually implemented using finite-precision floating-point formats both in practical applications and in theoretical investigations. In the digital domain, the size of the state space is finite and every trajectory after a finite number of iterations reaches a cycle. It is therefore important to study the influence of rounding errors and the finiteness of the state space on properties of nonlinear maps such as the number of cycles, their periods, sizes of basins of attraction of cycles, and average convergence times. In this work, a thorough analysis of the dynamics of finite-precision implementations of the Hénon map is carried out. Six computational formulas and three popular finite-precision floating-point formats are considered. An exhaustive search is performed to find all cycles existing for single-precision floating-point implementations. Interval methods are used to reduce the number of initial conditions that must be considered. An efficient graph-based algorithm is designed to find basins of attraction. For the double-precision and extended-precision implementations, statistical methods are utilized to find cycles and to estimate sizes of their basins of attraction. Properties of observed cycles and corresponding dynamical phenomena are thoroughly discussed.

In this work, a rigorous numerical study of the Chua’s circuit double-hook attractor is carried out. The existence of a trapping region enclosing the double-hook attractor is proved. It is also proved that the double-hook attractor is chaotic in the topological sense.

The existence of periodic windows in the parameter space is a common feature of nonlinear systems capable to produce chaotic behavior. Detection of positions of periodic windows is important both from the theoretical and practical points of view. In this work, a systematic method to detect periodic windows in bifurcation diagrams is proposed. The search method is a combination of the trajectory monitoring approach to find unstable periodic orbits and the continuation method to calculate positions of periodic windows. The method is applied to the Chua's circuit with a smooth nonlinearity. It is shown that the proposed method outperforms standard methods to find periodic windows in bifurcation diagrams.

Chua's circuit is an example of a simple electronic circuit displaying a variety of dynamical phenomena. In this work, we study the existence of periodic windows in the Chua's circuit with a cubic nonlinearity. It is shown that the standard method to construct bifurcation diagrams is not capable to reveal its complete structure. A continuation based method is proposed to systematically search for periodic windows. Computational examples are presented to show the usefulness of the proposed approach.

In simulations of the Chua's circuit with a smooth nonlinearity for certain parameter values one observes the double scroll attractor. This attractor contains an unstable equilibrium, and typical trajectories belonging to the attractor may pass arbitrarily close to this equilibrium. In consequence, it is impossible to compute trajectories over the whole attractor using standard rigorous numerical integration procedures. This is due to the existence of trajectories which spend arbitrarily long time in a neighborhood of the equilibrium. In this work, a method to find enclosures of trajectories passing arbitrarily close to an unstable fixed point of spiral type is presented. This method is used to prove the existence of a trapping region enclosing the double scroll attractor for the Chua's circuit with a cubic nonlinearity.

We study the problem of optimum placement of sectionalizing switches in radial distribution networks to minimize the average undelivered energy due to power outages. We show that existing algorithms do not guarantee finding the optimum solution. We present a fast tree structure based method for the computation of the average undelivered energy and an efficient algorithm to find the optimum allocation of switches to minimize the undelivered energy. The performance of algorithms is tested using an example radial distribution network.

The problem of existence of chaotic dynamics in the Chua's circuit with a smooth nonlinearity is studied rigorously by means of interval arithmetic methods. For different parameter values lower bounds of the topological entropy of a corresponding return map are found and it is proved that the system is chaotic in the topological sense. The trapping region enclosing the spiral attractor is constructed. It is discussed how to prove the existence of a trapping region for the case of the double-scroll attractor.

We describe a model of power networks which may be useful for studying power blackouts. The model is a combination of the admittance model of the network and the probabilistic model of faults of its components. To compute the probability of a power outage possible failure events are considered and in each case the possibility of a cascading failure is studied. Graph representation of the network is used to detect connected components in the network, and then network equations are solved separately in each component. We show how to use this model to compute the probability of a power blackout in power networks and how this model can be used to develop suggestions for improving power network designs. The IEEE 188 bus is considered to show the usefulness of this approach.

A systematic method to find all low-period windows for the logistic map fa(x) = ax(1 - x) is proposed. The method is used to obtain very good approximations of positions of periodic windows with periods p ≤ 20. For each window using the forward shooting based interval Newton operator we confirm its existence by proving the existence of a sink for several parameter values inside the window and compute a very accurate rigorous lower bound of the window width.

Zero-order holder discretization effects in sliding mode control systems with an input delay are studied. Conditions for the existence of periodic solutions are derived and the existence of periodic steady states is investigated. The influence of the discretization step and the delay on the period and the amplitude of steady state oscillations is studied. Simulation results are presented to show the structure of basins of attraction of periodic orbits with different switching patterns.

The problem of existence of stable periodic orbits (sinks) for the Hénon map in a neighborhood of classical parameter values is studied numerically. Several parameter values which sustain a sink are found. It is shown rigorously that the sinks exist. Regions of existence in the parameter space of the sinks are located using the continuation method.

It is shown that a certain set is positively invariant for the return map associated with the Chua's circuit. The set contains the intersection of the numerically observed double-scroll attractor and the planes defining the return map. The proof is based on rigorous numerics.

We show how to rigorously integrate piece-wise linear systems in regions containing trajectories tangent to hyperplanes separating the linear regions. The method is applied to compute enclosures of solutions for the Chua's circuit with parameter values where the attractor contains such trajectories.

Discretization effects in single input equivalent control based SMC systems are studied. A procedure for construction of basins of attraction for periodic orbits corresponding to a given symbol sequence is proposed. Several examples illustrating the proposed technique are included.

It is shown that the problem of existence of periodic orbits can be studied rigorously by means of a symbolic dynamics approach combined with interval methods. Symbolic dynamics is used to find approximate initial positions of periodic points and interval operators are used to prove the existence of periodic orbits in a neighborhood of the computer generated solution. As an example the Lorenz system is studied. All 2536 periodic orbits of the Poincare map with the period n les 14 are found.

The existence of short periodic orbits for the Lorenz system is studied rigorously. We describe a method for finding all short cycles embedded in a chaotic singular attractor (i.e. an attractor containing an equilibrium). The method uses an interval operator for proving the existence of periodic orbits in regions where it can be evaluated, and bounds for the return time in other regions. The six shortest periodic orbits for the Lorenz system are found.

Discretization effects in single input equivalent control based SMC systems of arbitrary dimension are studied. A bound for the number of iterates a trajectory may stay on one side of the sliding surface is found. Conditions for existence of periodic orbits are formulated. It is confirmed in simulations that even for very small values of discretization steps complex behaviors including periodic solutions of various length may be observed.

Two discretization methods for a simple sliding mode control system are studied in detail. It is shown that for arbitrary discretization step the zero-order holder discretization is equivalent to the Euler discretization with a smaller value of the discretization step. A complete diagram of admissible short periodic orbits is found and structure of periodic orbits is analyzed.

In this work, an analysis of Chua's circuit in terms of short periodic orbits is carried out. The circuit is considered with two sets of parameter values, for which the Roessler-type and the double-scroll attractors are observed. Using the number of short periodic orbits and their periods, we find estimates for the topological entropy of the Poincare map and the flow

Effects of finite switching frequency on the behaviour of a simple sliding mode control system are studied. We show that if the switching frequency is high enough then the system converges to a small neighborhood of the origin. Even for arbitrarily fast switchings there exist periodic orbits with infinitely long periods and complex aperiodic trajectories. We also prove that if the switching frequency is below a certain threshold, the system sustains periodic oscillations with arbitrarily large amplitudes

In this paper we investigate the influence of system non-uniformity on the existence and stability of synchronous motion in an array of bi-directionally coupled electronic circuits. In computer simulations we find the level of non-uniformity for which synchronous behavior is sustained. We also present several examples of attractors, which appear when the synchronous motions is no longer stable.

We propose an enhanced version of the DCSK (differential chaos shift keying) scheme where the chaotic carrier is exploited for conveying useful information. This is achieved by means of a pseudo-chaotic encoder which spreads the input sequence, approximating the dynamics of the Bernoulli shift. The information encoded in the chaotic carrier is retrieved by means of standard maximum-likelihood detection methods.