Border collision bifurcations in discontinuous one-dimensional linear-hyperbolic maps

In this paper we consider a discontinuous one-dimensional map, which is linear on one side of a generic point and hyperbolic on the other side, coming from economic applications. However this kind of piecewise smooth models is widely used also in other different applied contexts, and is characterized by border collision bifurcations. The simple formulation of the functions involved in the model allows for analytical results and the border collision bifurcations curves associated with the attracting cycles of the model are here determined. Also coexistence of two attracting cycles is shown to occur in a family of cycles of even period whose periodicity regions are overlapped in pair.     

Border collision bifurcations and chaotic sets in a two-dimensional piecewise linear map

A two-dimensional piecewise linear continuous model is analyzed. It reflects the dynamics occurring in a circuit proposed as chaos generator, in a simplified case. The parameter space is investigated in order to classify completely regions of existence of stable cycles, and regions associated with chaotic behaviors. The border collision bifurcation curves are analytically detected, as well as the degenerate flip bifurcations of k-cycles and the homoclinic bifurcations occurring in cyclic chaotic regions leading to chaos in one-piece.     

Border collision and fold bifurcations in a family of one-dimensional discontinuous piecewise smooth maps: divergence and bounded dynamics

In this work we continue the study of a family of 1D piecewise smooth maps, defined by a linear function and a power function with negative exponent, proposed in engineering studies. The range in which a point on the right side is necessarily mapped to the left side, and chaotic sets can only be unbounded, has been already considered. In this work we are characterizing the remaining ranges, in which more iterations of the right branch are allowed and in which divergent trajectories occur. We prove that in some regions a bounded chaotic repellor always exists, which may be the only non-divergent set, or it may coexist with an attracting cycle. In another range, in which divergence cannot occur, we prove that unbounded chaotic sets always exist. The role of particular codimension-two points is evidenced, associated with fold bifurcations and border collision bifurcations (BCBs), related to cycles having the same symbolic sequences. We prove that they exist related to the border collision of any admissible cycle. We show that each BCB, each fold bifurcation and each homoclinic bifurcation is a limit set of infinite families of other BCBs.     

Border collision bifurcation curves and their classification in a family of 1D discontinuous maps

In this paper we consider a one-dimensional piecewise linear discontinuous map in canonical form, which may be used in several physical and engineering applications as well as to model some simple financial markets. We classify three different kinds of possible dynamic behaviors associated with the stable cycles. One regime (i) is the same existing in the continuous case and it is characterized by periodicity regions following the period increment by 1 rule. The second one (ii) is the regime characterized by periodicity regions of period increment higher than 1 (we shall see examples with 2 and 3), and by bistability. The third one (iii) is characterized by infinitely many periodicity regions of stable cycles, which follow the period adding structure, and multistability cannot exist. The analytical equations of the border collision bifurcation curves bounding the regions of existence of stable cycles are determined by using a new approach.     

On the density of the minimal subspaces generated by discrete linear Hamiltonian systems

This paper focuses on the density of the minimal subspaces generated by a class of discrete linear Hamiltonian systems. It is shown that the minimal subspace is densely defined if and only if the maximal subspace is an operator; that is, it is single valued. In addition, it is shown that, if the interval on which the systems are defined is bounded from below or above, then the minimal subspace is non-densely defined in any non-trivial case.     

Renormalization and conjugacy of piecewise linear Lorenz maps

We investigate the uniform piecewise linearizing question for a family of Lorenz maps. Let f be a piecewise linear Lorenz map with different slopes and positive topological entropy, we show that f is conjugate to a linear mod one transformation and the conjugacy admits a dichotomy: it is either bi-Lipschitz or singular depending on whether f is renormalizable or not. f is renormalizable if and only if its rotation interval degenerates to be a rational point. Furthermore, if the endpoints are periodic points with the same rotation number, then the conjugacy is quasisymmetric.     

A characterization of multidimensional multi-knot piecewise linear spectral sequence and its applications

We characterize a class of piecewise linear spectral sequences. Associated with the spectral sequence, we construct an orthonormal exponential bases for L2（[0,1）d）, which are called generalized Fourier bases. Moreover, we investigate the convergence of Bochner-Riesz means of the generalized Fourier series.     

Limit cycles of discontinuous piecewise differential systems formed by linear centers in R2 and separated by two circles

We show that discontinuous planar piecewise differential systems formed by linear centers and separated by two concentric circles can have at most three limit cycles. Usually is a difficult problem to provide the exact upper bound that a class of differential systems can exhibit. Here we also provide examples of such systems with zero, one, two, or three limit cycles.     

On a class of new and practical performance indexes for approximation of fold bifurcations of nonlinear power flow equations

Efficient measurement of the performance index (the distance of a loading parameter from the voltage collapse point) is one of the key problems in power system operations and planning and such an index indicates the severity of a power system with regard to voltage collapse. There exist many interesting methods and ideas to compute this index. However, some successful methods are not yet mathematically justified while other mathematically sound methods are often proposed directly based on the bifurcation theory and they require the initial stationary state to be too close to the unknown turning point to make the underlying methods practical.This paper first gives a survey of several popular methods for estimating the fold bifurcation point including the continuation methods, bifurcation methods and the test function methods (Seydel's direct solution methods, the tangent vector methods and the reduced Jacobian method) and discuss their relative advantages and problems. Test functions are usually based on scaling of the determinant of the Jacobian matrix and it is generally not clear how to determine the behaviour of such functions. As the underlying nonlinear equations are of a particular type, this allows us to do a new analysis of the determinants of the Jacobian and its submatrices in this paper. Following the analysis, we demonstrate how to construct a class of test functions with a predictable analytical behaviour so that a suitable index can be produced. Finally, examples of two test functions from this class are proposed. For several standard IEEE test systems, promising numerical results have been achieved.     

On fractal properties of non-normal numbers with respect to Rényi f-expansions generated by piecewise linear functions

The paper is devoted to the study of fractal properties of subsets of the set of non-normal numbers with respect to Rényi f  -expansions generated by continuous increasing piecewise linear functions defined on [0,+∞)$[0,+\infty )$. All such expansions are expansions for real numbers generated by infinite linear IFS f={_f0,_f1,…,_fn,…}$f=\{f_0,f_1,\dots ,f_n,\dots \}$ with the following list of ratios _Q∞=(_q0,_q1,…,_qn,…)$Q_{\infty }=(q_0,q_1,\dots ,q_n,\dots )$.     

Automorphism groups of Cayley graphs generated by block transpositions and regular Cayley maps

This paper deals with the Cayley graph $\mathrm{Cay}({\mathrm{Sym}}_n,T_n)$, where the generating set consists of all block transpositions. A motivation for the study of these particular Cayley graphs comes from current research in Bioinformatics. As the main result, we prove that $\text{Aut}\left(\mathrm{Cay}({\mathrm{Sym}}_n,T_n)\right)$ is the product of the left translation group and a dihedral group ${\mathsf{D}}_{n+1}$ of order $2(n+1)$. The proof uses several properties of the subgraph $\Gamma$ of $\mathrm{Cay}({\mathrm{Sym}}_n,T_n)$ induced by the set $T_n$. In particular, $\Gamma$ is a $2(n?2)$-regular graph whose automorphism group is ${\mathsf{D}}_{n+1},$ $\Gamma$ has as many as $n+1$ maximal cliques of size $2$, and its subgraph $\Gamma \left(V\right)$ whose vertices are those in these cliques is a 3-regular, Hamiltonian, and vertex-transitive graph. A relation of the unique cyclic subgroup of ${\mathsf{D}}_{n+1}$ of order $n+1$ with regular Cayley maps on ${\text{Sym}}_n$ is also discussed. It is shown that the product of the left translation group and the latter group can be obtained as the automorphism group of a non-$t$-balanced regular Cayley map on ${\text{Sym}}_n$.     

On the dynamics of a periodic Colpitts oscillator forced by periodic and chaotic signals

In this paper, we have examined effects of forcing a periodic Colpitts oscillator with periodic and chaotic signals for different values of coupling factors. The forcing signal is generated in a master bias-tuned Colpitts oscillator having identical structure as that of the slave periodic oscillator. Numerically solving the system equations, it is observed that the slave oscillator goes to chaotic state through a period-doubling route for increasing strengths of the forcing periodic signal. For forcing with chaotic signal, the transition to chaos is observed but the route to chaos is not clearly detectable due to random variations of the forcing signal strength. The chaos produced in the slave Colpitts oscillator for a chaotic forcing is found to be in a phase-synchronized state with the forced chaos for some values of the coupling factor. We also perform a hardware experiment in the radio frequency range with prototype Colpitts oscillator circuits and the experimental observations are in agreement with the numerical simulation results.     

Sensitivity and Hessian matrix analysis of power spectral density functions for uniformly modulated evolutionary random seismic responses

This paper describes a numerical method for calculation of the sensitivity and Hessian matrix of the response PSD functions of structures subjected to uniformly modulated evolutionary random seismic excitation. The method is formulated based on the pseudo excitation method and Newmark method. The evolutionary non-stationary random response analysis is converted into step-by-step integration computations using the pseudo excitation method. The formulas of the pseudo responses, their first and second derivatives with respect to the structural design variables are derived based on the Newmark method. The PSD functions, their sensitivity and Hessian matrix are calculated using the pseudo responses, their first and second derivatives, respectively. Then the computation procedure of sensitivity and Hessian matrix of PSD functions is given in detail. Finally, the PSD functions’ sensitivity and Hessian matrix analysis of a three-story, two-bay planar frame subjected to the uniformly modulated evolutionary random earthquake ground motion has been studied to elucidate the proposed method.     

Asymptotic and spectral properties of operator‐valued functions generated by aircraft wing model

The present paper is devoted to the asymptotic and spectral analysis of an aircraft wing model in a subsonic air flow. The model is governed by a system of two coupled integro‐differential equations and a two parameter family of boundary conditions modelling the action of the self‐straining actuators. The differential parts of the above equations form a coupled linear hyperbolic system; the integral parts are of the convolution type. The system of equations of motion is equivalent to a single operator evolution–convolution equation in the energy space. The Laplace transform of the solution of this equation can be represented in terms of the so‐called generalized resolvent operator, which is an operator‐valued function of the spectral parameter. More precisely, the generalized resolvent is a finite‐meromorphic function on the complex plane having a branch‐cut along the negative real semi‐axis. Its poles are precisely the aeroelastic modes and the residues at these poles are the projectors on the generalized eigenspaces. The dynamics generator of the differential part of the system has been systematically studied in a series of works by the second author. This generator is a non‐selfadjoint operator in the energy space with a purely discrete spectrum. In the aforementioned series of papers, it has been shown that the set of aeroelastic modes is asymptotically close to the spectrum of the dynamics generator, that this spectrum consists of two branches, and a precise spectral asymptotics with respect to the eigenvalue number has been derived. The asymptotical approximations for the mode shapes have also been obtained. It has also been proven that the set of the generalized eigenvectors of the dynamics generator forms a Riesz basis in the energy space. In the present paper, we consider the entire integro‐differential system which governs the model. Namely, we investigate the properties of the integral convolution‐type part of the original system. We show, in particular, that the set of poles of the adjoint generalized resolvent is asymptotically close to the discrete spectrum of the operator that is adjoint to the dynamics generator corresponding to the differential part. The results of this paper will be important for the reconstruction of the solution of the original initial boundary‐value problem from its Laplace transform and for the analysis of the flutter phenomenon in the forthcoming work. Copyright © 2004 John Wiley & Sons, Ltd.     

A probe into the chaotic nature of daily streamflow time series by correlation dimension and largest Lyapunov methods

Two chaotic indicators namely the correlation dimension and the Lyapunov exponent methods are investigated for the daily river flow of Kizilirmak River. A delay time of 60 days used for the reconstruction is chosen after examining the first minimum of the average mutual information of the data. The sufficient embedding dimension is estimated using the false nearest neighbor algorithm, which has a value of 11. Based on these embedding parameters the correlation dimension of the resulting attractor is calculated, as well as the average divergence rate of nearby orbits given by the largest Lyapunov exponent. The presence of chaos in the examined river flow time series is evident with the low correlation dimension (2.4) and the positive value of the largest Lyapunov exponent (0.0061).     

Generalized resolvents of linear relations generated by a nonnegative operator function and a differential elliptic-type expression

We study invertible extensions of the minimal relation generated by a nonnegative operator function and a differential elliptic-type expression. We prove that the operators inverse to such extensions are integral operators and describe such integral operators. We obtain a formula for generalized resolvents of the minimal relation.     

Conditions for the algebraic independence of certain series involving continued fractions and generated by linear recurrences

The main theorem of this paper, proved using Mahler's method, gives a necessary and sufficient condition for the values Θ(x,a,q) at any distinct algebraic points to be algebraically independent, where Θ(x,a,q) is an analogue of a certain q-hypergeometric series and generated by a linear recurrence whose typical example is the sequence of Fibonacci numbers. Corollary 1 gives Θ(x,a,q) taking algebraically independent values for any distinct triplets (x,a,q) of nonzero algebraic numbers. Moreover, Θ(a,a,q) is expressed as an irregular continued fraction and Θ(x,1,q) is an analogue of q-exponential function as stated in Corollaries 3 and 4, respectively.     

Normalized integral table algebras generated by a faithful real element of degree 2 and having 4 linear elements

We completely classify the normalized integral table algebra (A, B) generated by a faithful real element of degree 2 and having four linear elements.     

Examples of discontinuous maximal monotone linear operators and the solution to a recent problem posed by B.F. Svaiter

In this paper, we give two explicit examples of unbounded linear maximal monotone operators. The first unbounded linear maximal monotone operator S on ?² is skew. We show its domain is a proper subset of the domain of its adjoint S^∗, and −S^∗ is not maximal monotone. This gives a negative answer to a recent question posed by Svaiter. The second unbounded linear maximal monotone operator is the inverse Volterra operator T on L²[0,1]. We compare the domain of T with the domain of its adjoint T^∗ and show that the skew part of T admits two distinct linear maximal monotone skew extensions. These unbounded linear maximal monotone operators show that the constraint qualification for the maximality of the sum of maximal monotone operators cannot be significantly weakened, and they are simpler than the example given by Phelps-Simons. Interesting consequences on Fitzpatrick functions for sums of two maximal monotone operators are also given.