Cycles of Chaotic Intervals in a Time-Delayed Chua's Circuit

Maistrenko, Yuri; Maistrenko, Volodymyr; Chua, Leon O.

doi:10.1142/9789812798855_0055

Cited by 29 publications

(58 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…These two maps, characterized by a decreasing jump and an increasing jump, respectively, have been recently studied in [7] where an analysis of the socalled "principal tongues" ( [28], [29], [30], [12], [13], [4], [1], [2]) or "tongues of first degree" (Leonov,[26], [27], Mira [31], [32]) is provided. We recall that the principal tongues are regions, in the parameters' space, where a periodic cycle of period k exists, with one periodic point on a side of the discontinuity point and the remaining (k − 1) points on the other side (for any integer k > 1).…”

Section: A Family Of Piecewise Linear Maps With Two Discontinuitiesmentioning

confidence: 99%

“…For example, Leonov ([26], [27]) described several bifurcations of that kind, and gave a recursive relation to find the analytic expression of the sequence of bifurcations occurring in a one-dimensional piecewise linear map with one discontinuity point, which is still almost unknown, except for a limited number of researchers among which Mira ([31], [32]), Maistrenko et al [28], [29], [30]. See also the results obtained by Feigen in 1978, re-proposed in di Bernardo et al [12], [13].…”

Section: Introductionmentioning

confidence: 99%

“…In fact, several papers on piecewise smooth dynamical systems and border collision bifurcations have been motivated by the study of models used to describe particular electrical circuits or systems for the transmission of signals ( [28], [29], [30], [12], [13], [3], [4], [5], [25], [14], [15], [20], [50], [51], [52], [1], [2]). …”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Periodic Cycles and Bifurcation Curves for One-Dimensional Maps with Two Discontinuities

Bischi¹,

Gardini²,

Merlone³

2009

Journal of Dynamical Systems and Geometric Theories

View full text Add to dashboard Cite

Abstract. Starting from a family of discontinuous piece-wise linear one-dimensional maps, recently introduced as a dynamic model in social sciences, we propose a geometric method for finding the analytic expression of the bifurcation curves, in the space of the parameters, that bound the regions characterized by the existence of stable periodic cycles of any period. The conditions for the creation and the destruction of periodic cycles, as well as the analytic expressions of the bifurcation conditions, are obtained by studying the occurrence of border-collision bifurcations. In this paper we consider the case of maps formed by three linear portions separated by two discontinuity points. After summarizing the bifurcation structure associated with one-dimensional maps with only one discontinuity point, we show how this is modified by the introduction of a second discontinuity point. Finally we show how the considered map can be obtained as the limit case of a family of continuous maps as a parameter is increased without bounds, and we show how the low period cycles, which are typical of the discontinuous map we consider, emerge from the more complex (i.e. chaotic) behaviors observed in the continuous maps when a parameter value is large enough. From the point of view of the social application the increasing values of the parameter can be interpreted as higher degrees of impulsivity of the agents involved in binary decisions.

show abstract

Section: A Family Of Piecewise Linear Maps With Two Discontinuitiesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Periodic Cycles and Bifurcation Curves for One-Dimensional Maps with Two Discontinuities

Bischi¹,

Gardini²,

Merlone³

2009

Journal of Dynamical Systems and Geometric Theories

View full text Add to dashboard Cite

show abstract

“…In particular, a one-dimensional (1D for short) continuous piecewise linear map with one border point, known as skew tent map, depending on the parameters values can have attracting cycles of any period as well as cyclic chaotic intervals of any period, also called -bands chaotic attractors, which have the relevant property of being robust (as introduced in [4]) with respect to parameter perturbations. The bifurcation structure of the skew tent map has been completely described (see, e.g., [5][6][7][8]). Moreover, the skew tent map can be used as a normal form for a socalled border collision bifurcation (BCB for short) which is characteristic in piecewise smooth maps [9,10].…”

Section: Introductionmentioning

confidence: 99%

Bifurcation Structure in a Bimodal Piecewise Linear Business Cycle Model

Аврутин

Sushko

Tramontana

2014

Abstract and Applied Analysis

View full text Add to dashboard Cite

We study the bifurcation structure of the parameter space of a 1D continuous piecewise linear bimodal map which describes dynamics of a business cycle model introduced by Day-Shafer. In particular, we obtain the analytical expression of the boundaries of several periodicity regions associated with attracting cycles of the map (principal cycles and related fin structure). By crossing these boundaries the map displays robust chaos.

show abstract

“…Recently, variants of this circuit have been investigated, to enlarge the class of nonlinear phenomena that can be generated by relatively simple circuits [1]- [8].…”

mentioning

confidence: 99%

Qualitative analysis of the dynamics of the time-delayed Chua's circuit

Biey

Bonani

Gilli

et al. 1997

IEEE Trans. Circuits Syst. I

View full text Add to dashboard Cite

Abstract-Several variants of the Chua's circuit have been recently proposed in order to enlarge the class of nonlinear phenomena that can be generated by relatively simple circuits. In particular, Sharkovsky et al. proposed the so called timedelayed Chua's circuit (TDCC), where the original lumped LC resonator is substituted by an ideal transmission line, thereby generating an infinite dimensional system. The TDCC has been studied in details in the absence of the capacitor C; the only lumped dynamic element left in the circuit. This paper studies the effects of the presence of C on the dynamics of the circuit. After recasting the circuit equations in a suitable normalized form, their characteristic equation is theoretically investigated and the regions in the parameter space where all the eigenvalues have negative real part are exactly evaluated along with all the possible qualitative eigenvalue distributions. This analysis allows for a qualitative description of the TDCC dynamics in presence of the capacitor C: In particular, it is shown that, for particular sets of circuit parameters, an even small value of C; e.g., a parasitic element, may completely change the behavior of TDCC and that, on the other hand, any TDCC, exhibiting the periodadding phenomenon for C = 0; still continues to present this phenomenon even if a small capacitor C is added to the circuit.

show abstract

Cycles of Chaotic Intervals in a Time-Delayed Chua's Circuit

Cited by 29 publications

References 0 publications

Periodic Cycles and Bifurcation Curves for One-Dimensional Maps with Two Discontinuities

Periodic Cycles and Bifurcation Curves for One-Dimensional Maps with Two Discontinuities

Bifurcation Structure in a Bimodal Piecewise Linear Business Cycle Model

Qualitative analysis of the dynamics of the time-delayed Chua's circuit

Contact Info

Product

Resources

About