BIFURCATIONS OF ATTRACTING CYCLES FROM TIME-DELAYED CHUA’S CIRCUIT

1995 ◽  
Vol 05 (03) ◽  
pp. 653-671 ◽  
Author(s):  
Yu. L. MAISTRENKO ◽  
V.L. MAISTRENKO ◽  
S.I. VIKUL ◽  
L.O. CHUA
Keyword(s):  
Parameter Space ◽  
Rotation Number ◽  
Piecewise Linear ◽  
Three Dimensional ◽  
Homoclinic Bifurcation ◽  
Dimensional Parameter ◽  
Tent Map ◽  
One Dimensional ◽  
Skew Tent Map ◽  
Border Collision Bifurcation

We study the bifurcations of attracting cycles for a three-segment (bimodal) piecewise-linear continuous one-dimensional map. Exact formulas for the regions of periodicity of any rational rotation number (Arnold’s tongues) are obtained in the associated three-dimensional parameter space. It is shown that the destruction of any Arnold’s tongue is a result of a border-collision bifurcation, and is followed by the appearance of a cycle of intervals with the same rotation number, whose dynamics is determined by a skew tent map. Finally, for the interval cycle the merging bifurcation corresponds to a homoclinic bifurcation of some point cycle.

BIFURCATION STRUCTURES IN A BIMODAL PIECEWISE LINEAR MAP: REGULAR DYNAMICS

2013 ◽  
Vol 23 (12) ◽  
pp. 1330040 ◽  
Author(s):  
ANASTASIIA PANCHUK ◽  
IRYNA SUSHKO ◽  
BJÖRN SCHENKE ◽  
VIKTOR AVRUTIN
Keyword(s):  
Parameter Space ◽  
Piecewise Linear ◽  
Periodic Points ◽  
Tent Map ◽  
Linear Map ◽  
Piecewise Linear Map ◽  
Skew Tent Map ◽  
Replacement Technique ◽  
Bifurcation Structures ◽  
Analytical Expressions

We consider a family of piecewise linear bimodal maps having the outermost slopes positive and less than one. Three types of bifurcation structures incorporated into the parameter space of the map are described. These structures are formed by periodicity regions related to attracting cycles, namely, the skew tent map structure is associated with periodic points on two adjacent branches of the map, the period adding structure is related to periodic points on the outermost branches, and the fin structure is contiguous with the period adding structure and is associated with attracting cycles having at most one point on the middle branch. Analytical expressions for the periodicity region boundaries are obtained using the map replacement technique.

The Volume of Motion: Introduction, Derivation, and an Application Comparing Various Spinal Fixation Devices

1993 ◽  
Vol 115 (1) ◽  
pp. 43-46 ◽  
Author(s):  
J. J. Crisco
Keyword(s):  
Rigid Body ◽  
Three Dimensional ◽  
Spinal Fixation ◽  
Dimensional Parameter ◽  
Directional Information ◽  
One Dimensional ◽  
Fixation Devices ◽  
Spinal Fixation Devices ◽  
Mathematical Derivation

Range of motion (ROM), the displacement between two limits, is one of the most common parameters used to describe joint kinematics. The ROM is a one-dimensional parameter, although the motion at many normal and pathological joints is three-dimensional. Certainly, the ROM yields vital information, but an overall measure of the three-dimensional mobility at a joint may also be useful. The volume of motion (VOM) is such a measure. The translational VOM is the volume defined by all possible ROMs of a point on a rigid body. The rotational VOM, although its interpretation is not as tangible as the translational VOM, is a measure of the three-dimensional rotational mobility of a rigid body. The magnitude of the VOM is proportional to mobility; the VOM is a scaler, which does not contain any directional information. Experimental determination of the VOM is not practical since it would require applying loads in an infinite number of directions. The mathematical derivation given here allows the VOM to be calculated, with the assumption of conservative elasticity, from the resultant displacements of three distinct load vectors of equal magnitude. An example of the VOM is presented in the comparison of the biomechanical stabilizing potential of various spinal fixation devices.

Nonlocality of one-dimensional bilinear hardening–softening elastoplastic axial lattices

2019 ◽  
Vol 25 (2) ◽  
pp. 475-497
Author(s):  
Vincent Picandet ◽  
Noël Challamel
Keyword(s):  
Difference Equations ◽  
Differential Operators ◽  
Scale Effects ◽  
Piecewise Linear ◽  
Three Dimensional ◽  
Continuous Model ◽  
Lattice System ◽  
Discrete Problem ◽  
Linear Difference Equations ◽  
One Dimensional

The static behaviour of an elastoplastic axial lattice is studied in this paper through both discrete and nonlocal continuum analyses. The elastoplastic lattice system is composed of piecewise linear hardening–softening elastoplastic springs connected between each other via nodes, loaded by concentrated tension forces. This inelastic lattice evolution problem is ruled by some difference equations, which are shown to be equivalent to the finite difference formulation of a continuous elastoplastic bar problem under distributed tension load. Exact solutions of this inelastic discrete problem are obtained from the resolution of this piecewise linear difference equations system. Localization of plastic strain in the first elastoplastic spring, connected to the fixed end, is observed in the softening range. A continuous nonlocal elastoplastic theory is then built from the lattice difference equations using a continualization process, based on a rational asymptotic expansion of the associated pseudo-differential operators. The continualized lattice-based model is equivalent to a distributed nonlocal continuous elastoplastic theory coupled to a cohesive elastoplastic model, which is shown to capture efficiently the scale effects of the reference axial lattice. The hardening–softening localization process of the nonlocal elastoplastic continuous model strongly depends on the lattice spacing, which controls the size of the nonlocal length scales. An analogy with the one-dimensional lattice system in bending is also shown. The considered microstructured elastoplastic beam is a Hencky bar-chain connected by elastoplastic rotational springs. It is shown that the length scale calibration of the nonlocal model strongly depends on the degree of the difference equations of each lattice model (namely axial or bending lattice). These preliminary results valid for one-dimensional systems allow possible future developments of new nonlocal elastoplastic models, including two- or even three-dimensional elastoplastic interactions.

Bifurcation Structures in a Bimodal Piecewise Linear Map: Chaotic Dynamics

2015 ◽  
Vol 25 (03) ◽  
pp. 1530006 ◽  
Author(s):  
Anastasiia Panchuk ◽  
Iryna Sushko ◽  
Viktor Avrutin
Keyword(s):  
Parameter Space ◽  
Chaotic Dynamics ◽  
Piecewise Linear ◽  
Chaotic Attractors ◽  
Bifurcation Structure ◽  
Tent Map ◽  
Linear Map ◽  
Piecewise Linear Map ◽  
Replacement Technique ◽  
Bifurcation Structures

In this work, we investigate the bifurcation structure of the parameter space of a generic 1D continuous piecewise linear bimodal map focusing on the regions associated with chaotic attractors (cyclic chaotic intervals). The boundaries of these regions corresponding to chaotic attractors with different number of intervals are identified. The results are obtained analytically using the skew tent map and the map replacement technique.

Long-Time Solutions to Heat-Conduction Transients with Time-Dependent Inputs

1980 ◽  
Vol 102 (1) ◽  
pp. 115-120 ◽  
Author(s):  
H. T. Ceylan ◽  
G. E. Myers
Keyword(s):  
Heat Conduction ◽  
Lower Cost ◽  
Piecewise Linear ◽  
Three Dimensional ◽  
Linear Functions ◽  
Time Dependent ◽  
Time Interval ◽  
Piecewise Linear Functions ◽  
One Dimensional ◽  
Long Time

An economical method for obtaining long-time solutions to one, two, or three-dimensional heat-conduction transients with time-dependent forcing functions is presented. The conduction problem is spatially discretized by finite differences or by finite elements to obtain a system of first-order ordinary differential equations. The time-dependent input functions are each approximated by continuous, piecewise-linear functions each having the same uniform time interval. A set of response coefficients is generated by which a long-time solution can be carried out with a considerably lower cost than for conventional methods. A one-dimensional illustrative example is included.

scholarly journals Effects of generalized pooling on binocular disparity selectivity of neurons in the early visual cortex

2016 ◽  
Vol 371 (1697) ◽  
pp. 20150266 ◽  
Author(s):  
Daisuke Kato ◽  
Mika Baba ◽  
Kota S. Sasaki ◽  
Izumi Ohzawa
Keyword(s):  
Parameter Space ◽  
Three Dimensional ◽  
Stereoscopic Vision ◽  
Dimensional Parameter ◽  
Space Domain ◽  
Spatial Pooling ◽  
Proposed Model ◽  
Early Visual Cortex ◽  
Object Features ◽  
Theoretical Analyses

The key problem of stereoscopic vision is traditionally defined as accurately finding the positional shifts of corresponding object features between left and right images. Here, we demonstrate that the problem must be considered in a four-dimensional parameter space; with respect not only to shifts in space ( X , Y ), but also spatial frequency (SF) and orientation (OR). The proposed model sums outputs of binocular energy units linearly over the multi-dimensional V1 parameter space ( X , Y , SF, OR). Theoretical analyses and physiological experiments show that many binocular neurons achieve sharp binocular tuning properties by pooling the output of multiple neurons with relatively broad tuning. Pooling in the space domain sharpens disparity-selective responses in the SF domain so that the responses to combinations of unmatched left–right SFs are attenuated. Conversely, pooling in the SF domain sharpens disparity selectivity in the space domain, reducing the possibility of false matches. Analogous effects are observed for the OR domain in that the spatial pooling sharpens the binocular tuning in the OR domain. Such neurons become selective to relative OR disparity. Therefore, pooling allows the visual system to refine binocular information into a form more desirable for stereopsis. This article is part of the themed issue ‘Vision in our three-dimensional world’.

scholarly journals CLOSED CURVES OF GLOBAL BIFURCATIONS IN CHUA'S EQUATION: A MECHANISM FOR THEIR FORMATION

2003 ◽  
Vol 13 (03) ◽  
pp. 609-616 ◽  
Author(s):  
ANTONIO ALGABA ◽  
MANUEL MERINO ◽  
FERNANDO FERNÁNDEZ-SÁNCHEZ ◽  
ALEJANDRO J. RODRÍGUEZ-LUIS
Keyword(s):  
Parameter Space ◽  
Three Dimensional ◽  
Global Bifurcations ◽  
Dimensional Parameter ◽  
Closed Curves

In this work, the presence of closed bifurcation curves of homoclinic and heteroclinic connections has been detected in Chua's equation. We have numerically found and qualitatively described the mechanism of the formation/destruction of such closed curves. We relate this phenomenon to a failure of transversality in a curve of T-points in a three-dimensional parameter space.

Imaginary Scators Bound Set Under the Iterated Quadratic Mapping in 1 + 2 Dimensional Parameter Space

2016 ◽  
Vol 26 (01) ◽  
pp. 1630002 ◽  
Author(s):  
M. Fernández-Guasti
Keyword(s):  
Parameter Space ◽  
Complex Structure ◽  
Three Dimensional ◽  
Periodic Points ◽  
Mandelbrot Set ◽  
Three Dimensions ◽  
Quadratic Mapping ◽  
Two Dimensional ◽  
Dimensional Parameter ◽  
Nilpotent Elements

The quadratic iteration is mapped within a nondistributive imaginary scator algebra in [Formula: see text] dimensions. The Mandelbrot set is identically reproduced at two perpendicular planes where only the scalar and one of the hypercomplex scator director components are present. However, the bound three-dimensional S set projections change dramatically even for very small departures from zero of the second hypercomplex plane. The S set exhibits a rich fractal-like boundary in three dimensions. Periodic points with period [Formula: see text], are shown to be necessarily surrounded by points that produce a divergent magnitude after [Formula: see text] iterations. The scator set comprises square nilpotent elements that ineluctably belong to the bound set. Points that are square nilpotent on the [Formula: see text]th iteration, have preperiod 1 and period [Formula: see text]. Two-dimensional plots are presented to show some of the main features of the set. A three-dimensional rendering reveals the highly complex structure of its boundary.

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