Coexistence of Multiple Attractors and Crisis Route to Chaos in a Novel Chaotic Jerk Circuit

2016 ◽  
Vol 26 (05) ◽  
pp. 1650081 ◽  
Author(s):  
J. Kengne ◽  
Z. T. Njitacke ◽  
A. Nguomkam Negou ◽  
M. Fouodji Tsostop ◽  
H. B. Fotsin
Keyword(s):  
Equilibrium Points ◽  
Three Dimensional ◽  
Period Doubling ◽  
Chaotic Oscillations ◽  
Multiple Attractors ◽  
Coexistence Of Multiple Attractors ◽  
Theoretical Predictions ◽  
Route To Chaos ◽  
Model Equations ◽  
Good Agreement

In this paper, a novel autonomous RC chaotic jerk circuit is introduced and the corresponding dynamics is systematically investigated. The circuit consists of opamps, resistors, capacitors and a pair of semiconductor diodes connected in anti-parallel to synthesize the nonlinear component necessary for chaotic oscillations. The model is described by a continuous time three-dimensional autonomous system with hyperbolic sine nonlinearity, and may be viewed as a linear transformation of model MO15 previously introduced in [Sprott, 2010]. The structure of the equilibrium points and the discrete symmetries of the model equations are discussed. The bifurcation analysis indicates that chaos arises via the usual paths of period-doubling and symmetry restoring crisis. One of the key contributions of this work is the finding of a region in the parameter space in which the proposed (“elegant”) jerk circuit exhibits the unusual and striking feature of multiple attractors (i.e. coexistence of four disconnected periodic and chaotic attractors). Laboratory experimental results are in good agreement with the theoretical predictions.

Asymmetry-Induced Dynamics for a Class of Diode-Based Chaotic Circuits: A Case Study

2020 ◽  
pp. 2150077 ◽  
Author(s):  
Léandre Kamdjeu Kengne ◽  
Jacques Kengne ◽  
Nicole Adelaïde Kengnou Telem ◽  
Justin Roger Mboupda Pone ◽  
Hervé Thierry Kamdem Tagne
Keyword(s):  
Complex Dynamics ◽  
Period Doubling ◽  
Multiple Attractors ◽  
Asymmetric Mode ◽  
Nonlinear Component ◽  
Realistic Situation ◽  
Coexistence Of Multiple Attractors ◽  
Chaotic Circuits ◽  
Route To Chaos ◽  
Antiparallel Diodes

We consider the modeling and asymmetry-induced dynamics for a class of chaotic circuits sharing the same feature of an antiparallel diodes pair as the nonlinear component. The simple autonomous jerk circuit of [J. Kengne, Z. T. Njitacke, A. N. Nguomkam, M. T. Fouodji and H. B. Fotsin, Coexistence of multiple attractors and crisis route to chaos in a novel chaotic jerk circuit, Int. J. Bifurcation Chaos Appl. Sci. Eng.26 (2016) 1650081] is used as the prototype. In contrast to current approaches where the diodes are assumed to be identical (and thus a perfect symmetric circuit), we examine the more realistic situation where the diodes have different electrical properties in spite of unavoidable scattering of parameters. In this case, the nonlinear component formed by the diodes pair displays an asymmetric current–voltage characteristic which induces asymmetry of the whole circuit. The model is described by a continuous-time 3D autonomous system (ODEs) with exponential nonlinearities. We examine the chaos mechanism with respect to system parameters both in the symmetric and asymmetric modes of operation by using bifurcation diagrams and phase space trajectory plots as the main indicators. Period doubling route to chaos, merging crisis, and multiple coexisting (i.e., two, four, or six) mutually symmetric attractors are reported in the symmetric mode of oscillation. In the asymmetric mode, several unusual nonlinear behaviors arise such as coexisting bifurcations, hysteresis, asymmetric double-band chaotic attractor, crisis, and coexisting multiple (i.e., two, three, four, or five) asymmetric attractors for some suitable ranges of parameters. Theoretical analyses and circuit experiments show a very good agreement. The results obtained in this work let us conjecture that chaotic circuits with antiparallel diodes pair are capable of much more complex dynamics than what is reported in the current literature and thus should be reconsidered accordingly in spite of the approach followed in this work.

scholarly journals Autonomous Jerk Oscillator with Cosine Hyperbolic Nonlinearity: Analysis, FPGA Implementation, and Synchronization

2018 ◽  
Vol 2018 ◽  
pp. 1-12 ◽  
Author(s):  
Karthikeyan Rajagopal ◽  
Sifeu Takougang Kingni ◽  
Gaetan Fautso Kuiate ◽  
Victor Kamdoum Tamba ◽  
Viet-Thanh Pham
Keyword(s):  
Control Method ◽  
Sliding Mode ◽  
Equilibrium Points ◽  
Period Doubling ◽  
Fpga Implementation ◽  
Phase Portraits ◽  
Adaptive Sliding Mode Control ◽  
Coexistence Of Attractors ◽  
Route To Chaos ◽  
Two Parameters

A two-parameter autonomous jerk oscillator with a cosine hyperbolic nonlinearity is proposed in this paper. Firstly, the stability of equilibrium points of proposed autonomous jerk oscillator is investigated by analyzing the characteristic equation and the existence of Hopf bifurcation is verified using one of the two parameters as a bifurcation parameter. By tuning its two parameters, various dynamical behaviors are found in the proposed autonomous jerk oscillator including periodic attractor, one-scroll chaotic attractor, and coexistence between chaotic and periodic attractors. The proposed autonomous jerk oscillator has period-doubling route to chaos with the variation of one of its parameters and reverse period-doubling route to chaos with the variation of its other parameter. The proposed autonomous jerk oscillator is modelled on Field Programmable Gate Array (FPGA) and the FPGA chip statistics and phase portraits are derived. The chaotic and coexistence of attractors generated in the proposed autonomous jerk oscillator are confirmed by FPGA implementation of the proposed autonomous jerk oscillator. A good qualitative agreement is illustrated between the numerical and FPGA results. Finally synchronization of unidirectional coupled identical proposed autonomous jerk oscillators is achieved using adaptive sliding mode control method.

Studies on the Three-Dimensional Temperature Transients in the Canine Prostate During Transurethral Microwave Thermal Therapy

2000 ◽  
Vol 122 (4) ◽  
pp. 372-379 ◽  
Author(s):  
Jing Liu ◽  
Liang Zhu ◽  
Lisa X. Xu
Keyword(s):  
Three Dimensional ◽  
Blood Perfusion ◽  
Thermal Therapy ◽  
Bioheat Transfer ◽  
Target Tissue ◽  
Urethral Wall ◽  
Theoretical Predictions ◽  
Closed Form Analytical Solution ◽  
Good Agreement

Thermal therapy of benign prostatic hyperplasia requires accurate prediction of the temperature distribution induced by the heating within the prostatic tissue. In this study, the Pennes bioheat transfer equation was used to model the transient heat transfer inside the canine prostate during transurethral microwave thermal therapy. Incorporating the specific absorption rate of microwave energy in tissue, a closed-form analytical solution was obtained. Good agreement was found between the theoretical predictions and in-vivo experimental results. Effects of blood perfusion and the cooling at the urethral wall on the temperature rise were investigated within the prostate during heating. The peak intraprostatic temperatures attained by application of 5, 10, or 15 W microwave power were predicted to be 38°C,41°C, and 44°C. Results from this study will help optimize the thermal dose that can be applied to target tissue during the therapy. [S0148-0731(00)01004-9]

Antimonotonicity, Chaos and Multiple Attractors in a Novel Autonomous Jerk Circuit

2017 ◽  
Vol 27 (07) ◽  
pp. 1750100 ◽  
Author(s):  
J. Kengne ◽  
A. Nguomkam Negou ◽  
Z. T. Njitacke
Keyword(s):  
Three Dimensional ◽  
Period Doubling ◽  
Basins Of Attraction ◽  
Electrical Circuit ◽  
Semiconductor Diode ◽  
Chaotic Attractors ◽  
Coexisting Attractors ◽  
Multiple Attractors ◽  
Systematic Analysis ◽  
The Stability

We perform a systematic analysis of a system consisting of a novel jerk circuit obtained by replacing the single semiconductor diode of the original jerk circuit described in [Sprott, 2011a] with a pair of semiconductor diodes connected in antiparallel. The model is described by a continuous time three-dimensional autonomous system with hyperbolic sine nonlinearity, and may be viewed as a control system with nonlinear velocity feedback. The stability of the (unique) fixed point, the local bifurcations, and the discrete symmetries of the model equations are discussed. The complex behavior of the system is categorized in terms of its parameters by using bifurcation diagrams, Lyapunov exponents, time series, Poincaré sections, and basins of attraction. Antimonotonicity, period doubling bifurcation, symmetry restoring crises, chaos, and coexisting bifurcations are reported. More interestingly, one of the key contributions of this work is the finding of various regions in the parameters’ space in which the proposed (“elegant”) jerk circuit experiences the unusual phenomenon of multiple competing attractors (i.e. coexistence of four disconnected periodic and chaotic attractors). The basins of attraction of various coexisting attractors display complexity (i.e. fractal basins boundaries), thus suggesting possible jumps between coexisting attractors in experiment. Results of theoretical analyses are perfectly traced by laboratory experimental measurements. To the best of the authors’ knowledge, the jerk circuit/system introduced in this work represents the simplest electrical circuit (only a quadruple op amplifier chip without any analog multiplier chip) reported to date capable of four disconnected periodic and chaotic attractors for the same parameters setting.

Backward Extrusion of Internally Shaped Tubes From Round Billets

1984 ◽  
Vol 106 (2) ◽  
pp. 143-149 ◽  
Author(s):  
D. Y. Yang ◽  
C. H. Han
Keyword(s):  
Pure Aluminum ◽  
Three Dimensional ◽  
Mapping Function ◽  
Velocity Transformation ◽  
Backward Extrusion ◽  
Commercially Pure ◽  
Theoretical Predictions ◽  
Experimental Values ◽  
Punch Pressure ◽  
Good Agreement

An analytical method is proposed for estimating the steady-state punch pressure for three-dimensional backward extrusion (or piercing) of complicated internally shaped tubes from circular billets. A kinematically admissible velocity field is derived to formulate an upper-bound solution using velocity transformation and mapping function. The configuration of deforming boundary surfaces are determined by minimizing the extrusion power with respect to some chosen parameters. Experiments are carried out with commercially pure aluminum billets for internally shaped tubes at various reductions of area by using different sizes of shaped punches, such as square and regular hexagons. It is shown that the theoretical predictions for extrusion load are in good agreement with the experimental values.

BIFURCATION AND CHAOS OF THE SINUSOIDALLY-DRIVEN CHUA'S CIRCUIT

1991 ◽  
Vol 01 (02) ◽  
pp. 369-384 ◽  
Author(s):  
K. MURALI ◽  
M. LAKSHMANAN
Keyword(s):  
Bifurcation Diagram ◽  
Piecewise Linear ◽  
Period Doubling ◽  
Periodic Excitation ◽  
Multiple Attractors ◽  
Bifurcation And Chaos ◽  
Bifurcation Phenomena ◽  
Coexistence Of Multiple Attractors ◽  
Two Parameter ◽  
Periodic Bifurcation

The effect of an external sinusoidal signal on the familiar Chua's piecewise-linear autonomous circuit is experimentally investigated. Under the influence of such a periodic excitation an immense variety of bifurcation phenomena like period doubling, period adding, quasiperiodicity, and intermittency has been observed. Also, equal periodic bifurcation, hysteresis, and coexistence of multiple attractors are reported. A two-parameter (drive amplitude–drive frequency) bifurcation diagram is constructed, which classifies the observed rich bifurcation sequences and chaotic structures.

Observations of Bifurcation and Chaos in Plant Physiological Responses to Light

1997 ◽  
Vol 24 (1) ◽  
pp. 91 ◽  
Author(s):  
Sergey Shabala ◽  
Robert Delbourgo ◽  
Ian Newman
Keyword(s):  
Higher Plants ◽  
Physiological Responses ◽  
Period Doubling ◽  
The Other ◽  
Chaotic Oscillations ◽  
Bifurcation And Chaos ◽  
Intact Plants ◽  
Theoretical Predictions ◽  
Successive Period ◽  
Temperature Responses

Although some theoretical predictions have been made, no experimental evidence of chaotic behaviour in plant physiological responses has been reported. Here we present observations of period- doubling and tripling in higher plants. For leaf bioelectric and temperature responses of maize, tomato, and burweed plants to rhythmical light, two different routes to chaos were found experimentally. One was via successive period-doubling and the other via the formation of intermittently chaotic oscillations from a subharmonic synchronisation. Because these effects appeared in intact plants, under conditions close to those found in nature, they may have wide significance, including for plant phylogenesis.

Experimental Study of Propeller-Induced Vibratory Pressures on Simple Surfaces and Correlation With Theoretical Predictions

1964 ◽  
Vol 8 (05) ◽  
pp. 15-28
Author(s):  
J. P. Breslin ◽  
T. Kowalski
Keyword(s):  
Free Space ◽  
Three Dimensional ◽  
Theoretical Treatment ◽  
Plate Boundaries ◽  
Practical Application ◽  
Finite Area ◽  
Experimental Procedure ◽  
Theoretical Predictions ◽  
Pressure Changes ◽  
Good Agreement

Vibratory pressures exerted on cylindrical and flat-plate boundaries due to a model propeller were measured at three advance coefficients. A number of "free-space" measurements also were made. All measurements were made by driving a propeller past fixed pressure gages. This method yielded curves of pressure changes which are entirely free from background noise. The magnitudes of the free-space pressures were found to be larger than one half the corresponding magnitudes measured by gages mounted flush in a large plate at equal clearances from the propeller. By postulating that the finite area of the gage diaphragm produces a partial image of the propeller (and hence a larger pressure than that in free space) an experimental procedure was devised for correcting for this finite-area effect yielding results in good agreement with theory. A theoretical treatment of this effect of finite gage size is given in Appendix 2. The decay of maximum amplitudes of vibrating pressures is shown by means of three-dimensional plots. The pressures were found to become vanishingly small within approximately one propeller diameter fore and aft of the center of the propeller. The comparison with theoretically calculated pressures and forces gives very close agreement for free-space pressures and reasonable agreement for forces on a cylindrical surface. The agreement of both pressures and forces with theory is excellent for operation near the design advance ratio. A strong plea is made for further experiments with ship models in an effort to develop design criteria for practical application.

A Simplified Method of Using Four-Hole Probes to Measure Three-Dimensional Flow Fields

1985 ◽  
Vol 107 (1) ◽  
pp. 31-35 ◽  
Author(s):  
N. Sitaram ◽  
A. L. Treaster
Keyword(s):  
Flat Plate ◽  
Three Dimensional ◽  
Flow Fields ◽  
Three Dimensional Flow ◽  
Dimensional Flow ◽  
Application Procedure ◽  
Simplified Method ◽  
Theoretical Predictions ◽  
Gradient Effects ◽  
Good Agreement

A simplified method of using four-hole probes to measure three-dimensional flow-fields is presented. This method is similar to an existing calibration and application procedure used for five-hole probes. The new method is demonstrated for two four-hole probes of different geometry. These four-hole probes and a five-hole probe are used to measure the turbulent boundary layer on a flat plate. The results from the three probes are in good agreement with theoretical predictions. The major discrepancies occur near the surface of the flat plate and are attributed to wall vicinity and velocity gradient effects.

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