Analytical Study and Experimental Confirmation of SNA Through Poincaré Maps in a Quasiperiodically Forced Electronic Circuit

2015 ◽  
Vol 25 (08) ◽  
pp. 1530020 ◽  
Author(s):  
A. Arulgnanam ◽  
Awadesh Prasad ◽  
K. Thamilmaran ◽  
M. Daniel
Keyword(s):  
Analytical Study ◽  
Poincaré Map ◽  
Initial Conditions ◽  
Nonlinear Element ◽  
Phase Portraits ◽  
Poincare Map ◽  
Explicit Analytical Solution ◽  
Strange Nonchaotic Attractors ◽  
First Time ◽  
Bifurcation Process

Quasiperiodically forced series LCR circuit with simple nonlinear element is studied analytically and experimentally. To the best of our knowledge, this is the first time that strange nonchaotic attractors (SNAs) are studied analytically. From the explicit analytical solution, the bifurcation process is shown. With a single negative conduction region of the nonlinear element two routes namely, Heagy–Hammel and fractalization routes to the birth of SNA are identified. The analytical analysis are confirmed by laboratory hardware experiments. In addition, for the first time, a detailed stroboscopic Poincaré map is generated experimentally for two different frequencies, for the above two routes, which clearly confirm the presence of SNAs in these two routes. Also, from the experimental data of the corresponding attractors, we quantitatively confirm the presence of SNAs through singular-continuous spectrum analysis. The analytical results as well as experimental observations are characterized qualitatively in terms of phase portraits, Poincaré map, power spectrum, and sensitivity dependance on initial conditions.

A PHYSICAL INTERPRETATION OF MELNIKOV’S METHOD

1992 ◽  
Vol 02 (01) ◽  
pp. 1-9 ◽  
Author(s):  
YOHANNES KETEMA
Keyword(s):  
Physical Interpretation ◽  
Poincaré Map ◽  
Initial Conditions ◽  
Homoclinic Orbits ◽  
Poincare Map ◽  
Stable And Unstable Manifolds ◽  
Melnikov's Method ◽  
Melnikov’S Method ◽  
Sensitive Dependence ◽  
The Stability

This paper is concerned with analyzing Melnikov’s method in terms of the flow generated by a vector field in contrast to the approach based on the Poincare map and giving a physical interpretation of the method. It is shown that the direct implication of a transverse crossing between the stable and unstable manifolds to a saddle point of the Poincare map is the existence of two distinct preserved homoclinic orbits of the continuous time system. The stability of these orbits and their role in the phenomenon of sensitive dependence on initial conditions is discussed and a physical example is given.

Nonlinear Rolling Motion of a Statically Biased Ship Under the Effect of External and Parametric Excitation

1993 ◽  
Author(s):  
Isaac Esparza ◽  
Jeffrey Falzarano
Keyword(s):  
Steady State ◽  
Parametric Excitation ◽  
Poincaré Map ◽  
Initial Conditions ◽  
Global Analysis ◽  
Wave Train ◽  
Periodic Wave ◽  
Rolling Motion ◽  
Poincare Map ◽  
Safe Basin

Abstract In this work, global analysis of ship rolling motion as effected by parametric excitation is studied. The parametric excitation results from the roll restoring moment variation as a wave train passes. In addition to the parametric excitation, an external periodic wave excitation and steady wind bias are also included in the analysis. The roll motion is the most critical motion for a ship because of the possibility of capsizing. The boundaries in the Poincaré map which separate initial conditions which eventually evolve to bounded steady state solutions and those which lead to unbounded capsizing motion are studied. The changes in these boundaries or manifolds as effected by changes in the ship and environmental conditions are analyzed. The region in the Poincaré map which lead to bounded steady state motions is called the safe basin. The size of this safe basin is a measure of the vessel’s resistance to capsizing.

scholarly journals A Numerical and Analytical Study of Phase Synchronization in Coupled SC-CNN Based Non-Identical Chaotic Systems

2021 ◽  
Vol 16 (2) ◽  
Author(s):  
H. Shameem Banu ◽  
P.S. Sheik Uduman
Keyword(s):  
Analytical Study ◽  
Chaotic Systems ◽  
Phase Synchronization ◽  
Simulation Method ◽  
Cellular Neural Network ◽  
Phase Portraits ◽  
Complete Synchronization ◽  
Synchronous State ◽  
Explicit Analytical Solution ◽  
Proposed Model

This paper seeks to address the phase synchronization phenomenon using the drive-response concept, in our proposed model, State Controlled Cellular Neural Network (SC-CNN) based on variant of MuraliLakshmanan-Chua (MLCV) circuit. Using this unidirectionally coupled chaotic non autonomous circuits, we described the transition of unsynchronous to synchronous state, by numerical simulation method as well as the results are confirmed by solving explicit analytical solution. In this aspect, the system undergoes the new effect of phase synchronization (PS) phenomenon have been observed before complete synchronization (CS) state. To characterize these phenomena by the phase portraits and the time series plots. Also particularly characterize for PS by the method of partial Poincare section map using phase difference versus time, numerically and analytically. The study of dynamics involved in SC-CNN circuit systems, mainly applicable in the field of neurosciences and in telecommunication fields.

Second Order Poincaré Map to Study Dynamical Behavior of Nonsymmetrical Gyrostat Satellite

2000 ◽  
Author(s):  
K. H. Shirazi ◽  
M. H. Ghaffari-saadat
Keyword(s):  
Phase Space ◽  
Poincaré Map ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Dynamical Behavior ◽  
Two Dimensions ◽  
Second Order ◽  
Poincare Map ◽  
Runge Kutta Method ◽  
Final Space

Abstract The second order poincare’ map is described and used for investigation of the dynamical behavior of a gyrostat satellite. The normalized attitudinal equations of motion for a typical non-symmetric gyrostat satellite are considered. For different sets of initial conditions the equations simulated by Runge-Kutta method. The poincare’ section is used to dimension reduction of system phase space. By this map the dimension reduced from six to five. Using secondary map the dimension of phase space can be reduced to four and considering symmetry of phase space the final space has two dimensions that is presentable at the plane. Bifurcation in the attitudinal behavior can be demonstrated easily by the derived map.

STRANGE NONCHAOTIC ATTRACTORS FROM QUASIPERIODICALLY FORCED UEDA’S CIRCUIT

1996 ◽  
Vol 06 (07) ◽  
pp. 1383-1388 ◽  
Author(s):  
ZHONG LIU ◽  
ZHIWEN ZHU
Keyword(s):  
Lyapunov Exponents ◽  
Experimental Observation ◽  
Frequency Spectrum ◽  
Poincaré Map ◽  
Theoretical Study ◽  
Poincare Map ◽  
Winding Number ◽  
Strange Nonchaotic Attractors

In this paper, we discuss the existence of strange nonchaotic attractors for the Ueda’s circuit with two-frequency quasiperiodic excitation. We have calculated the Lyapunov exponents, Poincaré map, winding number and frequency spectrum of characterizing the attractors. The results show that the Ueda’s circuit does indeed exhibit strange nonchaotic attractors. Because of the simplicity, it is expected that the circuit would be useful for the theoretical study and experimental observation of the strange nonchaotic attractors.

scholarly journals A New No-Equilibrium Chaotic System and Its Topological Horseshoe Chaos

2016 ◽  
Vol 2016 ◽  
pp. 1-6
Author(s):  
Chunmei Wang ◽  
Chunhua Hu ◽  
Jingwei Han ◽  
Shijian Cang
Keyword(s):  
Numerical Simulation ◽  
Lyapunov Exponents ◽  
Chaotic System ◽  
Poincaré Map ◽  
Chaotic Behavior ◽  
Dynamical Behavior ◽  
Phase Portraits ◽  
Poincare Map ◽  
Topological Horseshoe ◽  
Simulation Techniques

A new no-equilibrium chaotic system is reported in this paper. Numerical simulation techniques, including phase portraits and Lyapunov exponents, are used to investigate its basic dynamical behavior. To confirm the chaotic behavior of this system, the existence of topological horseshoe is proven via the Poincaré map and topological horseshoe theory.

scholarly journals Holling-Tanner Predator-Prey Model with State-Dependent Feedback Control

2018 ◽  
Vol 2018 ◽  
pp. 1-18 ◽  
Author(s):  
Jin Yang ◽  
Guangyao Tang ◽  
Sanyi Tang
Keyword(s):  
Limit Cycle ◽  
Poincaré Map ◽  
Impulsive Control ◽  
Phase Portraits ◽  
Poincare Map ◽  
Complicated Shape ◽  
Predator Prey ◽  
Predator Prey Model ◽  
State Dependent ◽  
Existence And Stability

In this paper, we propose a novel Holling-Tanner model with impulsive control and then provide a detailed qualitative analysis by using theories of impulsive dynamical systems. The Poincaré map is first constructed based on the phase portraits of the model. Then the main properties of the Poincaré map are investigated in detail which play important roles in the proofs of the existence of limit cycles, and it is concluded that the definition domain of the Poincaré map has a complicated shape with discontinuity points under certain conditions. Subsequently, the existence of the boundary order-1 limit cycle is discussed and it is shown that this limit cycle is unstable. Furthermore, the conditions for the existence and stability of an order-1 limit cycle are provided, and the existence of order-k(k≥2) limit cycle is also studied. Moreover, numerical simulations are carried out to substantiate our results. Finally, biological implications related to the mathematical results which are beneficial for successful pest control are addressed in the Conclusions section.

An Analytical and Experimental Study of SC-CNN-Based Simple Nonautonomous Chaotic Circuit

2019 ◽  
Vol 14 (12) ◽  
Author(s):  
H. Shameem Banu ◽  
P. S. Sheik Uduman ◽  
K. Thamilmaran
Keyword(s):  
Neural Network ◽  
Circuit Simulation ◽  
Experimental Studies ◽  
Power Spectra ◽  
Cellular Neural Network ◽  
Phase Portraits ◽  
Explicit Analytical Solution ◽  
Periodic Dynamics ◽  
First Time ◽  
Good Agreement

Abstract In this study, we report an explicit analytical solution of state-controlled cellular neural network (SC-CNN) based second-order nonautonomous system. The proposed system is modeled with an aid of a generalized two-state-controlled cellular neural network (CNN) equations and experimentally realized by imposing a suitable connection of simple two-state-controlled generalized CNN cells following the report of Swathi et al. [2014]. The chaotic and quasi-periodic dynamics observed from this system have been investigated through an analytical approach for the first time. The intriguing dynamics observed from the system where further substantiated by phase portraits, Poincaré map, power spectra, and “0−1 test.” We trace the transition of the system from periodic to chaos through analytical solutions, which are in good agreement with hardware experiments. Additionally, we show PSpice circuit simulation results for validating our analytical and experimental studies.

Strange Nonchaotic Attractors with Cellular Neural Networks

1998 ◽  
Vol 08 (07) ◽  
pp. 1551-1556 ◽  
Author(s):  
Zhiwen Zhu ◽  
Shi Wang ◽  
Henry Leung
Keyword(s):  
Neural Network ◽  
Dynamical System ◽  
Poincaré Map ◽  
Amplitude Spectrum ◽  
Cellular Neural Network ◽  
Poincare Map ◽  
Fourier Amplitude ◽  
Sinusoidal Input ◽  
Strange Nonchaotic Attractors ◽  
Fourier Amplitude Spectrum

In this paper, we investigate the existence of strange nonchaotic attractors in a nonautonomous cellular neural network (CNN). The network consists of two cells, each cell being driven by a sinusoidal input and the frequencies of two inputs being incommensurate. Numerical analyses based on Lyapunov exponent, Poincare map, double Poincare map, Fourier amplitude spectrum, and information dimension show the existence of strange nonchaotic attractors in a substantial parameter space. It appears that the CNN is the first dynamical system with separate excitations having the strange nonchaotic attractors.

Behavior of a Self-Sustained Electromechanical Transducer and Routes to Chaos

2005 ◽  
Vol 128 (3) ◽  
pp. 282-293 ◽  
Author(s):  
J. C. Chedjou ◽  
K. Kyamakya ◽  
I. Moussa ◽  
H.-P. Kuchenbecker ◽  
W. Mathis
Keyword(s):  
Electronic Circuit ◽  
Initial Conditions ◽  
Dynamical Behavior ◽  
Period Doubling ◽  
Electromechanical System ◽  
Phase Portraits ◽  
Coupling Coefficients ◽  
Oscillatory Solutions ◽  
Electromechanical Transducer ◽  
Design Engineers

This paper studies the dynamics of a self-sustained electromechanical transducer. The stability of fixed points in the linear response is examined. Their local bifurcations are investigated and different types of bifurcation likely to occur are found. Conditions for the occurrence of Hopf bifurcations are derived. Harmonic oscillatory solutions are obtained in both nonresonant and resonant cases. Their stability is analyzed in the resonant case. Various bifurcation diagrams associated to the largest one-dimensional (1-D) numerical Lyapunov exponent are obtained, and it is found that chaos can appear suddenly, through period doubling, period adding, or torus breakdown. The extreme sensitivity of the electromechanical system to both initial conditions and tiny variations of the coupling coefficients is also outlined. The experimental study of̱the electromechanical system is carried out. An appropriate electronic circuit (analog simulator) is proposed for the investigation of the dynamical behavior of the electromechanical system. Correspondences are established between the coefficients of the electromechanical system model and the components of the electronic circuit. Harmonic oscillatory solutions and phase portraits are obtained experimentally. One of the most important contributions of this work is to provide a set of reliable analytical expressions (formulas) describing the electromechanical system behavior. These formulas are of great importance for design engineers as they can be used to predict the states of the electromechanical systems and respectively to avoid their destruction. The reliability of the analytical formulas is demonstrated by the very good agreement with the results obtained by both the numeric and the experimental analysis.

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