Studying the Effect of Initial Conditions and System Parameters on the Behavior of a Chaotic Duffing System (original) (raw)
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This study exploited the computation accuracy of governing equations of linearly or periodically behaves dynamic system with fourth and fifth order Runge-Kutta algorithms to develop chaos diagrams of harmonically excited Duffing oscillator. The study adopt the fall to tolerance of absolute deviation between two independently sought solutions of governing equation to characterise excitation frequencies and amplitude parameter point of Duffing oscillator as either chaotic or not. Displacement and Velocity time history, Phase plot and Poincare were used to validate FORTRAN coded programmes used for this study and chaos diagrams developed at two different damping coefficients. The validation results agreed perfectly with those obtained in the literature. The chaos diagrams predicted by computation at two different damp coefficient levels conforms generally in trend to literature results by and qualitatively the same for three different combination of constant time based Runge-Kutta algorithms. The chances of chaotic behaviour of Duffing oscillator under the combined driven force of parameters becomes more than double at 0.0168 damping coefficient when compared with corresponding results at 0.168 damping coefficient. The probability that selected excitation frequencies and amplitudes will drive Duffing oscillator chaotically at 0.168 damp coefficients is 29.4%, 27.8% and 29.4% respectively. This study demonstrated the significant utility of numerical techniques in dealing with real-world problems that are dominantly nonlinear and shows that in addition to being sensitive to initial conditions, chaos is equally sensitivity to appropriate simulation time steps. In addition, the present chaos diagram generating numerical tool is uniquely characterised by being faster and predicting reliably than that earlier reported by the authors.
Chaotic cryptosystem based on inverse duffing circuit
2006
We have studied experimentally a chaotic cryptosystem, which is based on the inverse system approach. We applied this method to a second order nonlinear circuit (master circuit), which is described by a Duffing equation. We present the implementation of the slave circuit with the inverse system approach and we demonstrate the decryption when the information signal has several forms (sinusoidal and rectangular). By appropriate choice of the parameters of the nonlinear oscillator, and the signal oscillator, it is possible to have a cryptosystem capable of transmitting information securely and recovering it accurately.
A chaotic secure communication scheme based on duffing oscillators and frequency estimation
2013
This work presents a new technique to securely transmit and retrieve a message signal via chaotic systems. In our system, a two-valued message signal modulates the frequency of a Duffing oscillator sinusoidal term. An observer is used in the receiver side to retrieve the sinusoidal signal that contains the message and a novel frequency estimator is then used to reproduce an approximated estimation of the message signal. The performance of the system is analyzed by means of numerical simulations performed in Matlab.
New periodic-chaotic attractors in slow-fast Duffing system with periodic parametric excitation
Scientific Reports, 2019
A new type of responses called as periodic-chaotic motion is found by numerical simulations in a Duffing oscillator with a slowly periodically parametric excitation. The periodic-chaotic motion is an attractor, and simultaneously possesses the feature of periodic and chaotic oscillations, which is a new addition to the rich nonlinear motions of the Duffing system including equlibria, periodic responses, quasi-periodic oscillations and chaos. In the current slow-fast Duffing system, we find three new attractors in the form of periodic-chaotic motions. These are called the fixed-point chaotic attractor, the fixed-point strange nonchaotic attractor, and the critical behavior with the maximum Lyapunov exponent fluctuating around zero. The system periodically switches between one attractor with a fixed single-well potential and the other with time-varying two-well potentials in every period of excitation. This behavior is apparently the mechanism to generate the periodic-chaotic motion.
Comparative Analysis of Numerically Computed Chaos Diagrams in Duffing Oscillator
This study utilised optimum fractal disk dimension algorithms to characterize the evolved strange attractor (Poincare section) when adaptive time steps Runge-Kutta fourth and fifth order algorithms are employed to compute simultaneously multiple trajectories of a harmonically excited Duffing oscillator from very close initial conditions. The challenges of insufficient literature that explore chaos diagrams as visual aids in dynamics characterization strongly motivate this study. The object of this study is to enable visual comparison of the chaos diagrams in the excitation amplitude versus frequency plane. The chaos diagrams obtained at two different damp coefficient levels conforms generally in trend to literature results[1] and qualitatively the same for all algorithms. The chances of chaotic behaviour are higher for combined higher excitation frequencies and amplitudes in addition to smaller damp coefficient. Fourth and fifth order Runge-Kutta algorithms indicates respectively 62.3% and 53.3% probability of chaotic behaviour at 0.168 damp coefficient and respectively 77.9% and 78.9% at 0.0168 damp coefficient. The chaos diagrams obtained by fourth order algorithms is accepted to be more reliable than its fifth order counterpart, its utility as tool for searching possible regions of parameter space where chaotic behaviour/motion exist may require additional dynamic behaviour tests.
Experimental demonstration of a chaotic cryptographic scheme
2006
In this paper we demonstrate an experimental chaotic cryptosystem. This cryptosystem is based on the inverse system approach, which we apply to a second order nonlinear circuit, described by a Duffing equation. We present the implementation of the electronic circuit obeying Duffing's equation, and the slave circuit which is implemented with the inverse system approach. By choosing several forms for the information signal (sinusoidal, triangular and rectangular) we demonstrate the good performance of the proposed cryptosystem. Finally, we conclude that the method works well enough, when the system oscillates with frequencies that are much higher than the characteristic frequencies of the information signal.
Communications in Nonlinear Science and Numerical Simulation, 2014
This work presents a new technique to securely transmit and retrieve a message signal via chaotic systems. The main contribution of this paper is twofold: the way that the message signal is encrypted in the frequency of a sinusoidal term and a novel frequency estimator for retrieving the message. In our system, a two-valued message signal modulates the frequency of the Duffing oscillator sinusoidal term. Then, two chaotic signals generated by the oscillator are encrypted with a Delta modulator and sent through a noisy channel. A Lyapunov-based observer is used in the receiver side to retrieve the sinusoidal term that contains the message and a novel frequency estimator is then used to retrieve the confidential message signal. The system was implemented in Matlab/Simulink in order to analyze its performance.
A Novel Chaotic System for Secure Communication Applications
Information Technology And Control, 2015
The secure communication using synchronization between identical chaotic systems have been introduced in literature for a long time. A well-known practical application of chaotic synchronized systems is the Pecora and Carroll (P-C) secure communication method. In this paper, the P-C secure communication algorithm is applied to a novel three dimensional, autonomous chaotic attractor. Having a 45 ○ slope between sub-driver and subreceiver circuits of a novel chaotic attractor clearly demonstrates that it can be used for the purpose of secure communications.
Frequency–driven chaos in the electrical circuit of Duffing-Holmes oscillator and its control
Iranian Journal of Physics Research, 2018
Accurate detection of weak periodic signals within noise and the possibility of secure messaging have made Duffing oscillator (DO) highly important in the field of communication. Investigation on the properties of DO is, thus, very important. An elegant approach to accomplish this is to fabricate electronic circuit simulating DO non-linear equation and to study the effect of input signal amplitude (Vin) and frequency (f), disentangling these two from each other. Recently, Vin-driven chaotic dynamics has been studied by constructing a simple Duffing-Holmes (DH) oscillator circuit. However, the f-driven characteristics of the oscillator remain unknown at constant Vin. The present work is based on the MATLAB simulation of the f-driven chaotic dynamics of the DH equation. Similar output, mixed with chaos and non-chaos, is obtained by constructing the circuit, both in lab and by PSPICE simulation. The circuit moves into complete chaos at f=270 Hz, while period-2 bifurcation appears at f=680 Hz for the constant Vin 0.9V. The chaos control is also achieved by two simple methods. In the first method, the variation of the circuit parameter (capacitance) induces chaos control. In the second one, synchronization is achieved by coupling two similar oscillators. These two methods, though apparently simple, could be highly beneficial for using DH in secure communication.
The Analysis of Periodic Signal Detection Method Based on Duffing System Chaotic Dynamics
Vìsnik Nacìonalʹnogo tehnìčnogo unìversitetu Ukraïni "Kììvsʹkij polìtehnìčnij ìnstitut", 2018
This article presents the analysis of periodic signal detection method based on Duffing system sensitivity to weak influences. The described signal detection method is developed with using of Duffing system that oscillates in chaotic state, without transitions to periodic state. The main advantage of such method is the absence of periodic oscillation modes with low sensitivity. The divergence of Duffing system phase trajectories is investigated with influences of different periodic signals under low signal-to-noise ratio values. The estimation of phase trajectories divergence is performed with using of numeric integration. The signal detection method is analyzed with different forms of input signal: sinusoidal, square, triangle. The analysis shows that a reliable detection of periodic signal can be performed for any of the three presented forms of signal with repeating frequency near the frequency of the driving signal. The obtained results show wide capabilities of Duffing system applications for detection of weak periodic signals.