Literature review

The topic of the coupled chaotic system has become very common in scientific research. The synchronization process of the coupled chaotic system has been in use in the last several years. The mathematical exploration of the dynamic system that includes interior crisis transition and periodic doubling revealed several scenarios of the bifurcations. Whenever there is coupling that is magnetic in nature, there is an observation of other kinds of dynamic phenomena including the coexistence of the solution, transient chaos, and multistability.

There is normally a provision for the reasons considered theoretical for the observation besides the experimental confirmations through the practical measurements that exist in the wireless transfer. In most of the cases, the illustrations are the possibility of getting the mechanism that controls and also synchronizes the system in a very simple and also effective manner that uses the high-frequency oscillations.

Chen; (2015). Robust finite-time chaos synchronization of uncertain permanent magnet synchronous motors. ISA transactions, 58, 262-269.

Several proposals have been put forward according to Chen to assist in the study of the dynamic and synchronization of the coupled systems by the use of the cheap, simple and easy techniques. The specific part that needs to be addressed includes the impacts of the magnetic coupling on the process of the synchronization of the two Colpitts oscillators. The description of the dynamic system is achieved by the use of the smooth model of the mathematics. There is also an investigation that seeks to establish the stability of the state that is balanced or that is in the state of equilibrium. The system that is coupled is normally accompanied by the chaos and the hyperchaos in the exact range of the measurements.

Deng, Z.(2015). Synchronization controller design of two coupling permanent magnet synchronous motors system with nonlinear constraints. ISA transactions, 59, 243-255.

 In the case of the nonlinear dynamics and in the electronics the modeling of the oscillators is done without the use of the coupling from the external magnetic fielding. This is according to Deng. In some cases, there may be interactions that involve the systems which have been coupled physically. This particular case is common in specific conditions of the experiment. In this particular study, the attention is given to the impacts of the magnetic coupling on the characteristics of the Colpitts coupled oscillator.

Kana, L. K., Fomethe, A., Fotsin, H. B., Wembe, E. T., & Moukengue, A. I. (2017). Complex Dynamics and Synchronization in a System of Magnetically Coupled Colpitts Oscillators. Journal of Nonlinear Dynamics, 2017.

Chen; (2015). Robust finite-time chaos synchronization of uncertain permanent magnet synchronous motors. ISA transactions, 58, 262-269.

In this book, the chaos synchronization refers to the characteristic behavior whereby more than two systems which have been coupled together indicate the very high index of similarity of the chaotic oscillations. While considering the effects of similar systems of the dynamics, the loss of the synchronization that is found between the smaller sub-systems is attributed to the Lyapunov exponents of the entire system globally. As per the article of the “Renewable and sustainable energy,” the process or the technique of the synchronization has assisted in the energy saving generally. Whenever there is a loss in the synchrony an automatic transition from chaotic to the hyperchaotic behavior.

Kengne, J., Chedjou, J. C., Kenne, G., & Kyamakya, K. (2012). Dynamical properties and chaos synchronization of improved Colpitts oscillators. Communications in Nonlinear Science and Numerical Simulation, 17(7), 2914-2923.

There have been proposals of several solutions in this book in connection to the challenge some of which include adaptive control, active control, active-backstepping and adaptive backstepping. According to Kengne et al most of these proposals are characterized by controllers and design aspect that is very difficult and very complex hence not easy to achieve. This has been illustrated in the book of the “Transaction on Industrial Electronics” .

Due to the already identified complexities and the difficulties in the existing systems, there is needed to introduce a method that is numerically and experimentally approved to be simple. The research seeks to put forward a very simple, cheap and easy technique that can be used effectively in the dynamic and subsequent synchronization of the Colpitts coupled oscillator.

Li, L. B., Sun, L. L., Zhang, S. Z., & Yang, Q. Q. (2015). Speed tracking and synchronization of multiple motors using ring coupling control and adaptive sliding mode control. ISA transactions, 58, 635-649.

 In the general set up, the metastable systems are characterized by very high level of the complexity in the behavior of the varying systems. The books illustrates this effect comprehensively.

This kind of dynamic system is effectively elaborated in the topic of the “Transactions for Vehicular technology” Normally these results from the interactions among the coexisting connections. The connections are sometimes referred to the attractions. Due to the existence of various interactions that exist among the attractors, the metastable system is known to be extremely sensitive when they are exposed to the original conditions. In addition, the existence of the different attractors and complex fractal basins make the influence of the system to emanate from the state considered to be very insignificant.

Deng, Z.(2015). Synchronization controller design of two coupling permanent magnet synchronous motors system with nonlinear constraints. ISA transactions, 59, 243-255.

Rabbi, S. F., & Rahman, M. A. (2014). Critical criteria for successful synchronization of line-start IPM motors. IEEE Journal of Emerging and Selected Topics in Power Electronics, 2(2), 348-358.

On the other hand, the characteristics of the system will tend to change qualitatively whenever there is a change in the parameters of the system. The occurrence of the attractors is as a result of the minor intervals within the system boundary. Any slight variation in the already set parameter may lead to rapid improvement in the system of the attractors. The sensitivity of the metastable to the noise levels is very high as indicated in this particular book.

Ruviaro, M., Runcos, F., Sadowski, N., & Borges, I. M. (2012). Analysis and test results of a brushless doubly fed induction machine with rotary transformer. IEEE Transactions on Industrial Electronics, 59(6), 2670-2677.

This is effectively illustrated in the book “Transaction on the power” The high level of noise may sometimes result into the popping effect that exists between different attractors. In the designed system, the parameters in the design were influenced by the hyperchaos and the chaos. In any selection of the magnetic coupling, the constant value has to be equivalent to, K=0.1. However, whenever the value of K translates to 0.6, there will be a generation of the phenomenon of the transient chaos which is equally coupled with the Colpitts oscillator.

Tchitnga, R., Fotsin, H. B., Nana, B., Fotso, P. H. L., & Woafo, P. (2012). Hartley’s oscillator: The simplest chaotic two-component circuit. Chaos, Solitons & Fractals, 45(3), 306-313.

Fundamental bits of knowledge that are associated with the sets considered attractive may be required to exist together in the framework. This can be picked up by accomplishing an assessment of the current volume of the oscillator as explained bin this book.

As indicated by the condition that is demonstrated as follows, it is apparent that any underlying volume of the component will ceaselessly be shrunk by the streaming straightforward terms, every one of the volume components will really psychologist to zero as the time slipped by. Likewise, all the framework circles will be restricted to an explicit limit that is subset to zero volume in the periods of the accessible space. This will at long last make an asymptotic movement to meet at the purpose of the beginning. The equation illustrates the boundary conditions for the dynamic characteristics. 

Vaidyanathan, S., Volos, C. K., & Pham, V. T. (2015). Analysis, control, synchronization and SPICE implementation of a novel 4-D hyperchaotic Rikitake dynamo system without equilibrium. Journal of Engineering Science and Technology Review, 8(2), 232-244.

The solutions of the systems are done numerically in order to establish the routes that link to the chaos within the model as explained in this book. The setting of the time base in most of the model has been at 0.005 in most of the parameters used in the system. The procedures are computed using the variables that are within the extension of the model. In every parameter, there is an intergradation of the system for a relatively long time. The effect of setting a long time is that it produces the process of discarding the transient. In general, there are basically two indicators which are used in the identification of the kind of the transition. The bifurcation diagram is regarded as the first indicator. The second indicator is the dimensional numerical Lyapunov exponent whose definition is dependent on the individual parameters. 

Vasiljevic, N., Courtney, M., & Mann, J. (2014). A time-space synchronization of coherent Doppler scanning lidars for 3D measurements of wind fields.

For the sensitivity of the oscillator to be investigated in the cases of the minor changes, there is a need to carry out the process of the scanning. The monitoring of the bifurcation is done using the specific range. The diagram of the bifurcation is achieved by plotting of the coordinates of the minima and maxima along the selected lines. The results obtained clearly shows that any system can just experience the behavior that is striking and this includes the chaotic motion and also harmonic motion. This can effectively be used in the monitoring of the control parameters.

References

Chen, Q., Ren, X., & Na, J. (2015). Robust finite-time chaos synchronization of uncertain permanent magnet synchronous motors. ISA transactions, 58, 262-269.

Deng, Z., Shang, J., & Nian, X. (2015). Synchronization controller design of two coupling permanent magnet synchronous motors system with nonlinear constraints. ISA transactions, 59, 243-255.

Kana, L. K., Fomethe, A., Fotsin, H. B., Wembe, E. T., & Moukengue, A. I. (2017). Complex Dynamics and Synchronization in a System of Magnetically Coupled Colpitts Oscillators. Journal of Nonlinear Dynamics, 2017.

Kengne, J., Chedjou, J. C., Kenne, G., & Kyamakya, K. (2012). Dynamical properties and chaos synchronization of improved Colpitts oscillators. Communications in Nonlinear Science and Numerical Simulation, 17(7), 2914-2923.

Li, L. B., Sun, L. L., Zhang, S. Z., & Yang, Q. Q. (2015). Speed tracking and synchronization of multiple motors using ring coupling control and adaptive sliding mode control. ISA transactions, 58, 635-649.

Rabbi, S. F., & Rahman, M. A. (2014). Critical criteria for successful synchronization of line-start IPM motors. IEEE Journal of Emerging and Selected Topics in Power Electronics, 2(2), 348-358.

Ruviaro, M., Runcos, F., Sadowski, N., & Borges, I. M. (2012). Analysis and test results of a brushless doubly fed induction machine with rotary transformer. IEEE Transactions on Industrial Electronics, 59(6), 2670-2677.

Tchitnga, R., Fotsin, H. B., Nana, B., Fotso, P. H. L., & Woafo, P. (2012). Hartley’s oscillator: The simplest chaotic two-component circuit. Chaos, Solitons & Fractals, 45(3), 306-313.

Vaidyanathan, S., Volos, C. K., & Pham, V. T. (2015). Analysis, control, synchronization and SPICE implementation of a novel 4-D hyperchaotic Rikitake dynamo system without equilibrium. Journal of Engineering Science and Technology Review, 8(2), 232-244.

Vasiljevic, N., Courtney, M., & Mann, J. (2014). A time-space synchronization of coherent Doppler scanning lidars for 3D measurements of wind fields.