Research Methodology on Magnetically Coupled Colpitts Oscillators

This report provides an overview of research on synchronization and complex dynamics in magnetically coupled Colpitts oscillators. It explores the mathematical modeling of dynamic systems, including bifurcation scenarios and the impact of magnetic coupling on synchronization. The study investigates the stability of equilibrium states and the occurrence of chaos and hyperchaos. The report discusses various techniques for studying coupled systems, with a focus on the effects of magnetic coupling on Colpitts oscillators and the challenges in designing effective controllers. It highlights the sensitivity of metastable systems to initial conditions and noise levels, emphasizing the importance of understanding these dynamics. The report also references various publications that support these findings and illustrates the relationship between the systems' parameters and their characteristics, and the importance of these factors on the system's behavior. It aims to present a simple and effective technique for studying the dynamics and subsequent synchronization of the Colpitts coupled oscillator.

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A topic for the Research
Synchronization and Complex Dynamics in a system of Magnetically Coupled Colpitts
Oscillators
Introduction
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.
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.
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.
Figure 1: Bifurcation diagram [3]
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.

Figure 2: Bifurcation graphs

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.