A frequency-domain-based master stability function for synchronization in nonlinear periodic oscillators
TL;DR: The problem of investigating synchronization properties on periodic trajectories is reduced to an eigenvalue problem by means of the joint application of master stability function and harmonic balance techniques, and the method permits to exploit the periodicity of trajectories.
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Abstract: SUMMARY
An efficient methodology to study conditions for stable in-phase synchronization in networks of periodic identical nonlinear oscillators is proposed. The problem of investigating synchronization properties on periodic trajectories is reduced to an eigenvalue problem by means of the joint application of master stability function and harmonic balance techniques. The proposed method permits to exploit the periodicity of trajectories, reducing computational time with respect to traditional time-domain approaches (which were designed to deal with generic attractors) and good accuracy. In addition, such method can easily deal with networks of nonlinear periodic oscillators described by differential-algebraic equations, and then both static and dynamic coupling could be studied. Copyright © 2011 John Wiley & Sons, Ltd.
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Figures
Figure 4. Msf for Chua’s oscillator with dynamic coupling, with the 9 different configurations defined by (19). Panel i, j, in the row i from the top and column j from the left, refers to the case of component j influencing the component i, so that, for example, the panel (1, 2) refers to xn′2 influencing xn1. Only the frequency version obtained with Algorithm 1 is presented. 
Figure 3. Msf for Chua’s oscillator with static coupling, with the 9 different configurations defined by (19). Panel i, j, in the row i from the top and column j from the left, refers to the case of component j influencing the component i, so that, for example, the panel (1, 2) refers to xn′2 influencing xn1. Time domain version in solid line, frequency version in dashed line. 
Figure 8. Mean square error for a fully connected network of 11 Chua’s oscillators coupled both on the derivative and the right-hand side. The left panel refers to the case (1, 2), the central panel refers to the case (2, 2), and the case (3, 3) is considered in the right panel. 
Figure 7. MSF for Chua’s oscillator with static and dynamic coupling, with the 9 different configurations defined by (19). Panel i, j, in the row i from the top and column j from the left, refers to the case of component j influencing the component i, so that, for example, the panel (1, 2) refers to x2n′ influencing x1n. 
Figure 5. Mean square error, in a fully connected network of Chua’s oscillators with dynamic couplings, as a function of the coupling strength. The left panel refers to the second state variable influencing the first, first state variable influencing the second one on the right panel. 
Figure 6. Mean square error, in ring connected network of Chua’s oscillators with dynamic coupling, as a function of the coupling strength. The left panel refers to the second state variable influencing the first, first state variable influencing the second one on the right panel.
Citations
A fast technique for calculating master stability function
Shirin Panahi,Sajad Jafari +1 more
TL;DR: Investigating the stability of the synchronization manifold is a critical topic in the field of complex dynamical networks and Master stability function (MSF) is known as a powerful and efficient tool.
11
Synchronization analysis of networks of identical and nearly identical Chua's oscillators
Igor Mishkovski,Miroslav Mirchev,Fernardo Corinto,Mario Biey +3 more
- 20 May 2012
TL;DR: It is shown that losses can lower down the coupling bounds for which a given network of oscillators synchronizes, and by using the extended MSF losses reduce the synchronization error when the oscillators are nearly identical.
3
Synchronization in Complex Networks: Properties and Tools
Mario Biey,Fernando Corinto,Igor Mishkovski,Marco Righero +3 more
- 01 Jan 2013
TL;DR: In this chapter, the subject of synchronization is introduced and discussed considering the effects due to network topology, several of the most popular topologies are considered, showing their influence on network synchronizability.
Analysis of a four-wing fractional-order chaotic system via frequency-domain and time-domain approaches and circuit implementation for secure communication
TL;DR: A new chaos circuit based on the four-wing fractional-order chaotic systems is designed to implement the synchronization scheme, and the effectiveness and feasibility of the proposed synchronization scheme are verified by the new analog circuit.
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