HSND-2015, IPR. Department of Physics, University of Burdwan, Burdwan, West Bengal.
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1 New kind of deaths: Oscillation Death and Chimera Death HSND-2015, IPR Dr. Tanmoy Banerjee Department of Physics, University of Burdwan, Burdwan, West Bengal.
2 Points to be discussed Oscillation suppression Amplitude death (AD) Oscillation death (OD) Nontrivial AD (newly discovered) AD-OD Transition: First Experimental evidence AD-NAD Transition (newly discovered) Chimera death Conclusion
3 Collective behavior Natural systems are rarely isolated Coupled Oscillators provide a useful paradigm for the study of collective behavior of large complex systems Coupled oscillators show a plethora of complex collective behaviors such as: Synchronization Phase-locking Oscillation quenching Weak coupling: phase effect Strong coupling: Amplitude effect Newton s law of motion
4 Amplitude Death (AD) Amplitude death (AD) is one of the intriguing phenomena that occurs in coupled oscillators when they interact in such a way as to suppress each others oscillations and collectively go to the stable fixed point that was unstable otherwise. In many practical situations AD is desirable, as for example in laser applications and neuronal activity where a constant output is needed and fluctuations should be suppressed. Bar-Eli who showed that two interacting model continuous stirred tank reactors can stop oscillations and arrive at steady states if coupled diffusively. Bar Eli effect
5 Consider two diffusively coupled Stuart-Landau oscillators Earlier studies of coupled Landau-Stuart oscillators by Aronson et al. have shown that amplitude death is only possible for instantaneous coupling when the intrinsic frequencies are disparate Thus to achieve AD parameter mismatch is essential in the instantaneous coupling.
6 Time-delay coupling: AD is possible in identical oscillators Pioneering work group of Prof. Sen [Reddy et al, PRL]. showed that with time delay, death occurs even for identical oscillators with Time delay is ubiquitous in real systems due to finite propagation speed of signals, finite reaction times of Chemical reactions, finite response time of synapses, etc.
7 Geometrical interpretation The current state P(t) is pulled towards the retarded state Q(t-τ) of the other oscillator and vice-versa. For appropriate values of K and time delay both oscillations will spiral inwards and die out. (Reddy, Sen and Johnston, Phys. Rev. Letts. 80 (1998) 5109; Physica D 129 (1999) 15 )
8 Experimental verification carried out on coupled nonlinear oscillators (Reddy, Sen, Johnston, PRL, 85 (2000) 3381)
9 Instantaneous coupling ALSO can induce AD in identical oscillator Conjugate coupling Dynamic coupling Mean-field coupling Mean-field coupling is one of the most widely studied topics because of its presence in many natural phenomena in the field of biology, physics, and engineering. In the genetic oscillators interacting through a quorum-sensing mechanism, the occurrence of OD is shown where the concentration of the autoinducer molecule that can diffuse through the cell membrane contains a mean-field term. In three dimensional Ising spin model the mean-field interaction explains ferromagnetism.
10 Amplitude Death Vs. Oscillation Death
11 There are two distinct types of oscillation quenching processes: amplitude death (AD) and oscillation death (OD). In AD coupled oscillators arrive at a common stable steady state which was unstable otherwise and thus form a stable homogeneous steady state (HSS). But, in the case of OD, oscillators populate different coupling dependent steady states and thus gives rise to stable inhomogeneous steady states (IHSS); in the phase space OD may coexist with limit cycle oscillations.
12 AD is important in the case of control applications where suppression of unwanted oscillations is necessary e.g., in Laser application, neuronal systems, etc. On the other hand, OD is a much more complex phenomenon because it induces inhomogeneity in a rather homogeneous system of oscillators that has strong connections and importance in the field of biology (e.g., synthetic genetic oscillator, cellular differentiation), physics, etc.
13 Although, AD and OD are two structurally different phenomena--their genesis and manifestations are different, but for many years they are (erroneously) treated in the same footing. Only recently pioneering works by Koseska et al established the much needed distinctions between AD and OD. Although an extensive research work has been reported on AD, but the phenomenon of OD is a less explored topic.
14 Koseska et al show: AD and OD can simultaneously occur in diffusively coupled Stuart-Landau oscillators. An important transition phenomenon, namely the transition between AD to OD in Stuart-Landau oscillators with parameter mismatch. They established that the transition occurs due to the interplay between the heterogeneity and the coupling parameter that is analogous with the Turing type bifurcation in spatially extended systems. The group of Dr. Shyamal Dana shows the transition between AD and OD in identical nonlinear oscillators that are coupled diffusively and perturbed by a symmetry breaking repulsive coupling link.
15 In the previous studies... It was believed that the mean-field coupling in oscillators can induce AD only For the first time we show that Mean-field coupling can induce OD and AD-OD transition even in identical oscillators.
16
17 We consider N Stuart-Landau oscillators interacting through mean-field diffusive coupling Equilibrium points
18 Eigenvalues of the trivial fixed point Two pitchfork bifurcations Two Hopf bifurcations
19 Bifurcation diagram using XPPAUT
20 A NEW AD state! NTAD state is unique in comparison with its conventional counterpart in, at least, two way: First, unlike AD that has two possible routes: Hopf and saddle-node bifurcation, the NTAD state is born via a subcritical pitchfork bifurcation. Second, in sharp contrast with the AD, which is supported or enhanced by parameter mismatch, the NT-AD state is completely destroyed by parameter mismatch. In the NT-AD state we have two different solutions: x1=x2, and -x1=-x2; occurrence of one of this two states is determined by the initial conditions. This initial condition dependent amplitude death state is not observed earlier.
21
22
23 Effect of parameter mismatch NTAD vanishes for ANY parameter mismatch
24 256 Stuart- Landau oscillators
25 The FIRST experimental observation of the AD to OD transition T. Banerjee and D. Ghosh, Experimental observation of a transition from amplitude to oscillation death in coupled oscillators, Phys. Rev. E, 89, 2014.
26 Although OD and AD have been separately observed previously in a number of different dynamical systems, to the best of our knowledge, hitherto no experimental confirmation of the predicted AD-OD transition has been reported. We report the first experimental evidence of AD-OD transition in coupled oscillators. For this we consider two Van der Pol oscillators in their stable oscillation mode coupled via mean-field diffusion.
27 a=0.35
28
29 AD OD For Q<Q*, AD to OD transition occurs Experimental OD For Q>Q* no AD, thus only limit cycle to OD Experimental
30 Theoretical and experimental results: close matching in parameter space
31 Direct-Indirect Coupling Induced AD, OD and NAD D. Ghosh and Tanmoy Banerjee (2014) "Transitions among the diverse oscillation quenching states induced by the interplay of direct and indirect coupling", Physical Review E, 90(6), DOI: /PhysRevE (arxiv preprint: [nlin.cd], 2014).
32 This coupling scheme was introduced by Resmi et al. [Phys. Rev. E, 84, (2011)] as a general scheme to induce amplitude death (AD) in nonlinear oscillators. We show that apart from AD, it can induce OD and a new AD state, namely NAD state and the following transitions: AD-OD transition AD-NAD transition This NAD state has not only a nonzero homogeneous steady state but, more significantly, in this state the system becomes bi-stable
33 Stuart-Landau oscillator Environmental or indirect coupling Diffusive or direct coupling
34
35 d=1.5
36 AD-OD AD-NAD
37 More fine structures
38 Experimental set up
39 Experimental observation
40 Chimera death Tanmoy Banerjee (2014) "Chimera death induced by the mean-field diffusive coupling", arxiv preprint: arxiv: [nlin.cd], 2014.
41 Recently a novel dynamical state, called the chimera death, is discovered in a network of non locally coupled identical oscillators [A. Zakharova, M. Kapeller, and E. Sch\"{o}ll, Phy.Rev.Lett. 112, (2014)], which is defined as the coexistence of spatially coherent and incoherent oscillation death state. This state arises due to the interplay of non locality and symmetry breaking and thus bridges the gap between two important dynamical states, namely the chimera and oscillation death. The chimera is an intriguing spatio-temporal dynamical state where the synchronous and asynchronous behavior are observed simultaneously in a network of coupled identical oscillators. After its discovery in phase oscillators the chimera state attracts immediate attention due to its surprisingly complex behavior and a possible connection with many real world phenomena such as unihemispheric sleep in certain species, the multiple time scales of sleep dynamics, etc.
42 Although both chimera and oscillation death state induce inhomogeneity in a rather homogeneous network of oscillators, but their inter connection was not identified until the recent pioneering work by Zakharova et al. The authors reported an important discovery of a new state, which they called the chimera death, and established the much speculated connection between the chimera and oscillation death. According to Zakharova et al, chimera death (CD) is the steady state version of chimera, i.e., the population of oscillators in a network splits into incongruous coexisting domains of spatially coherent OD (where neighboring nodes attain essentially the same branch of the inhomogeneous steady state) and spatially incoherent OD (where the neighboring nodes jump among the different branches of inhomogeneous steady state in a completely random sequence).
43 For the first time, we show that a nonlocal coupling is not an essential ingredient to achieve Chimera death.
44 IPS Global AD CIMERA DEATH
45 Multi-cluster Chimera death
46 It is noteworthy that the choice of initial condition is crucial for obtaining chimera but still it is not well understood (see, e.g., Ref.\cite{raj} where it is commented ``the role of initial conditions in the emergence of chimera states and the existence of multistability needs to be explored in the future"). We believe that this study will initiate the search for the chimera death in other coupling schemes that are not non local in topology and at the same time open up the research to engineer multi-cluster chimera death state in electronic and biological networks.
47 Conclusion OD is a new state in comparison with AD New kind of oscillation suppression states (NAD, Chimera death, etc.) await further exploration Studies on OD is still in its infancy and need further studies Continuation software package like XPPAUT is indispensible to study OD
48 Acknowledgements Prof. B. C. Sarkar (University of Burdwan) Research scholars: Mr. Debabrata Biswas Mrs. Debarati Ghosh Mr. Bishwajit Paul Mr. Biswajit Karmakar SERB (DST) for providing FAST TRACK RESEARCH PROJECT for young scientists
49 Thank you
DR. TANMOY BANERJEE. Contact Information
DR. TANMOY BANERJEE Assistant Professor Department of Physics University of Burdwan Burdwan 713104 West Bengal India Email: tbanerjee@phys.buruniv.ac.in, tanbanrs@yahoo.co.in URL: http://www.researchgate.net/profile/tanmoy_banerjee
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