In Conjugate-coupled Nonlinear Oscillator Networks, Stability of Amplitude Death

Susan Kimberly^*
*Correspondence: Susan Kimberly, Department of Physics, University of Texas, TX, USA, Email:

Introduction

Numerous unexpected phenomena may manifest when nonlinear dynamical systems are connected, depending on the dynamics of the underlying systems and how the connection is set up. An important emergent phenomenon in this context is the total suppression of oscillations, often known as amplitude death. As a result of the contact, the entire system's oscillations stop, resulting in stationary behaviour [1]. The uncoupled system's otherwise unstable fixed points or brand-new stationary points may be the fixed points that the coupling stabilises. Such behaviour has use in a variety of fields, including laser physics and the dynamics of biological systems. The characteristics of the various coupling tactics and scenarios that result in AD in a range of different contexts are covered.

Description

A key topic of interest in many fields of science is the nature of the dynamics of coupled nonlinear systems. Oscillators are used to simulate a wide range of natural phenomena, including the movement of pendulums and springs, lasers, fluid flow, ecological systems, population growth, and more. Depending on the situation, these may be linear or nonlinear, and as is well known, the motion may be periodic or chaotic [2]. Numerous unique phenomena can be observed when systems that are capable of complicated behaviour when operating alone are connected. These depend on both the characteristics of the isolated systems, the type of nonlinearity, the type of motion chaotic or regular, the existence of equilibria, etc., as well as the suppression of oscillations, which is formally known as amplitude death as a result of the interaction oscillations of the entire system stop, leading to stationarity, is a significant emergent event in this setting. Such behaviour is emergent in the sense that stationary dynamics are not present in the isolated or uncoupled systems. In a formal sense, amplitude death is an example of a more general process that encompasses both the suppression of amplitude changes and the termination of oscillations as a result of coupling. This is the stabilising or development of simpler attractors by coupling from chaotic oscillations.

There are two outcomes that could happen after oscillations halt. The occurrence of amplitude death implies that, if the connected systems have exactly one equilibrium, then this equilibrium becomes asymptotically stable. The stabilisation of novel fixed points, i.e., those that are unstable or may not even exist in the uncoupled system, can occur in coupled systems, though, as they are capable of having more than one stationary state [3]. We start by taking into account the various known instances in which is known to occur, along with limitations and prerequisite conditions. Identifying the various interactions that facilitate is one area of interest. This is especially important in natural systems where it may be impossible to access the parameters of individual systems and where amplitude death can be attained primarily by choosing the right types of interaction. Additionally, it's likely that when internal settings are changed, a specific kind of the interaction happens. As a comparison, think of the synchronisation of two systems, a related process.

Systems synchronise as a result of specific types of contact, and for a particular type of coupling diffusive, say synchronicity is only achievable when the coupling strength is large enough. Consideration of the situation where the time delay itself varies follows naturally from the assumption that signals from one subsystem must travel at a finite speed to reach the other. In cases where the delay itself is modelling stochastic effects, this is very crucial [4]. Examined the impact of a time delay distribution and shown how even a tiny spread in the delay distribution can significantly expand the range of parameters for which amplitude death occurs. Since this AD zone might become unbounded if the variance of the distribution is beyond a certain threshold, the extension of the AD region with a distribution of delays is a significant effect with many applications. Once more, the effect is fairly generic and occurs for arbitrary delay spread, for various distributional shapes, and for arbitrary numbers of oscillators. The scenario in which systems react to cumulative signals, i.e., by integrating data obtained over a period of time, is closely related to the case of distributed delays. In addition to causing the region of to expand forever, this happens when systems have a finite intrinsic response time [5].

Conclusion

The study of many features of both fixed and distributed delays, including a deterministic time variable interaction, has been done since time-delay interactions are frequently seen in studies. As an example, the stabilisation of fixed the concept of amplitude death has also been used to examine velocity delayed coupling. Mixed interaction structure, with immediate recent investigations have looked at interactions in one system and time delay in the other, or unequal time delays in several directions to mimic spatial heterogeneity and one way ring time delay. The study of nonlinear dynamics revolves around aiming for a certain steady state, especially when stabilising low-dimensional dynamics like fixed points or periodic orbits. Targeting fixed point solutions by taking into account certain interactions has been discussed in considerable detail. If one wants to exploit the phenomenon in actual applications, this will be necessary. Targeted fixed power outputs in linked laser systems can be useful for laser welding and manufacture, laser surgery, and other laserrelated fields. Is to be avoided in various practical applications, such as in brain function; we have also evaluated strategies for the trustworthy avoidance of. However, due to the fact that many natural systems have specialised and limited coupling patterns, such as the either the diffusive interaction between nonlinear chemical systems or synaptic coupling between neurons.

References

1. Han, Wenchen, Hongyan Cheng, Qionglin Dai and Haihong Li, Ping Ju."Amplitude death, oscillation death, wave, and multistability in identical Stuart–Landau oscillators with conjugate coupling." Commun Nonli Sci Numeri Simula 39 (2016): 73-80.

Google Scholar, Crossref

2. Karnatak, Rajat, Ram Ramaswamy and Awadhesh Prasad. "Amplitude death in the absence of time delays in identical coupled oscillators." Phys Rev E 76 (2007): 035201.

Indexed at, Google Scholar, Crossref

3. Saxena, Garima, Awadhesh Prasad and Ram Ramaswamy. "Amplitude death: The emergence of stationarity in coupled nonlinear systems." Phys Rep 521 (2012): 205-228.

Google Scholar, Crossref

4. Xiao, Rui, Ling Wei Kong, Zhong-Kui Sun and Ying Cheng Lai. "Predicting amplitude death with machine learning." Phys Rev E 104 (2021): 014205.

Indexed at, Google Scholar, Crossref

5. Sugitani, Yoshiki, Keiji Konishi and Naoyuki Hara. "Experimental verification of amplitude death induced by a periodic time-varying delay-connection." Nonli Dyn 70 (2012): 2227-2235.

Indexed at, Google Scholar, Crossref