Recent Trends of Controlling Chaotic Resonance and Future Perspectives

Stochastic resonance is a phenomenon in which the effects of additive noise strengthen the signal response against weak input signals in non-linear systems with a specific barrier or threshold. Recently, several studies on stochastic resonance have been conducted considering various engineering applications. In addition to additive stochastic noise, deterministic chaos causes a phenomenon similar to the stochastic resonance, which is known as chaotic resonance. The signal response of the chaotic resonance is maximized around the attractor-merging bifurcation for the emergence of chaos-chaos intermittency. Previous studies have shown that the sensitivity of chaotic resonance is higher than that of stochastic resonance. However, the engineering applications of chaotic resonance are limited. There are two possible reasons for this. First, the stochastic noise required to induce stochastic resonance can be easily controlled from outside of the stochastic resonance system. Conversely, in chaotic resonance, the attractor-merging bifurcation must be induced via the adjustment of internal system parameters. In many cases, achieving this adjustment from outside the system is difficult, particularly in biological systems. Second, chaotic resonance degrades owing to the influence of noise, which is generally inevitable in real-world systems. Herein, we introduce the findings of previous studies concerning chaotic resonance over the past decade and summarize the recent findings and conceivable approaches for the reduced region of orbit feedback method to address the aforementioned difficulties.

Single-cell Networks Reorganise to Facilitate Whole-brain Supercritical Dynamics During Epileptic Seizures

Excitation-inhibition (EI) balance may be required for the organisation of brain dynamics to a phase transition, criticality, which confers computational benefits. Brain pathology associated with EI imbalance may therefore occur due to a deviation from criticality. However, evidence linking critical dynamics with EI imbalance-induced pathology is lacking. Here, we studied the effect of EI imbalance-induced epileptic seizures on brain dynamics, using in vivo whole-brain 2-photon imaging of GCaMP6s larval zebrafish at single-neuron resolution. We demonstrate the importance of EI balance for criticality, with EI imbalance causing a loss of whole-brain critical statistics. Using network models we show that a reorganisation of network topology drives this loss of criticality. Seizure dynamics match theoretical predictions for networks driven away from a phase transition into disorder, with the emergence of chaos and a loss of network-mediated separation, dynamic range and metastability. These results demonstrate that EI imbalance drives a pathological deviation from criticality.

Probing the edge between integrability and quantum chaos in interacting few-atom systems

Interacting quantum systems in the chaotic domain are at the core of various ongoing studies of many-body physics, ranging from the scrambling of quantum information to the onset of thermalization. We propose a minimum model for chaos that can be experimentally realized with cold atoms trapped in one-dimensional multi-well potentials. We explore the emergence of chaos as the number of particles is increased, starting with as few as two, and as the number of wells is increased, ranging from a double well to a multi-well Kronig-Penney-like system. In this way, we illuminate the narrow boundary between integrability and chaos in a highly tunable few-body system. We show that the competition between the particle interactions and the periodic structure of the confining potential reveals subtle indications of quantum chaos for 3 particles, while for 4 particles stronger signatures are seen. The analysis is performed for bosonic particles and could also be extended to distinguishable fermions.

Chaotic Dynamics and Complexity in Real and Physical Systems

Emergence of chaos and complex behavior in real and physical systems has been discussed within the framework of nonlinear dynamical systems. The problems investigated include complexity of Child’s swing dynamics , chaotic neuronal dynamics (FHN model), complex Food-web dynamics, Financial model (involving interest rate, investment demand and price index) etc. Proper numerical simulations have been carried out to unravel the complex dynamics of these systems and significant results obtained are displayed through tables and various plots like bifurcations, attractors, Lyapunov exponents, topological entropies, correlation dimensions, recurrence plots etc. The significance of artificial neural network (ANN) framework for time series generation of some dynamical system is suggested.

Nonlinear phenomena in models of the circadian clock

The mammalian circadian clock is well-known to be important for our sleep–wake cycles, as well as other daily rhythms such as temperature regulation, hormone release or feeding–fasting cycles. Under normal conditions, these daily cyclic events follow 24 h limit cycle oscillations, but under some circumstances, more complex nonlinear phenomena, such as the emergence of chaos, or the splitting of physiological dynamics into oscillations with two different periods, can be observed. These nonlinear events have been described at the organismic and tissue level, but whether they occur at the cellular level is still unknown. Our results show that period-doubling, chaos and splitting appear in different models of the mammalian circadian clock with interlocked feedback loops and in the absence of external forcing. We find that changes in the degradation of clock genes and proteins greatly alter the dynamics of the system and can induce complex nonlinear events. Our findings highlight the role of degradation rates in determining the oscillatory behaviour of clock components, and can contribute to the understanding of molecular mechanisms of circadian dysregulation.

The Emergence of Chaos in Quantum Mechanics

Nonlinearity in Quantum Mechanics may have extrinsic or intrinsic origins and is a liable route to a chaotic behaviour that can be of difficult observations. In this paper, we propose two forms of nonlinear Hamiltonian, which explicitly depend upon the phase of the wave function and produce chaotic behaviour. To speed up the slow manifestation of chaotic effects, a resonant laser field assisting the time evolution of the systems causes cumulative effects that might be revealed, at least in principle. The nonlinear Schrödinger equation is solved within the two-state approximation; the solution displays features with characteristics similar to those found in chaotic Classical Mechanics: sensitivity on the initial state, dense power spectrum, irregular filling of the Poincaré map and exponential separation of the trajectories of the Bloch vector σ in the Bloch sphere.

Modelling chance and necessity in natural systems

Nearly 30 years ago, emerged the concept of deterministic chaos. With it came sensitivity to initial conditions, nonlinearities, and strange attractors. This constituted a paradigm shift that profoundly altered how numerical modellers approached dynamic systems. It also provided an opportunity to resolve a situation of mutual misunderstanding between scientists and non-scientists about uncertainties and predictability in natural systems. Our proposition is that this issue can be addressed in an original way which involves modelling based on the principles of chance and necessity (CaN). We outline the conceptual and mathematical principles of CaN models and present an application of the model to the Barents Sea food-web. Because CaN models rely on concepts easily grasped by all actors, because they are explicit about knowns and unknowns and because the interpretation of their results is simple without being prescriptive, they can be used in a context of participatory management. We propose that, three decades after the emergence of chaos theories, CaN can be a practical step to reconcile scientists and non-scientists around the modelling of structurally and dynamically complex natural systems, and significantly contribute to ecosystem-based fisheries management.