Detection meeting control: Unstable steady states in high-dimensional nonlinear dynamical systems

Adaptive delayed feedback control for stabilizing unstable steady states

Delayed feedback control is a commonly used control method for stabilizing unstable periodic orbits and unstable steady states. The present paper proposes an adaptive tuning delay time rule for delayed feedback control focused on stabilizing unstable steady states. The rule is designed to slowly vary the delay time, increasing the difference between the past and current states of dynamical systems, which induces the delay time to automatically fall into the stability region. We numerically confirm that the tuning rule works well for the Stuart-Landau oscillator, FitzHugh-Nagumo model, and Lorenz system.

Modulating Biological Rhythms: A Noncomputational Strategy Harnessing Nonlinearity and Decoupling Frequency and Amplitude

Understanding and achieving concurrent modulation of amplitude and frequency, particularly adjusting one quantity and simultaneously sustaining the other at an invariant level, are of paramount importance for complex biophysical systems, including the signal pathway where different frequency indicates different upstream signal yielding a certain downstream physiological function while different amplitude further determines different efficacy of a downstream output. However, such modulators with clearly described and universally valid mechanisms are still lacking. Here, we rigorously propose an easy-to-use control strategy containing only one or two steps, leveraging the nonlinearity in the modulated systems to decouple frequency and amplitude in a noncomputational manner. The strategy's efficacy is demonstrated using representative biochemical systems and, thus, it could be potentially applicable to modulating rhythms in experiments of biochemistry and synthetic biology.

Closed-Loop Control of Complex Networks: A Trade-Off between Time and Energy

Controlling complex nonlinear networks is largely an unsolved problem at the present. Existing works focus either on open-loop control strategies and their energy consumptions or on closed-loop control schemes with an infinite-time duration. We articulate a finite-time, closed-loop controller with an eye toward the physical and mathematical underpinnings of the trade-off between the control time and energy as well as their dependence on the network parameters and structure. The closed-loop controller is tested on a large number of real systems including stem cell differentiation, food webs, random ecosystems, and spiking neuronal networks. Our results represent a step forward in developing a rigorous and general framework to control nonlinear dynamical networks with a complex topology.

Basin stability measure of different steady states in coupled oscillators

In this report, we investigate the stabilization of saddle fixed points in coupled oscillators where individual oscillators exhibit the saddle fixed points. The coupled oscillators may have two structurally different types of suppressed states, namely amplitude death and oscillation death. The stabilization of saddle equilibrium point refers to the amplitude death state where oscillations are ceased and all the oscillators converge to the single stable steady state via inverse pitchfork bifurcation. Due to multistability features of oscillation death states, linear stability theory fails to analyze the stability of such states analytically, so we quantify all the states by basin stability measurement which is an universal nonlocal nonlinear concept and it interplays with the volume of basins of attractions. We also observe multi-clustered oscillation death states in a random network and measure them using basin stability framework. To explore such phenomena we choose a network of coupled Duffing-Holmes and Lorenz oscillators which are interacting through mean-field coupling. We investigate how basin stability for different steady states depends on mean-field density and coupling strength. We also analytically derive stability conditions for different steady states and confirm by rigorous bifurcation analysis.