Nonlinear dynamical systems far from thermodynamic equilibrium often exhibit self-organization, i.e., the spontaneous emergence of temporal, spatial, or spatiotemporal dissipative structures from the inherent nonlinear cooperative dynamics. Examples of such structures (or dynamical states) are periodic orbits, spatiotemporal patterns, but also non-equilibrium steady states. Dissipative structures in self-organizing nonlinear systems are indeed widespread in physics, chemistry, and biology. In recent years, much progress has been achieved in not only describing these intriguing dynamical structures but rather developing novel approaches to control such systems. A major aim of control is to select one of several, often coexisting states, and to stabilize it deliberately.

The idea and usage of control has a long tradition in engineering and applied mathematics. At earlier times, however, control concepts had then been considered by quite separate communities, including the nonlinear dynamics control community,1 which originally emerged from chaos control,2,3 the classical mathematical control and optimization community,4 and the coherent quantum control community.5 Due to this separation, a broader picture of the applicability and potential of control concepts in different disciplines was markedly absent. Only recently, different control communities began to exchange and mutually benefit. At the same time, applications of concepts of control have been substantially broadened toward new areas such as complex networks, neural systems, but also hard and soft condensed matter, and excitable media such as cardiac tissue.

This Focus Issue provides a collection of 22 papers describing examples of the vibrant research in this area from the theoretical side. The articles highlight the present understanding of the field, but they also address novel developments and open problems.

Many of the studies have been done, or initiated, in the framework of the Collaborative Research Center (CRC) 910 “Control of self-organizing nonlinear system: Theoretical methods and concepts of application,” a large scale research initiative in the area of Berlin/Germany funded by the German Research Foundation (DFG) from 2011 to 2022. This initiative has bundled the expertise of researchers from theoretical physics, mathematics, and theoretical neuroscience in the Berlin area. At the same time, there were strong interactions with other national and international experts. This was enabled via an extended guest program and, in particular, via biennial international conferences, one of which (Warnemünde 2016) has led to a book6 in the Springer Series Understanding Complex Systems collecting cutting-edge research up to that time. Altogether, the CRC 910 has played a significant role in merging communities and putting the field forward. Indeed, control of self-organizing systems is a timely, still developing, interdisciplinary topic attracting many scientists in the world. In Sec. II of this Editorial, we summarize the areas covered in this Focus Issue. We close Sec. III with some conclusions and emerging problems in adjacent fields.

In the following, we shortly present the contributions to this Focus Issue and put them into a wider context. To this end, we group the articles according to the following themes: patterns and multiscale systems (Sec. II A); complex networks and synchronization (Sec. II B); and applications to neural systems, neural networks, and machine learning (Sec. II C).

Within the last years, concepts of control have been put forward for a variety of patterns in heterogeneous and multiscale systems, including systems with delay and stochasticity. The strongly varying spatial and temporal scales of self-organized patterns pose a particular methodological challenge for control. Paradigmatic examples of patterns that have been, and are, investigated are waves of various shapes, solitons, and fronts [largely described on the basis of partial differential equations (PDEs)], but also synchrony in systems of excitable elements.

To analyze these topics from the methodological side, an arsenal of partially overlapping competencies from mathematics and theoretical physics side is used, including analysis of multiscale systems, nonlocal evolution equations, reaction–diffusion systems, bifurcation theory, reduction through symmetries, design of time-delayed spatiotemporal feedback controls, differential-algebraic model hierarchies, and network theory. Based on this pool of methods, a number of universal concepts have been advanced, such as the interaction of coupled elements by instantaneous or time-delayed, local or nonlocal coupling and feedback, pattern-selection, forcing by external fields, interaction of competing time scales, influence of heterogeneities, fluctuating parameters, and noise.

In the following Sec. II A 1, we first summarize contributions dealing with the analysis and control of PDEs. We then proceed in Sec. II A 2 with effects of time delay, time-delayed feedback control, and the analysis of systems with stochasticity.

1. PDEs

Partial differential equations (PDEs), such as reaction–diffusion systems, evolution equations for complex fluid flows, or wave equations for optical pulses are natural playgrounds for the investigation of self-organized spatiotemporal behavior and its control.7–10 Many of these patterns, however, exhibit instability, posing significant challenges for observation. This instability renders the solutions particularly susceptible to errors, whether in experimental setups or in numerical simulations, leading to effectively invisible patterns. As a result, extensive research has focused on how to effectively control and stabilize these patterns.11–13 

In their Review Paper, Mielke and Schindler14 examine the intricacies of self-similarity in reaction–diffusion systems, categorizing its occurrence into three distinct types: exact solutions, asymptotic behavior, and constrained self-similarity. They propose a novel perspective by interpreting these self-similar profiles as non-equilibrium steady states.

Fiedler and Rocha15 examine Sturm global attractors in scalar parabolic PDEs on the unit interval. Focusing on Sturm meanders with three “noses,” they precisely describe the connection graphs and reveal that despite being based on irreversible diffusion, the connection graph displays time reversibility on the boundary of the global attractor.

Schneider and Dai16 introduce new noninvasive feedback controls rooted in symmetry groupoids to address instability in reaction–diffusion patterns, exemplified by the Chafee–Infante equation. Instead of traditional symmetry-based methods, their strategy utilizes added symmetries to design innovative convolution controls. Through a Fourier basis, they showcase the capability of symmetry groupoids for spatiotemporal pattern stabilization.

In their research, Veerman and Schneider17 examine the potential of a noninvasive Pyragas-inspired feedback control to stabilize singularly perturbed reaction–diffusion systems with two components. They successfully stabilize pulse solutions in a significant region of parameter space. The effectiveness of this control technique is proven via spectrum analysis using Evans function techniques.

Bandara et al.18 explore solitons in a generalized nonlinear Schrödinger equation modeling optical waveguide pulses with quartic dispersion. By transforming this into a Hamiltonian fourth-order ODE, they uncover novel soliton types, specifically generalized solitons with non-decaying oscillating tails and multi-oscillation solitons featuring varied oscillation episodes. These findings shed light on the intricate phenomenon of truncated homoclinic snaking.

2. Delay and stochasticity

In control of spatiotemporal behavior, there is often a time delay between the observation, i.e., collection of information, and the controlling action. More generally, delays are essentially omnipresent in dynamical systems involving signaling and communication as it is often the case in optical sciences and engineering but also in living matter.19 Traditionally, time delays have often been considered a nuisance hindering a proper implementation of a feedback control loop or an experiment. However, there is now an emerging understanding of the delicately constructive contribution of certain delays toward the active design of self-organizing systems with desirable spatiotemporal behavior.20–22 

Keane et al.23 investigate clustering patterns that emerge in a unidirectional ring of four delay-coupled electrochemical oscillators. By means of bifurcation analysis, the authors show how the stable cluster states, corresponding to experimental observations, lose their stability via an assortment of bifurcation types.

Inspired by time-delayed feedback control, the paper by Just24 shows that synchronization of non-identical systems can be achieved by mutual time-delayed feedback with an asymptotically vanishing interaction. The author develops an analytic perturbation scheme, which uncovers the merits as well as the constraints of such an approach.

Kyrychko and Blyuss25 model dynamics of pediatric vaccination as an imitation game, using a delay distribution to represent how the memory of past side effects influences current perception of risk. It is not just the mean duration of this memory, but also the actual distribution that determines whether the disease will be maintained in the population.

Stöhr and Wolfrum26 explore the dynamics of temporal dissipative solitons in the Morris–Lecar model under time-delayed feedback. Utilizing classical homoclinic bifurcation theory, they analyze pulse solutions in systems with large delays. Their findings reveal that a homoclinic orbit flip can destabilize multi-pulse solutions, prompting the appearance of stable pulse packages linked to the model’s excitability properties.

Koch et al.27 analyze the dynamics of vertically emitting micro-cavities with semiconductor quantum-wells under strong time-delayed optical feedback. They uncover multistable dark and bright temporal localized states and square-waves with specific periodicity. Their findings reveal the potential for generating complex temporal patterns, holding significance for nonlinear dynamics and applications.

Blyuss and Kyrychko28 derive and analyze a mathematical model of a sexual response. Solutions corresponding to the dynamics associated with the Masters–Johnson sexual response cycle are represented by “canard”-like trajectories. Importantly, the authors also investigate a stochastic version of their model.

The dynamics of complex networks is a still emerging, interdisciplinary field receiving strong international attention.29–33 Major current topics, whose importance is recognized even among non-scientists, are power grids, brain, climate, epidemics, and social systems. As a consequence, the advancement of control of coupled systems and networks from the theoretical side is a very active field.

An important class of network models are coupled oscillators (e.g., phase oscillators) and large, dynamical networks of time-continuous oscillators. These models are paradigmatic systems in which phenomena like synchronization, chimeras (i.e., spatially coexisting domains of synchronized and desynchronized dynamics),34 and other partial synchronization patterns can be studied nowadays even experimentally.35 However, there are also many recent theoretical works on chemical and metabolic networks, laser networks, cavity quantum networks, network models for epidemics (see, e.g., Refs. 36 and 37), and interareal networks of neurons (see Sec. II C). We note that the topological and structural properties of networks, including irregular, random, small-world, or scale-free types, have been thoroughly studied in the network community already for some time. However, the dynamics of complex networks with time delays and/or noise has been, and still is, a very challenging and important issue. For example, control of networks with processing latencies has important implications such as stabilization of power grids under varying loads and supply. Other important, recent developments are multilayer networks38,39 and adaptive networks.40 With these, researchers seek to explore alternative and more realistic approaches for the modeling of complex interactions in real-world systems.

Conceptually, the design of feedback loops to control the dynamics, and to judiciously select different modes such as synchronous oscillations, cluster states, and chimeras, has been extremely fruitful in various types of networks.

Jüttner and Martens41 study a minimal model of Kuramoto phase oscillators including a general adaptive learning rule with three parameters (strength of adaptivity, adaptivity offset, and adaptivity shift), mimicking learning paradigms based on spike-time-dependent plasticity. They perform a detailed bifurcation analysis for the case of N = 2 units and numerically study the case of N = 50.

Thiele et al.42 present a framework for describing the emergence of recurrent synchronization in complex networks with adaptive interactions. The authors identify asymmetric adaptation rules and temporal separation between the adaptation and the dynamics of individual nodes as key features for the emergence of recurrent synchronization.

Rybalova et al.43 disclose regimes, such as the coexistence of a multichimera state/a traveling wave and a solitary state in a ring network of nonlocally coupled units, each given by a discrete-time model of the van der Pol oscillator. The majority of the observed chimera/solitary states are transient toward a purely traveling wave mode.

Semenov and Zakharova44 show that multiplex structure of a network can be used to control noise-induced dynamics of such a network in the regime of stochastic resonance. The authors illustrate this phenomenon on an example of two- and multi-layer networks of bistable overdamped oscillators.

Ferri et al.45 study a social network, specifically, an agent-based opinion model that allows us to explore the interplay between opinion formation and mobility. Their results, based on numerical simulations, show that in general, mobility promotes the global consensus, especially for extreme opinions.

Along with the investigation of mechanisms underlying the complex phenomena in dynamical networks, and designing methods for their control, it is crucial to explore and develop concepts of their applications.

Here we focus, first, on neural systems. Neural systems and networks, similar to many other complex systems, self-organize into a large variety of states that are vital for the proper functioning of the brain. For instance, synchronous spatiotemporal patterns play an important role in neuronal communication and plasticity and in various cognitive processes. Developing realistic concepts of modeling the interactions between the neurons and understanding the effects of different network structures and coupling setups on the resulting dynamical behavior are the crucial challenges from the application point of view. We then move toward more abstract neural network models and, finally, touch the field of machine learning.

Bukh et al.46 find that the emergence of oscillatory activity in delay-coupled excitable FitzHugh–Nagumo neurons is observed for smaller values of the coupling strength as the dissipation parameter decreases. This can provide the possibility of controlling the spatiotemporal behavior of the considered neural networks.

Salfenmoser and Obermayer47 explore methods from optimal control theory to study efficient, stimulating control signals designed to make the neural activity match desired targets. Efficiency is quantified by a cost function, which trades control strength against closeness to the target activity. The authors apply this method to a Wilson–Cowan model of coupled excitatory and inhibitory neural populations.

Aristides et al.48 analyze small groups of coupled Izhikevich neurons and show that the unscented Kalman filter (UKF) allows inferring the connectivity between the neurons, even for heterogeneous, directed, and temporally evolving networks. The UKF can also recover the parameters of a single neuron, even when the parameters vary in time.

Stannat et al.49 present an algorithm that uses artificial neural networks to approximate optimal controls for stochastic reaction–diffusion equations. This method significantly reduces computational complexity and is effective against the curse of dimensionality.

Köster et al.50 propose a new approach to dynamical system forecasting called data-informed-reservoir computing. Reservoir computing is a type of machine learning in which an input drives a dynamical system (the “reservoir”). The approach in this paper, while solely being based on data, yields increased accuracy, reduced computational cost, and mitigates tedious hyper-parameter optimization of the reservoir computer. The approach is tested on the Lorenz system and the Kuramoto–Sivashinsky system.

This Focus Issue brings together important theoretical advances within and around the interdisciplinary field of control of self-organizing nonlinear systems. The articles cover a range of topics and directions as outlined above. From a fundamental point of view, the articles also give an impression as to which concepts of control are particularly versatile. These include (i) feedback control (closed-loop control) where unstable states are stabilized adaptively by using the internal dynamics to adjust the control force, including the special case of time-delayed feedback control, (ii) variation of an external stimulus to select a desired dynamic or stationary nonequilibrium state, and (iii) optimal control,4 with respect to the forcing or feedback protocol in order to minimize a suitably defined cost functional. These methodological developments complement theoretical work in related areas such as control of delay-differential equations,51 analysis of port-Hamiltonian systems,52 and of nonlinear evolution equations53 not covered here.

The above-mentioned universal concepts of control have been strongly influential, and beneficial, in various areas of application, even though the concrete analysis of specific nonlinear systems often remains challenging. In this Focus Issue, we have focused on applications on networks and patterns. Already here there are several emerging themes, whose importance is more and more acknowledged, such as control of power grids and epidemics. But also beyond the topics covered in this Focus Issue, there are new areas where the application of control has been established and plays an increasingly important role. Here, we would like to mention two particularly important examples from the physics side.

In the quantum area, since the seminal works of Wiseman and Milburn,5 quantum feedback control has developed into a major tool to manipulate quantum dynamics in the theory and experiment.54 Applications include quantum-optical control, quantum information, quantum gases, and non-reciprocal media.55, Markovian feedback control is nowadays used in a variety of contexts, including, e.g., the realization of a Maxwell demon in a quantum nanostructure,56 the preparation of targeted eigenstates in optical lattices,57 and the stabilization of a topological superconducting phase.58 Theoretical progress in this area is complemented by an increasing stream of experiments of combined experimental–theoretical research in quantum optics and solid state physics where quantum feedback control concepts are employed, such as for shot noise suppression in quantum nanostructures59 or closed-loop control of interferiometry.60 Regarding non-Markovian (delayed) feedback control, the extension of corresponding concepts for classical systems toward the control of nonlinear phenomena dominated by quantum fluctuations provides a severe challenge. In particular, to establish coherent quantum feedback, the non-Markovian interaction between a quantum system and its spectrally structured environment has to include its full entanglement.61 

Soft condensed matter such as colloidal suspensions (i.e., micrometer-sized particles in a thermally fluctuating bath) and complex (e.g., polymeric) fluids easily respond to external perturbations and thus show diverse nonlinear phenomena and instabilities on different length- and timescales. In particular, control of colloidal systems has become a prime example of deliberate manipulation of a classical noisy systems.62 Recent theoretical works have focused on the description of colloidal feedback control via stochastic thermodynamics.63 An emphasis was on the important case of time-delayed feedback64,65 that had formerly played the role of a curiosity. Many experimental realizations of colloids under control involve active, self-propelled particles.66,67 Indeed, in the last 20 years, active matter composed of artificial or biological active constituents has developed into one of the most important topics of (classical) non-equilibrium statistical physics. It is, therefore, not surprising that there is an increasing number of studies applying not only feedback but also other techniques such as optimal control68 and control by external fields69 on active systems.70 This includes, on a more coarse-grained level, control of patterns in reaction–diffusion models of biophysical active media, such as active poroelastic media and cardiac tissue.71 Another direction in the field of soft matter concerns the understanding of how adaption of external stimuli can be used to optimize certain dynamical patterns in complex fluid flow.72 

We thank all of the authors who contributed with their exciting recent works in this Focus Issue. We also would like to express our thanks to all the referees for their careful reading and constructive comments. Special thanks go to the editor-in-chief, Jürgen Kurths, for his support and helpful advice. Many of the works have been funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) in the framework of the Collaborative Research Center SFB 910 Control of self-organizing nonlinear systems: Theoretical methods and concepts of application (Project No. 163436311-SFB 910). This initiative was founded by Eckehard Schöll, whom we cordially thank for all his support and engagement in the field over decades.

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