In this Focus Issue, we explore recent advances in complexity and machine learning applied to nonlinear and emergent phenomena in mathematics, physics, and interdisciplinary science. The collected papers address topics, including chaos, turbulence, dissipative solitons, active matter, complex networks, stochastic dynamics, bifurcations, and quantum systems. Several contributions investigate self-organization, synchronization, multistability, and pattern formation in physical, biological, ecological, and socioeconomic systems, while others employ neural networks, physics-informed learning, reservoir computing, and data-driven approaches to model, predict, and analyze complex dynamical behaviors. These studies demonstrate the broad impact of complexity and machine learning across modern interdisciplinary research.

Koopman operators are linear, infinite-dimensional operators that can potentially capture the behavior of nonlinear dynamical systems, offering a unified description for prediction and control. For equivariant systems, estimating the operators while respecting the symmetry leads to expensive computations that scale with a power of the order of the symmetry group. Here, following the work of Salova et al. (2019) on radial basis functions, we construct a dictionary of complex polynomials that leverages the system's symmetry to yield a low-dimensional representation, thereby overcoming radial basis functions' computational limitations. We apply this method to a network of phase oscillators with global phase invariance, obtaining compressed descriptions, verified spectra, and good forecasting power.

Among the many properties of Orbital Angular Momentum (OAM) beams of light, one should mention the resilience under turbulent conditions. This feature has been exploited in the successful transmission of data using increasingly sophisticated encoding and decoding schemes. In this presentation, we use the language of Group Theory and focus on some OAM superpositions that show well-defined invariances with respect to reflections and rotations, and can be associated with subgroups of a large dihedral group. Based on numerical simulations, we use a selection of OAM superpositions that essentially look like light petals throughout the turbulent propagation and can be identified correctly despite some errors in the invariance tests. These ideas are extended to beams with nonuniform polarization such as vector vortex beams.

The symmetry properties of the attractors of equivariant dissipative dynamical systems can suffer symmetry-changing bifurcations, that can be detected and classified using well-established methods. Novel data-driven methods, such as the Koopman and Perron-Frobenius operators, besides reducing any nonlinear system to a linear one, can also be applied to the analysis of equivariant dynamical systems and the classification problem. In this article, we study matrix approximations of these infinite-dimensional operators that respect the original symmetry and introduce an aggregate matrix that has a clear interpretation. Its sparsity pattern reveals the presence of multiple conjugate attractors and indicates the structure of their symmetry subgroup. We apply these ideas to data generated by three nonlinear equivariant systems, finding attractors of nontrivial subgroups and detecting symmetry-changing bifurcations. The proposed method can be incorporated into existing computational processes for the analysis, prediction and control of nonlinear equivariant systems.

Laser communication link performance in free space depends heavily on atmospheric conditions present on the propagation path. Distortions due to atmospheric turbulence, such as scintillation and beam wander, greatly diminish signal detection and performance. With the final goal of improving the communication link, experimental measurements and analysis of turbulence strength are presented as metrics for determining the system's detection limits. Experimental optical trials were recorded over a 1-km horizontal path in order to study the intensity fluctuations, beam dance, and the spatial spectrum properties under different regimes. By taking into account wind speed and vibrations of the building where the measurements were carried on, correlations between variables are shown with the use of a photodetector, and a 2D lateral effect position sensor.

We describe a method for determining optimal selections of orbital angular momentum (OAM) superpositions for OAM signal modulation in free-space optical communications using a measure of distance in the context of the Optimal Transport theory. Within the range of topological charges $\ell=-20$ to $\ell=20$ we design OAM constellations using 16 to 128 symbols consisting of solos, duets, trios, and quartets of OAM modes. We propose a classification strategy requiring relatively low complexity to evaluate the performance of these constellations, achieving a classification error smaller than 1/1000 in weak to strong turbulence conditions for the 16-OAM constellation. We have found that the optimal set shows some dependence on the receiver's architecture, so we offer results for optical detectors based on the conjugate projection, the mode sorter, and the Shack-Hartmann sensor.

Nonlinear systems that model physical experiments often have many equilibrium configurations, and the number of these static solutions grows with the number of degrees of freedom and the presence of symmetries. It is impossible to know a priori how many equilibria exist and which ones are stable or relevant, therefore from the modeler’s perspective, an exhaustive search and symmetry classification in the space of solutions are necessary. With this purpose in mind, the method of deflation (introduced by Farrell as a modification of the classic Newton iterative method) offers a systematic way of finding every possible solution of a set of equations. In this contribution we apply deflated Newton and deflated continuation methods to a model of macroscopic magnetic rotors, and find hundreds of new equilibria that can be classified according to their symmetry. We assess the benefits and limitations of the method for finding branches of solutions in the presence of a symmetry group, and explore the high-dimensional basins of attraction of the method in selected 2-dimensional sections, illustrating the effect of deflation on the convergence.

An experimental campaign for the propagation of laser modes carrying orbital angular momentum (OAM) over 1 kilometer has been established at UANDES university campus. We describe our method for estimating OAM spectra and the average topological charge values from the images delivered by Shack-Hartmann sensor. For OAM beams transmitted with a single topological charge we analyze the average departure of the measured charge with respect to the intended one and the spread of these values as a function of turbulence strength.

Vector vortex beams (VVB) combine a nontrivial phase structure and a transverse polarization pattern that can be used for optical free-space or optical fiber communication links. In this presentation we will show, using numerical propagation, the effects of atmospheric turbulence on the estimation of the pixel-by-pixel Stokes parameters, and how the concepts of Optimal Transport allow an optimal selection of basis symbols and a more accurate detection of the VVBs in moderate turbulence.

We describe an experimental testbed to evaluate optical vortices and present a field study of single-OAM and dual-OAM vortices propagated over 900m. OAM detection is performed using a Shack-Hartman sensor and a fast-imaging camera. Beams are analyzed in terms of their topological charges and turbulence strength.

We propose the use of orbital angular momentum (OAM) states to form modulation symbols in a free-space laser communication link affected by atmospheric turbulence. A collection of superpositions of 2 and 4 active OAM modes are used for transmitting digital information. To sense and decode the data we compare three candidate architectures, based on a Mode Sorter, a Shack-Hartmann and phase flattening holograms. In this work we use concepts of Optimal Transport, particularly the Wasserstein distance and barycenter, for an optimal selection of OAM superpositions and a more appropriate processing of OAM spectra, leading to more accurate detection, achieving a classification error smaller than 1/1000 in intermediate to strong turbulence conditions.

We demonstrate an optical detection and decoding strategy to increase the information rate and spectral efficiency of free-space laser communication links affected by turbulence by means of dense orbital angular momentum (OAM) modulation. Using three candidate receiver architectures-based on a Shack-Hartmann sensor, a Mode Sorter, and a complex conjugate projection scheme as a base case-we demonstrate an algorithmic classification system based on the received OAM spectra produced by these architectures. This classification scheme allows low-error-rate data transmission in turbulence using 16-OAM, 32-OAM, and 64-OAM symbol constellations, with OAM states between -20 and 20. We evaluate and compare their performance under weak to strong atmospheric turbulence conditions using an accuracy metric and confusion matrices.

Reaction-diffusion systems are used in biology, chemistry, and physics to model the interaction of spatially distributed species. Particularly of interest is the spatial replacement of one equilibrium state by another, depicted as traveling waves or fronts. Their profiles and traveling velocity depend on the nonlinearities in the reaction term and on spatial diffusion. If the reaction occurs at regularly spaced points, the velocities also depend on lattice structures and the orientation of the traveling front. Interestingly, there is a wide region of parameters where the speeds become zero and the fronts do not propagate. In this paper, we focus on systems with three stable coexisting equilibrium states that are described by the butterfly bifurcation and study to what extent the three possible 1D traveling fronts suffer from propagation failure. We demonstrate that discreteness of space affects the three fronts differently. Regions of propagation failure add a new layer of complexity to the butterfly diagram. The analysis is extended to planar fronts traveling through different orientations in regular 2D lattices. Both propagation failure and the existence of preferred orientations play a role in the transient and long-time evolution of 2D patterns.

Macroscopic magnets can easily be manipulated and positioned so that interactions between themselves and with external fields induce interesting dynamics and equilibrium configurations. In this work, we use rotating magnets positioned in a line or at the vertices of a regular polygon. The rotation planes of the magnets can be modified at will. The rich structure of stable and unstable configurations is dictated by symmetry and the side of the polygon. We show that both symmetric solutions and their symmetry-breaking bifurcations can be explained with group theory. Our results suggest that the predicted magnetic textures should emerge at any length scale as long as the interaction is polar, and the system is endowed with the same symmetries.

We present an experimental campaign consisting on the propagation of laser modes with orbital angular momentum (OAM), carried out in our campus, for a total propagation distance of 1 kilometer. In this proceedings article we describe our experiment and describe some preliminary results, using topological charges up to |ℓ| = 45. We also demonstrate that a Shack-Hartmann sensor may be used for OAM sensing in the presence of weak to intermediate turbulence without the help from adaptive optics.

In previous publications, we showed that the incremental process of the chaotic diffusion of dissipative solitons in a prototypical complex Ginzburg-Landau equation, known, e.g., from nonlinear optics, is governed by a simple Markov process leading to an Anti-Persistent Random Walk of motion or by a more complex Hidden Markov Model with continuous output densities. In this article, we reveal the transition between these two models by studying the soliton dynamics in dependence on the main bifurcation parameter of the Ginzburg-Landau equation and identify the underlying hidden Markov processes. These models capture the non-trivial decay of correlations in jump widths and symbol sequences representing the soliton motion, the statistics of anti-persistent walk episodes, and the multimodal density of the jump widths. We demonstrate that there exists a physically meaningful reduction of the dynamics of an infinite-dimensional deterministic system to one of a probabilistic finite state machine and provide a deeper understanding of the soliton dynamics under parameter variation of the underlying nonlinear dynamics.

The increase of data rate and bandwidth efficiency of free-space optical communication links may be supported by the use of dense orbital angular momentum (OAM) states, carrying several information bits per transmission. Using machine-learning decoding, the performance of 32-OAM and 64-OAM signal constellations –designed using 4-state superpositions– are studied using numerical propagation models. Using two candidate architectures for detection –Shack-Hartmann and Mode Sorter– we evaluate the performance of the modulation in a simulated optical atmospheric channel by means of the detection accuracy.

The analysis of superpositions of Orbital Angular Momentum (OAM) modes is a challenging problem, particularly when atmospheric turbulence is present or when the phase structure of the wavefront is not available. In such conditions it is not possible to correct the distortions and reconstruct the vorticial phase structure: the rings and petals that characterize the intensity profiles of such beams become deformed and may even lose integrity. These artifacts may compromise the possibility of establishing free-space optical links based on OAM superpositions. We propose using a particular selection of Laguerre-Gauss modes and convolutional neural networks for a reliable classification of superpositions of two modes. The network (based on a pre-trained network AlexNet that combines convolutional and fully-connected layers) is trained as a classifier based on 2-d intensity profiles that can be obtained from a digital camera. For illustrating the proposed method, we used simulations of light beams propagated through L = 1 km with three levels of turbulence: C2n ∈ {2×10-15, 9.24×10-15, 2.9×10-14} m-2/3. The emitted beams are made up of 2 different Laguerre-Gauss modes with OAM between -15 and +15, and radial indices between 0 and 3. Classification results show that the radial index can be used effectively to enlarge the set of information symbols.

In classical mechanics, solutions can be classified according to their stability. Each of them is part of the possible trajectories of the system. However, the signatures of unstable solutions are hard to observe in an experiment, and most of the times if the experimental realization is adiabatic, they are considered just a nuisance. Here we use a small number of XY magnetic dipoles subject to an external magnetic field for studying the origin of their collective magnetic response. Using bifurcation theory we have found all the possible solutions being stable or unstable, and explored how those solutions are naturally connected by points where the symmetries of the system are lost or restored. Unstable solutions that reveal the symmetries of the system are found to be the culprit that shape hysteresis loops in this system. The complexity of the solutions for the nonlinear dynamics is analyzed using the concept of boundary basin entropy, finding that the damping time scale is critical for the emergence of fractal structures in the basins of attraction. Furthermore, we numerically found domain wall solutions that are the smallest possible realizations of transverse walls and vortex walls in magnetism. We experimentally confirmed their existence and stability showing that our system is a suitable platform to study domain wall dynamics at the macroscale.

When propagated through atmospheric turbulence, Orbital Angular Momentum (OAM) modes suffer a loss of orthogonality that can compromise their detection and classification. The problem is more challenging when user information encoded on multi-state OAM superpositions needs to be detected with high probability. Optical sensors like the Shack-Hartmann detector or the Mode Sorter are candidates for such task. We describe how OAM histograms derived from such detectors can be used for decoding the original data symbols. We propose Machine Learning strategies for a reliable classification of the histogram patterns obtained with 4-mode superpositions propagated over a 1 km range in weak to intermediate turbulence.