Nonlinear Sciences

The Boros-Moll map appears as a subsystem of a Landen transformation associated to certain rational integrals and its dynamics is related to the convergence of them. In the paper, we study the dynamics of a one-parameter family of maps which unfolds the Boros-Moll one, showing that the existence of an unbounded invariant chaotic region in the Boros-Moll map is a peculiar feature within the family. We relate this singularity with a specific property of the critical lines that occurs only for this special case. In particular, we explain how the unbounded chaotic region in the Boros-Moll map appears. Special attention is devoted to explain the main contact/homoclinic bifurcations that occur in the family. We also report some other bifurcation phenomena that appear in the considered unfolding.

We present a simple low-cost electronic circuit that is able to show two different dynamical regimens with oscillations of voltages and with constant values of them. This device is designed as a negative feedback three-node network inspired in the genetic repressilator. The circuit's behavior is modeled by a system of differential equations which is studied in several different ways by applying the dynamical system formalism, making numerical simulations and constructing and measuring it experimentally. We find that the most important characteristics of the Hopf bifurcation can be found and controlled. Particularly, a resistor value plays the role of the bifurcation parameter, which can be easily varied experimentally. As a result, this system can be employed to introduce many aspects of a research in a real physical system and it enables us to study one of the most important kinds of bifurcation.

We explore the chaotic dynamics and complexity of a neuro-system with respect to variable synaptic weights in both noise free and noisy conditions. The chaotic dynamics of the system is investigated by bifurcation analysis and 0-1 test. A multiscale complexity of the system is proposed based on the notion of recurrence plot density entropy. Numerical results support the proposed analysis. Impact of music on the aforesaid neuro-system has also been studied. The analysis shows that inclusion of white noise even with a minimal strength makes the neuro dynamics more complex, where as music signal keeps the dynamics almost similar to that of the original system. This is properly interpreted by the proposed multiscale complexity measure.

It is important to know the accurate trajectory of a free fall object in fluid (such as a spacecraft), whose motion might be chaotic in many cases. However, it is impossible to accurately predict its chaotic trajectory in a long enough duration by traditional numerical algorithms in double precision. In this paper, we give the accurate predictions of the same problem by a new strategy, namely the Clean Numerical Simulation (CNS). Without loss of generality, a free fall disk in water is considered, whose motion is governed by the Andersen-Pesavento-Wang model. We illustrate that convergent and reliable trajectories of a chaotic free fall disk in a long enough interval of time can be obtained by means of the CNS, but different traditional algorithms in double precision give disparate trajectories. Besides, unlike the traditional algorithms in double precision, the CNS can predict the accurate posture of the free fall disk near the vicinity of the bifurcation point of some physical parameters in a long duration. Therefore, the CNS can provide reliable prediction of chaotic systems in a long enough interval of time.

We describe a simple and systematic method for obtaining approximate sensitivity information from a chaotic dynamical system using a hierarchy of cumulant equations. The resulting forward and adjoint systems yield information about gradients of functionals of the system and do not suffer from the convergence issues that are associated with the tangent linear representation of chaotic systems. The functionals on which we focus are ensemble-averaged quantities, whose dynamics are not necessarily chaotic; hence we analyse the system's statistical state dynamics, rather than individual trajectories. The approach is designed for extracting parameter sensitivity information from the detailed statistics that can be obtained from direct numerical simulation or experiments. We advocate a data-driven approach that incorporates observations of a system's cumulants to determine an optimal closure for a hierarchy of cumulants that does not require the specification of model parameters. Whilst the sensitivity information from the resulting surrogate model is approximate, the approach is designed to be used in the analysis of turbulence, whose number of degrees of freedom and complexity currently prohibits the use of more accurate techniques. Here we apply the method to obtain functional gradients from low-dimensional representations of Rayleigh-Bénard convection.

Chaotic dynamics in systems ranging from low-dimensional nonlinear differential equations to high-dimensional spatio-temporal systems including fluid turbulence is supported by non-chaotic, exactly recurring time-periodic solutions of the governing equations. These unstable periodic orbits capture key features of the turbulent dynamics and sufficiently large sets of orbits promise a framework to predict the statistics of the chaotic flow. Computing periodic orbits for high-dimensional spatio-temporally chaotic systems remains challenging as known methods either show poor convergence properties because they are based on time-marching of a chaotic system causing exponential error amplification; or they require constructing Jacobian matrices which is prohibitively expensive. We propose a new matrix-free method that is unaffected by exponential error amplification, is globally convergent and can be applied to high-dimensional systems. The adjoint-based variational method constructs an initial value problem in the space of closed loops such that periodic orbits are attracting fixed points for the loop-dynamics. We introduce the method for general autonomous systems. An implementation for the one-dimensional Kuramoto-Sivashinsky equation demonstrates the robust convergence of periodic orbits underlying spatio-temporal chaos. Convergence does not require accurate initial guesses and is independent of the period of the respective orbit.

Advection-diffusion problems of magnetic field and tracer field are analyzed using the field theoretic perturbative renormalization group. Both advected fields are considered to be passive, i.e., without any influence on the turbulent environment, and advecting velocity field is generated by compressible version of stochastic Navier-Stokes equation. The model is considered in the vicinity of space dimension d=4 and is a continuation of previous work [N.V. Antonov et al., Phys. Rev. E 95, 033120 (2017)]. The perturbation theory near the special dimension d=4 is constructed within a double expansion scheme in y (which describes scaling behavior of the random force that enters a stochastic equation for the velocity field) and ϵ=4−d . We show that up to one-loop approximation both types of advected fields exhibit similar universal scaling behavior. In particular, we demonstrate this statement on the inertial range asymptotic behavior of the correlation functions of advected fields. The critical dimensions of tensor composite operators are calculated in the leading order of (y,ϵ) expansion.

We study the dynamics of a ring of patches with vegetation-prey-predator populations, coupled through interactions of the Lotka-Volterra type. We find that the system yields aperiodic, recurrent and rare explosive bursts of predator density in a few isolated spatial patches from time to time. Further, the collective predator biomass also exhibits sudden uncorrelated occurrences of large deviations as the coupled system evolves. The maximum value of the predator population in a patch, as well as the maximum value of the predator biomass, increases with coupling strength. These trends are further corroborated by fits to Generalized Extreme Value distributions, where the location and scale factor of the distribution increases markedly with coupling strength, indicating the crucial role of coupling interactions in the generation of extreme events. These results indicate how occurrences of extremely large predator populations can emerge in coupled population dynamics, and in a more general context they suggest a generic class of deterministic nonlinear systems that can naturally exhibit extreme events.

In this paper we provide an extension for the method of Discrete Lagrangian Descriptors with the purpose of exploring the phase space of unbounded maps. The key idea is to construct a working definition, that builds on the original approach introduced in Lopesino et al. (2015), and which relies on stopping the iteration of initial conditions when their orbits leave a certain region in the plane. This criterion is partly inspired by the classical analysis used in Dynamical Systems Theory to study the dynamics of maps by means of escape time plots. We illustrate the capability of this technique to reveal the geometrical template of stable and unstable invariant manifolds in phase space, and also the intricate structure of chaotic sets and strange attractors, by applying it to unveil the phase space of a well-known discrete time system, the Hénon map.

We discuss a specific model, which we refer to as RandLOE, of a large multi-agent network whose dynamic is prescribed via a combination of deterministic local laws and random exogenous factors. The RandLOE approach lies outside the framework of Stochastic Differential Equations, but lends itself to analytic examination as well as to stable simulation even for relatively large networks. RandLOE is based on the logistic operator equation (LOE), which is a multidimensional dynamical system extending the classical logistic equation via an operator-algebraic interaction term. The network is defined by interpreting the LOE variable as an adjacency matrix of a complete graph. Depending on the choice of parameters, it can display a number of essentially distinct dynamical characteristics: e.g. cycles of expansion and contraction.