Complex Dynamical Phenomena Research Articles

Bifurcation and chaotic behavior in the discrete BVP oscillator

In this work, a new class of discrete Bonhoeffer–van der Pol (BVP) system with an odd function is proposed and investigated. At first, the necessary and sufficient conditions on the existence and stability of the fixed points for this system are given. We then show the system passes through various bifurcations of codimension one, including pitchfork bifurcation, saddle–node bifurcation, flip bifurcation and Neimark–Sacker bifurcation under some certain parameter conditions. The center manifold theorem and bifurcation theory are the main tools in the analysis of the local bifurcations. Furthermore, we prove rigorously there exists Marotto’s chaos in this discrete BVP system, which means the fixed point eventually evolves into a snap-back repeller. Finally, numerical simulation evidences are provided not only to further demonstrate our theoretical analysis, but also to exhibit the complex dynamical phenomena, such as the period-9, -17, -18 orbits, attracting invariant cycles, quasi-periodic orbits, ten-coexisting chaotic attractors, etc. These phenomena illustrate relatively rich dynamical behaviors of the discrete BVP oscillator.

One family of 13315 stable periodic orbits of non-hierarchical unequal-mass triple systems

The three-body problem has been studied for more than three centuries [1,2], and has received much attention in recent years [3-5]. It shows complex dynamical phenomena due to the mutual gravitational interaction of the three bodies. Triple systems are common in astronomy, but all observed periodic triple systems are hierarchical up till now [6-8]. It is traditionally believed that bound non-hierarchical triple systems are almost unstable and disintegrate into a stable binary system and a single star [5], and thus stable periodic orbits of non-hierarchical triple systems are rather scarce. Here we report one family of 13315 stable periodic orbits of the non-hierarchical triple system with unequal mass. Compared with the narrow mass region (only 10E-5) of the stable figure-eight solution [9], our newly-found stable periodic orbits can have fairly large mass region. It is found that many of these newly-found stable periodic orbits have the mass ratios close to those of the hierarchical triple systems that have been measured by the astronomical observation. It implies that these stable periodic orbits of the non-hierarchical triple system with distinctly unequal masses can be quite possibly observed in practice. Our investigation also suggests that there should exist an infinite number of stable periodic orbits of non-hierarchical triple systems with distinctly unequal masses. Obviously, these stable periodic orbits of the non-hierarchical unequal-mass triple system have broad impact for the astrophysical scenario: they could inspire the theoretical and observational study of the non-hierarchical triple system, the formation of triple stars [6], the gravitational waves pattern [10] and the gravitational waves observation [11] of the non-hierarchical triple system.

Periodic solutions, chaos and bi-stability in the state-dependent delayed homogeneous Additive Increase and Multiplicative Decrease/Random Early Detection congestion control systems

The combination of Additive Increase and Multiplicative Decrease (AIMD) congestion control and Random Early Detection (RED) queue as a whole congestion control system plays a key role in the overwhelming success of the Internet. Thus it is important to investigate the periodic oscillations and complicated dynamics of the state-dependent delayed homogeneous Additive Increase and Multiplicative Decrease/Random Early Detection (AIMD/RED) congestion control system and its modified version fully in this paper. Firstly employing the semi-analytical method called as the harmonic balance method with alternating frequency/time (HB-AFT) domain technique, the approximate analytical expressions of periodic solutions of the generalized homogeneous-flow Additive Increase and Multiplicative Decrease/Random Early Detection (AIMD/RED) system with state-dependent round-trip delay are considered. We compare them with the results of numerical simulations by WinPP, they agree very well with each other. It demonstrates that the method employed here is versatile, valid, simple and effective. Then to the end of improving its modeling and performance, we modify the above model by taking an easy approximate dropping function. Furthermore, for the modified delayed homogeneous system, the approximate analytical expressions of periodic solutions are obtained accurately, and some complex dynamics are also presented. Four kinds of bi-stability, i.e., the coexistence of chaos and Period-3 solution, that of Period-1 and Period-2 solutions, that of Period-2 and Period-2 solutions, that of Period-4 and Period-2 solutions are disclosed. And a route to chaos, i.e., Period Doubling bifurcation to chaos, and the window of Period-3 to chaos are also discovered. The periodic oscillation can reduce the link utilization, induce the TCP stream synchronization services and further congestion. Chaotic oscillation may result in collapse. Therefore, all complex dynamical phenomena found in this paper are harmful and should be avoided. The obtained results can be very helpful for the researchers to have a better understanding of the mechanism of the network congestion control system, and they can select the parameters properly to improve the network stability and performance.

Special issue on dynamics of socioeconomic systems: systemic risk—finance

Over the past decades, complexity has emerged as an alternative way to understand and model complex dynamical phenomena in a number of fields. Instead of focusing on local and independent behaviors, complexity suggests that observed phenomena are the results of complex interactions between components of a system. The DySES conference, which took place at the Sorbonne in 2018, gave the unique opportunity to present the state of the art in the complexity field and in many areas including the socioeconomic ones. This special issue of Soft Computing presents papers in finance both in the field of complexity and beyond it. Many topics are covered. Jay et al. focus on portfolio management and introduce robust covariance matrices using recent developments in the random matrix theory (RMT). Levantesi et al. propose a novel approach, combining the random forest and the 2D P-spline in order to provide a more accurate mortality rate forecasting, therefore extending the Lee–Carter model. Chorro et al. focus on option pricing. They start from the inverse Gaussian GARCH model and build a new pricing kernel. Cerqueti et al. work on complex networks and introduce new influence measures based on vertex centrality. They apply their measure to the SP100 universe. Still within the complexity field, Gatfaoui et al. introduce a test of non-chaoticity when data are noisy. They show that financial data do not have a chaotic behavior. Fiori et al. build a model to capture systemic risk. In particular, they set the focus on the inequalities of wealth and income, using the well-known Gini coefficient. Di Paolo tackles the problem of the increase in life expectancy with regard to pension funds and annuity providers. The concept of observed survival probabilities is extensively used. Within a related area, D’Amato et al. propose a de-risking strategy model for LTC insurers that face demographic changes implying disability risks. They suggest new methodologies. Baione et al. consider non-life risk premium and study the diversification effect. They use quantile regressions to estimate the individual conditional loss. Kaucic et al. study investment strategies known as enhanced indexing. They introduce an improved version of the particle swarm optimization algorithm (PSO) and present an example using the Euro Stoxx. At last, Kudlak et al. question the importance of dedicated local budgets concerning crisis management. As guest editors, we would like to thank Soft Computing for giving us the opportunity to have this special issue done on all these topics.

A bistable nonvolatile locally-active memristor and its complex dynamics

This paper introduces a novel bistable nonvolatile locally-active memristor model based on Chua's unfolding theorem to explore the influence of the local activity on the complexity of nonlinear systems. It is shown that the memristor has two asymptotically stable equilibrium points in its power-off plot with a negative memductance and a positive memductance, namely, it is nonvolatile. It is then shown that the fast switching between the two stable equilibrium points can be implemented by applying an appropriate voltage pulse. It is found that the memristor possesses four locally-active regions in its DC V-I plot, and a small signal equivalent circuit associated with a locally-active operating point of the memristor is designed using the small-signal analysis method, to explore its locally-active characteristics. Based on the small signal equivalent circuit, it is furthermore demonstrated that the nonvolatile locally-active memristor, when connected in parallel with a linear capacitor, oscillates periodically around a locally-active operating point via Hopf bifurcation. The dynamics of the memristor and its periodic oscillation are analyzed using the theory of local activity, pole-zero analysis of admittance functions, Hopf bifurcation and the edge of chaos. Finally, it is found that a chaotic oscillation evolved from the periodic oscillation appears by adding an energy-storage element (inductor) on the periodic oscillating circuit, which leads to the chaotic oscillation, coexisting attractors and various complex dynamical phenomena.

Complicated oscillations and non-resonant double Hopf bifurcation of multiple feedback delayed control system of the gut microbiota

In this paper, we consider the complex dynamics of a novel mathematical model for a feedback control system of the gut microbiota, which was proposed by Dong, et al. The main work of the present paper is to study the effect of antibiotics injection on the gut microbiota through some dynamical methods, such as double Hopf bifurcation analysis and so on. We first use DDE-BIFTOOL to find the non-resonant double Hopf bifurcation points of the system, and draw the bifurcation diagram with two bifurcation parameters, τ1 and τ2, i.e., respective measurement delays. Then we study small perturbations of two delay differential equations at these double Hopf bifurcation points, and the method of multiple scales is employed to obtain two common complex amplitude equations. By analyzing the amplitude equations, we can derive the classification and unfolding of these double Hopf bifurcation points. Finally, we verify the results by numerical simulations. We find more complicated dynamic behaviors of the system via analytical method. For example, there exists stable equilibrium, stable periodic solution or even the co-existing stable periodic solutions in respective region. And the numerical simulations are consistent with the analytic results, meanwhile it implies that the MMS is effective and accurate. All complex dynamical phenomena found in the present paper can be very helpful for the researchers to understand the mechanism of the system of gut microbiota. And it is also very significant to microbiology and engineering control. It reveals that the measurement delays can induce the complicated dynamics in this system and to the ends of excellent performance, we should take the proper values of these delays.

Existence of Chaos in the Chen System with Linear Time-Delay Feedback

It is mathematically challenging to analytically show that complex dynamical phenomena observed in simulations and experiments are truly chaotic. The Shil’nikov lemma provides a useful theoretical tool to prove the existence of chaos in three-dimensional smooth autonomous systems. It requires, however, the proof of existence of a homoclinic or heteroclinic orbit, which remains a very difficult technical problem if contigent on data. In this paper, for the Chen system with linear time-delay feedback, we demonstrate a homoclinic orbit by using a modified undetermined coefficient method and we propose a spiral involute projection method. In such a way, we identify experimentally the asymmetrical homoclinic orbit in order to apply the Shil’nikov-type lemma and to show that chaos is indeed generated in the Chen circuit with linear time-delay feedback. We also identify the presence of a single-scroll attractor in the Chen system with linear time-delay feedback in our experiments. We confirm that the Chen single-scroll attractor is hyperchaotic by numerically estimating the finite-time local Lyapunov exponent spectrum. By means of a linear scaling in the coordinates and the time, such a method can also be applied to the generalized Lorenz-like systems. The contribution of this work lies in: first, we treat the trajectories corresponding to the real eigenvalue and the image eigenvalues in different ways, which is compatible with the characteristics of the trajectory geometry; second, we propose a spiral involute projection method to exhibit the trajectory corresponding to the image eigenvalues; third, we verify the homoclinic orbit by experimental data.

Hybrid symplectic integrators for planetary dynamics

Abstract Hybrid symplectic integrators such as MERCURY are widely used to simulate complex dynamical phenomena in planetary dynamics that could otherwise not be investigated. A hybrid integrator achieves high accuracy during close encounters by using a high-order integration scheme for the duration of the encounter while otherwise using a standard second-order Wisdom–Holman scheme, thereby optimizing both speed and accuracy. In this paper we reassess the criteria for choosing the switching function that determines which parts of the Hamiltonian are integrated with the high-order integrator. We show that the original motivation for choosing a polynomial switching function in MERCURY is not correct. We explain the nevertheless excellent performance of the MERCURY integrator and then explore a wide range of different switching functions including an infinitely differentiable function and a Heaviside function. We find that using a Heaviside function leads to a significantly simpler scheme compared to MERCURY , while maintaining the same accuracy in short-term simulations.

Stability and bifurcation analysis of a discrete predator\u2013prey system with modified Holling\u2013Tanner functional response

In this paper, we study a discrete predator–prey system with modified Holling–Tanner functional response. We derive conditions of existence for flip bifurcations and Hopf bifurcations by using the center manifold theorem and bifurcation theory. Numerical simulations including bifurcation diagrams, maximum Lyapunov exponents, and phase portraits not only illustrate the correctness of theoretical analysis, but also exhibit complex dynamical behaviors and biological phenomena. This suggests that the small integral step size can stabilize the system into the locally stable coexistence. However, the large integral step size may destabilize the system producing far richer dynamics. This also implies that when the intrinsic growth rate of prey is high, the model has bifurcation structures somewhat similar to the classic logistic one.

Sample-Based Methods of Analysis for Multistable Dynamical Systems

In this paper we present how sample based analysis can complement classical methods for analysis of dynamical systems. We describe how sample based algorithms can be utilized to obtain better understanding of complex dynamical phenomena, especially in multistable dynamical systems that are difficult for analytical investigations. Relying on the simple, direct numerical integration algorithms we are able to detect all possible solutions including hidden and rare attractors; investigate the ranges of stability in multiple parameters space; analyse the influence of parameters mismatch or model imperfections; assess the risk of dangerous or unwanted behaviour and reveal the structure of multidimensional phase space. For each mentioned application we present methodology, example on paradigmatic non-linear dynamical system and discuss practical applications. The presented methods of analysis can be applied to solve numerous of scientific problems originating from different disciplines. Moreover, their robustness and efficiency will grow with the upcoming increase of computational power.

Global dynamics of a mechanical system with dry friction

The global dynamics of a mechanical system with dry friction is completely analyzed. This mechanical system is a class of discontinuous and transcendental piecewise smooth differential systems. Moreover, it can exhibit rich and complex dynamical phenomena, such as Hopf bifurcation, grazing bifurcation, grazing-sliding bifurcation and bifurcations of limit cycles. Finally, all global phase portraits of the system are presented on the Poincaré disc.

Numerical Continuation of Limit Cycle Oscillations and Bifurcations in High-Aspect-Ratio Wings

This paper applies numerical continuation techniques to a nonlinear aeroelastic model of a highly flexible, high-aspect-ratio wing. Using continuation, it is shown that subcritical limit cycle oscillations, which are highly undesirable phenomena previously observed in numerical and experimental studies, can exist due to geometric nonlinearity alone, without need for nonlinear or even unsteady aerodynamics. A fully nonlinear, reduced-order beam model is combined with strip theory and one-parameter continuation is used to directly obtain equilibria and periodic solutions for varying airspeeds. The two-parameter continuation of specific bifurcations (i.e., Hopf points and periodic folds) reveals the sensitivity of these complex dynamics to variations in out-of-plane, in-plane and torsional stiffness and a ‘wash out’ stiffness coupling parameter. Overall, this paper demonstrates the applicability of continuation to nonlinear aeroelastic analysis and shows that complex dynamical phenomena, which cannot be obtained by linear methods or numerical integration, readily exist in this type of system due to geometric nonlinearity.

RETRACTED ARTICLE: Evaluation of the Filtered Noise Turbulent Inflow Generation Method

Flows around buildings and in urban areas have the ability to exchange mass and momentum through mixing layers. The complex dynamical phenomena arising in mixing layers can be studied using Large Eddy Simulation (LES). As mixing layers depend on the turbulence conditions upstream of the buildings or urban areas, appropriate turbulent inlet conditions have to be provided to a simulation. Due to the high efficiency and level of control, the filtered noise inflow method was selected. The control over the Reynolds stresses as well as nine length scales make this method suitable to replicate conditions measured in experiments. In this paper, a formal method to obtain the filter coefficients is presented. This is achieved by relating the spatial filtering to a Finite Impulse Response (FIR) filter and the temporal filtering to an Autoregressive (AR) model. Three closed-form solutions for the spatial filter coefficients are presented having a Gaussian, double-exponential and exponential correlation function. By means of an LES simulation of a turbulent wall-bounded flow, the input-output behaviour is investigated. It was found that a combination of a Gaussian filter with length scales that increase with increasing wall distance result in the fastest downstream development of the artificial turbulence and the smallest loss of turbulent kinetic energy.

Differences in the Vibrational Dynamics of H(2)O and D(2)O: Observation of Symmetric and Antisymmetric Stretching Vibrations in Heavy Water.

Water's ability to donate and accept hydrogen bonds leads to unique and complex collective dynamical phenomena associated with its hydrogen-bond network. It is appreciated that the vibrations governing liquid water's molecular dynamics are delocalized, with nuclear motion evolving coherently over the span of several molecules. Using two-dimensional infrared spectroscopy, we have found that the nuclear motions of heavy water, D2O, are qualitatively different than those of H2O. The nonlinear spectrum of liquid D2O reveals distinct O-D stretching resonances, in contrast to H2O. Furthermore, our data indicates that condensed-phase O-D vibrations have a different character than those in the gas phase, which we understand in terms of weakly delocalized symmetric and antisymmetric stretching vibrations. This difference in molecular dynamics reflects the shift in the balance between intra- and intermolecular couplings upon deuteration, an effect which can be understood in terms of the anharmonicity of the nuclear potential energy surface.

Observation of Aubry-type transition in finite atom chains via friction.

The highly nonlinear many-body physics of a chain of mutually interacting atoms in contact with a periodic substrate gives rise to complex static and dynamical phenomena, such as structural phase transitions and friction. In the limit of an infinite chain incommensurate with the substrate, Aubry predicted a transition with increasing substrate potential, from the chain's intrinsic arrangement free to slide on the substrate, to a pinned arrangement favouring the substrate pattern. So far, the Aubry transition has not been observed. Here, using spatially resolved position and friction measurements of cold trapped ions in an optical lattice, we observed a finite version of the Aubry transition and the onset of its hallmark fractal atomic arrangement. Notably, the observed critical lattice depth for few-ion chains agrees well with the infinite-chain prediction. Our results elucidate the connection between competing ordering patterns and superlubricity in nanocontacts-the elementary building blocks of friction.

A Network of Neural Oscillators for Fractal Pattern Recognition

Biological neural networks are high dimensional nonlinear systems, which presents complex dynamical phenomena, such as chaos. Thus, the study of coupled dynamical systems is important for understanding functional mechanism of real neural networks and it is also important for modeling more realistic artificial neural networks. In this direction, the study of a ring of phase oscillators has been proved to be useful for pattern recognition. Such an approach has at least three nontrivial advantages over the traditional dynamical neural networks: first, each input pattern can be encoded in a vector instead of a matrix; second, the connection weights can be determined analytically; third, due to its dynamical nature, it has the ability to capture temporal patterns. In the previous studies of this topic, all patterns were encoded as stable periodic solutions of the oscillator network. In this paper, we continue to explore the oscillator ring for pattern recognition. Specifically, we propose algorithms, which use the chaotic dynamics of the closed loops of Stuart---Landau oscillators as artificial neurons, to recognize randomly generated fractal patterns. We manipulate the number of neurons and initial conditions of the oscillator ring to encode fractal patterns. It is worth noting that fractal pattern recognition is a challenging problem due to their discontinuity nature and their complex forms. Computer simulations confirm good performance of the proposed algorithms.

New Discontinuity-Induced Bifurcations in Chua's Circuit

Chua's circuit, an archetypal example of nonsmooth dynamical systems, exhibits mostly discontinuous bifurcations. More complex dynamical phenomena of Chua's circuit are presented here due to discontinuity-induced bifurcations. Some new kinds of classical bifurcations are revealed and analyzed, including the coexistence of two classical bifurcations and bifurcations of equilibrium manifolds. The local dynamical behavior of the boundary equilibrium points located on switch boundaries is found to be determined jointly by the Jacobian matrices evaluated before and after switching. Some new discontinuous bifurcations are also observed, such as the coexistence of two discontinuous and one classical bifurcation.

The isolation characteristics of an archetypal dynamical model with stable-quasi-zero-stiffness

In this paper, a single-degree-of-freedom geometrically nonlinear oscillator with stable-quasi-zero-stiffness (SQZS) is presented, which can be extensively applied in vibration isolation due to its high static load bearing capacity and low dynamic stiffness. This model comprises a lumped mass denoeing the isolated object and a pair of horizontal springs providing negative stiffness paralleled with a vertical linear spring to bear the load. The equation of motion of the system is formulated with an originally irrational nonlinearity based upon SD oscillator instead of the conventionally approximate Duffing system of polynomial type, which will produce results with a high precision unquestionably, especially for the prediction of a large displacement behaviour. The frequency response characteristics particularly for transmissibility of the model, subjected to harmonic forcing and vibrating base, are obtained by using an extended averaging approach to achieve the parameter optimization for maximum frequency band of isolation. Furthermore, numerical simulations are carried out to detect the complex dynamical phenomena of periodic, chaotic motions and coexistence of multiple solutions, and so on, in addition to verifying the analytical results. Finally, an interesting strategy is proposed to extend the frequency band of isolation by controlling the initial value within the attraction basin of a small amplitude attractor, which can be utilized for an effective isolation. The results presented herein provide an insight of dynamics into the SQZS mechanism for its application in vibration engineering.

Bifurcation analysis of PID-controlled neuromuscular blockade in closed-loop anesthesia

The problem of PID-controlled neuromuscular blockade (NMB) in closed-loop anesthesia is considered. Contrary to the usual practice of designing PID-controllers for nonlinear systems on the basis of a linearized model and online tests, bifurcation analysis is utilized in this paper for that purpose. Two nonlinear Wiener models for the NMB are considered: a conventional pharmacokinetic/pharmacodynamic (PK/PD) model and a parsimonious model suitable for online parameter estimation. The models under a PID feedback are analyzed in order to discern the safe intervals of the controller parameters that are not subject to complex dynamical phenomena. The parsimony of the mathematical model is instrumental in minimizing the number of bifurcation parameters. The analyses show that the closed-loop systems undergo Andronov–Hopf bifurcation at a point in the model parameter space giving rise to nonlinear oscillations. For steeper, but still feasible slopes of the nonlinear function parameterizing the static nonlinearity of the Wiener models, deterministic chaos can arise in the closed loop for lower concentrations of the anesthetic drug. A model-based PID-controller tuning procedure is suggested that guarantees a certain settling time and robustness margin of the resulting loop with respect to the bifurcation. The tuning procedure is illustrated on mathematical models identified from patient data and the corresponding PID controllers.

Renormalization aspects of chaotic strings

Chaotic strings are a class of non-hyperbolic coupled map lattices, exhibiting a rich structure of complex dynamical phenomena with a surprising correspondence to physical contents. In this paper we introduce different types and models for chaotic strings, where 2B-strings with finite length are considered in detail. We demonstrate possibilities to extract renormalized quantities, which are expected to describe essential properties of the string.