Chaotic Oscillations Research Articles

Solitary routes to chimera states.

We show how solitary states in a system of globally coupled FitzHugh-Nagumo oscillators can lead to the emergence of chimera states. By a numerical bifurcation analysis of a suitable reduced system in the thermodynamic limit we demonstrate how solitary states, after emerging from the synchronous state, become chaotic in a period-doubling cascade. Subsequently, states with a single chaotic oscillator give rise to states with an increasing number of incoherent chaotic oscillators. In large systems, these chimera states show extensive chaos. We demonstrate the coexistence of many of such chaotic attractors with different Lyapunov dimensions, due to different numbers of incoherent oscillators.

Turbulence and heat release rate network structure in hydrogen-enriched combustion

Complex network theory is used to analyze the spatiotemporal dynamics of the PRECCINSTA swirl burner, operating on hydrogen-methane fuel blends. At a power setting of 15 kW with equivalence ratio of 0.8 and hydrogen fuel fraction (HFF) ranging from 0% to 80%, period-1 and period-2 limit cycle oscillations as well as chaotic oscillations were observed. A turbulence network was constructed from the vorticity data obtained with particle image velocimetry. In addition, a heat release rate (HRR) correlation network was constructed from chemiluminescence images. Although the two networks concern a common thermoacoustic system, both exhibit significant differences: the turbulence network displays power law degree distribution, maintains small world property for all HFFs and is scale-free only in the absence of hydrogen enrichment. The HRR correlation network does not feature these properties, but hints at an asymmetric coupling between the heat release rate and the acoustic pressure for all HFFs. Furthermore, the HRR network is responsive to changes in HFF and the corresponding shifts in the dynamical state as well as in the root-mean-square of acoustic pressure. On the contrary, the turbulence network displays no such sensitivity and its properties are almost constant in the upper HFF range. It exhibits stationary hub structures for all the fuel blends tested, whereas the HRR correlation network is hub-free.

Nonlinear optical effects and optomechanical oscillations in hollow Whispering Gallery Mode microresonators: Coexistence, suppression, amplification and route to chaos

Whispering Gallery Mode (WGM) hollow microcavities turn out to be the site of an extremely rich and complex phenomenological scenario when pumped with a continuous-wave laser source. The coexistence of numerous non-linear and optomechanical effects have been reviewed in this paper. In our previous works we have investigated and described non-linear emissions as the stimulated Brillouin and Raman scattering, the degenerated and non-degenerated Kerr effects, such as four wave mixing (FWM). These effects happened concomitantly to parametric optomechanical oscillations which are the consequence of the radiation pressure. We have confirmed the regenerative oscillation of acoustic eigenmodes of the cavity leading to parametric instabilities and the activation of optomechanical chaotic oscillations. Finally, we have demonstrated that the blue-side excitation of WGM resonances lead to the chaos transition with a spectral evolution depending on the cavity size.

A tristable nonlinear energy sink

The tristable characteristics are mostly used in the field of energy harvesting and vibration isolator, but are rarely devoted to in the field of vibration absorption. In this paper, a tristable nonlinear energy sink (TNES) model is proposed for vibration absorption. The TNES is constructed by fixing a magnet in the middle of the clamping pre-pressure beam, and setting a magnet on both sides of the fixed magnet. By adjusting the magnet position, the proposed TNES can be transformed into bistable NES (BNES) and monostable NES (MNES). The TNES is coupled with the linear oscillator (LO), and the differential equation of motion of the coupled system is obtained with the Hamiltonian principle. The approximate analytical method and the numerical method are mutually verified. The vibration reduction efficiency of the TNES under harmonic excitation is presented. The influence of the parameters of the TNES on the dynamics and vibration reduction efficiency is studied. The optimal parameters of the TNES are obtained by the particle swarm optimization (PSO) algorithm. Compared with the corresponding bistable NES and monostable NES, it shows that the tristable NES has obvious advantages in vibration suppression efficiency. The effects of the TNES barrier depths and the excitation amplitude on the vibration reduction efficiency are revealed. The TNES can dissipate energy through chaotic inter-well oscillation between three stable positions, and achieve the purpose of effectively suppressing vibration.

New memristor-less, resistor-less, two-OTA based grounded and floating meminductor emulators and their applications in chaotic oscillators

This paper presents a simple realization of the meminductor emulator using two operational transconductance amplifiers (OTAs) and two capacitors. To obtain different configurations of meminductor in the proposed design, one of the OTA has been replaced by a multi-output transconductance amplifier (MO-OTA). MO-OTA is a slightly modified version of conventional OTA that has multiple outputs. The design flexibilities offered by MO-OTA helped in achieving decremental as well as incremental configurations for both grounded and floating meminductor emulators. The proposed meminductor emulator realizations are very simple because these emulators do not employ an analog multiplier, resistor, or memristor. The pinched hysteresis curves have been acquired for the frequency range of 600 kHz–900 kHz for different values of capacitors. The proposed meminductor emulators have been designed and simulated with the help of the LTspice tool using TSMC 180 nm CMOS technology parameters. The pinched hysteresis loops, transient analysis, and non-volatility tests of the proposed meminductor emulators confirm their satisfactory behavior as meminductor. The functioning of proposed meminductor emulator circuits has also been tested by demonstrating their use in chaotic oscillators.

Energy harvesting of inverted piezoelectric flags in an oscillating flow

The energy harvesting potential of flexible structures (e.g. flags) made of piezoelectric materials has drawn rapidly increasing attention in recent years. In this work, we numerically study the energy harvesting performance of an inverted piezoelectric flag in an oscillating flow using an immersed boundary-lattice Boltzmann method for Reynolds number of 100, mass ratio of 2.9 and non-dimensional bending stiffness of 0.26 which correspond to the maximum flapping amplitude for a single inverted flag in a uniform flow. 2D simulations are conducted by varying the ellipticity (e), the ratio R of the frequency of the oscillating flow to the fundamental natural frequency of the flag and the horizontal velocity amplitude (Au) of the flow. Three coupling regimes at Au=0.5 are identified: chaotic oscillations regime I (0.1≤R≤1 ), large periodic and symmetric oscillation regime IIa1.1≤R≤1.5and IIb(2.1≤R≤3.0), and small periodic and asymmetric oscillation regime III(1.6≤R≤2). The maximum mean electrical power coefficient C¯P occurs in regime IIa at R=1.5 with Au=0.5, α (piezo-mechanical coupling parameter) = 0.5, and β (piezo-electric tuning parameter) = 1.5. C¯P is 0.10 for a single inverted flag, and is 148%, higher than that of the corresponding flag in the uniform flow. This improvement is attributed to the higher flapping angular amplitude (180°), higher ratio of the flapping frequency to the oscillating frequency (virtually constant at 0.5) of the flags, and constructive vortex interaction in regime IIa.

A new buffering theory of social support and psychological stress.

A dynamical model linking stress, social support, and health has been recently proposed and numerically analyzed from a classical point of view of integer-order calculus. Although interesting observations have been obtained in this way, the present work conducts a fractional-order analysis of that model. Under a periodic forcing of an environmental stress variable, the perceived stress has been analyzed through bifurcation diagrams and two well-known metrics of entropy and complexity, such as spectral entropy and C0 complexity. The results obtained by numerical simulations have shown novel insights into how stress evolves with frequency and amplitude of the perturbation, as well as with initial conditions for the system variables. More precisely, it has been observed that stress can alternate between chaos, periodic oscillations, and stable behaviors as the fractional order varies. Moreover, the perturbation frequency has revealed a narrow interval for the chaotic oscillations, while its amplitude may present different values indicating a low sensitivity regarding chaos generation. Also, the perceived stress has been noted to be highly sensitive to initial conditions for the symptoms of stress-related ill-health and for the social support received from family and friends. This work opens new directions of research whereby fractional calculus might offer more insight into psychology, life sciences, mental disorders, and stress-free well-being.

Use of Chaotic Oscillations for Precoding and Synchronization in OFDM

This paper proposes a novel linear precoding method for Orthogonal Frequency Division Multiplexing (OFDM) based on the employment of the chaotic waveforms generated by the fourth-order chaotic oscillator and orthonormalized by the Gram-Schmidt process. The proposed linear precoding method is aimed to increase resilience to the multipath propagation issues and reduce the Peak-to-Average Power Ratio (PAPR) of the transmitted signal. Moreover, the chaotic waveform enables novel timing synchronization methods to be implemented in the receiver. The modelling of baseband Linear Precoded OFDM (LP-OFDM) data transmission system with Rayleigh channel has been performed in Simulink environment to validate the proposed method and to compare the performance to the classic precoding methods, such as Walsh-Hadamard Transform (WHT). Experiments have shown that in a high Signal-to-Noise Ratio (SNR) scenario, the employment of the novel precoding scheme allows reducing Bit Error Ratio (BER) by several dB compared to non-precoded OFDM. The proposed precoding method leads to the reduction of PAPR; however, it is not as efficient as classical precoding schemes, such as WHT. Experimental evidence of synchronization of the chaotic oscillators within 50 samples long time interval is presented.

Experimental investigation on flow-induced vibration control of flexible risers fitted with new configuration of splitter plates

Experimental investigations within the Reynolds number range of 2600–9100 were performed to examine the effectiveness of serrated splitter plates in controlling the flow-induced vibration (FIV) of flexible risers. Conventional splitter plates were arranged for contrast to highlight the unparalleled features of the new splitter plates. Bending strains of FIV responses were measured and converted into critical parameters of displacement, dominant frequency, oscillation orbit, suppression efficiency, and fatigue damage for analysis. Test results illustrate that for flexible risers with conventional or serrated splitter plates, a galloping phenomenon occurs at a low reduced velocity, and standing wave characteristics invariably dominate their FIV responses. As the reduced velocity varies increasingly, the conventional splitter plate triggers a continuous monotonous increase in the FIV amplitudes of controlled risers. However, owing to the phase difference in the interaction of the shear layers between proximal plates and distal plates, the FIV amplitudes of the risers fitted with serrated splitter plates start to stay constant virtually and then distinctly decline with appearing chaotic oscillation trajectories. Moreover, the control performance of serrated splitter plates with uniformly distributed serrations in alleviating FIV amplitudes is optimal at high reduced velocities; whereas excessively dense or sparse serrations can reduce FIV inhibition.

A Cubic Memristive System with Two Twin Rössler-Type Chaotic Attractors Symmetrical About an Invariant Plane

Memristive circuits and systems have been widely studied in the last years due to their potential applications in several technological areas. They are capable of producing nonlinear periodic and chaotic oscillations, due to their locally-active characteristics. In this paper, we consider a cubic four-parameter differential system which models a memristive circuit consisting of three elements: a passive linear inductor, a passive linear capacitor and a locally-active current-controlled generic memristor. This system has a saddle-focus equilibrium point at the origin, whose global stable and unstable manifolds are, respectively, the [Formula: see text]-axis and the plane [Formula: see text], which are invariant sets where the dynamic is linear. We show that this structure can generate two twin Rössler-type chaotic attractors symmetrical with respect to the plane [Formula: see text]. We describe the mechanism of creation of these chaotic attractors, showing that, although being similar to the Rössler attractor, the twin attractors presented here have simpler structural mechanism of formation, since the system has no homoclinic or heteroclinic orbits to the saddle-focus, as presented by the Rössler system. The studied memristive system has the rare property of having chaotic dynamics and an invariant plane with linear dynamic, which is quite different from other chaotic systems presented in the literature that have invariant surfaces filled by equilibrium points. We also present and discuss the electronic circuit implementation of the considered system and study its dynamics at infinity, via the Poincaré compactification, showing that all the solutions, except the ones contained in the plane [Formula: see text], are bounded and cannot escape to infinity.

Dynamics of Fractional-Order Chaotic Rocard Relaxation Econometric System

The chaotic Rocard relaxation econometric system was proposed earlier than the Lorenz system. In this paper, the fractional-order Rocard system is investigated and its dynamical behaviors are analyzed by bifurcation diagram, Lyapunov exponent and spectral entropy (SE) complexity. The stability analysis is conducted and the conditions for the fractional-order system to maintain chaotic or periodic state are investigated. Compared with the integer-order counterpart, the fractional-order system generates more periodic parameter areas, and they play vital roles in economic cycles and economic chaos control. We also found that the fractional-order oscillator has a larger Lyapunov exponent than its integer-order counterpart. By applying the 0–1 test method, the minimum order for the system for chaotic oscillation is determined. The SE complexity evidence further confirms involving fractional calculus that enriches dynamical behaviors. These results indicate the promising applications in chaos control of the economic system.

Occasional coupling enhances amplitude death in delay-coupled oscillators.

This paper aims to study amplitude death in time delay coupled oscillators using the occasional coupling scheme that implies intermittent interaction among the oscillators. An enhancement of amplitude death regions (i.e., an increment of the width of the amplitude death regions along the control parameter axis) can be possible using the occasional coupling in a pair of delay-coupled oscillators. Our study starts with coupled limit cycle oscillators (Stuart-Landau) and coupled chaotic oscillators (Rössler). We further examine coupled horizontal Rijke tubes, a prototypical model of thermoacoustic systems. Oscillatory states are highly detrimental to thermoacoustic systems such as combustors. Consequently, a state of amplitude death is always preferred. We employ the on-off coupling (i.e., a square wave function), as an occasional coupling scheme, to these coupled oscillators. On monotonically varying the coupling strength (as a control parameter), we observe an enhancement of amplitude death regions using the occasional coupling scheme compared to the continuous coupling scheme. In order to study the contribution of the occasional coupling scheme, we perform a detailed linear stability analysis and analytically explain this enhancement of the amplitude death region for coupled limit cycle oscillators. We also adopt the frequency ratio of the oscillators and the time delay between the oscillators as the control parameters. Intriguingly, we obtain a similar enhancement of the amplitude death regions using the frequency ratio and time delay as the control parameters in the presence of the occasional coupling. Finally, we use a half-wave rectified sinusoidal wave function (motivated by practical reality) to introduce the occasional coupling in time delay coupled oscillators and get similar results.

Analysis of the onset of a regime shift and detecting early warning signs of major population changes in a two-trophic three-species predator-prey model with long-term transients.

Identifying early warning signs of sudden population changes and mechanisms leading to regime shifts are highly desirable in population biology. In this paper, a two-trophic ecosystem comprising of two species of predators, competing for their common prey, with explicit interference competition is considered. With proper rescaling, the model is portrayed as a singularly perturbed system with fast prey dynamics and slow dynamics of the predators. In a parameter regime near singular Hopf bifurcation, chaotic mixed-mode oscillations (MMOs), featuring concatenation of small and large amplitude oscillations are observed as long-lasting transients before the system approaches its asymptotic state. To analyze the dynamical cause that initiates a large amplitude oscillation in an MMO orbit, the model is reduced to a suitable normal form near the singular-Hopf point. The normal form possesses a separatrix surface that separates two different types of oscillations. A large amplitude oscillation is initiated if a trajectory moves from the "inner" to the "outer side" of this surface. A set of conditions on the normal form variables are obtained to determine whether a trajectory would exhibit another cycle of MMO dynamics before experiencing a regime shift (i.e. approaching its asymptotic state). These conditions serve as early warning signs for a sudden population shift as well as detect the onset of a regime shift in this ecological model.

Dynamics of forced and unforced autophoretic particles

Chemically active, or autophoretic, particles that isotropically emit or absorb solute molecules undergo spontaneous self-propulsion when their activity is increased beyond a critical Péclet number ($Pe$). Here, we conduct numerical computations, using a spectral element based method, of a rigid, spherical autophoretic particle in unsteady rectilinear translation. The particle can be freely suspended (or ‘unforced’) or subject to an external force field (or ‘forced’). The motion of an unforced particle progresses through four regimes as$Pe$is increased: quiescent, steady, stirring and chaos. The particle is stationary in the quiescent regime, and the solute profile is isotropic about the particle. At$Pe=4$the fore–aft symmetry in the solute profile is broken, resulting in its steady self-propulsion. Our computations indicate that the self-propulsion speed scales linearly with$Pe-4$near the onset of self-propulsion, as has been predicted in previous studies. A further increase in$Pe$gives rise to the stirring regime at$Pe\approx 27$, where the fluid undergoes recirculation, while the particle remains essentially stationary. As$Pe$is increased even further, the particle dynamics is marked by chaotic oscillations at$Pe\approx 55$and higher, which we characterize in terms of the mean square displacement and velocity autocorrelation of the particle. Our results for an autophoretic particle under a weak external force are in good agreement with recent asymptotic predictions (Saha, Yariv & Schnitzer,J. Fluid Mech., vol. 916, 2021, p. A47). Additionally, we demonstrate that the strength and temporal scheduling of the external force may be tuned to modulate the chaotic dynamics at large$Pe$.

A signature algorithm based on chaotic maps and factoring problems

There has been a great deal of interest in researching the behavior of chaotic systems over the past decade. They are distinguished by their sensitivity to initial conditions, continuous broad-band power spectrum, and similarity to random behavior. Chaos has possible applications in the encryption, compression, and modulation blocks of a digital communication system. The possibility of self-synchronization of chaotic oscillations has spawned a deluge of research on the cryptographic applications of chaos. FAC (Factoring) and CM (Chaotic maps) are used in this research to develop a new signature system (CM). Our system has a higher security level than any other based on a single hard number-theoretic problem since it is extremely unlikely that FAC and CM can be solved effectively at the same time. We further demonstrate that the scheme’s performance involves only minimal operations in signing and validating logarithms, and that it is impervious to attack.

Drastically reduced sensor hardware for solvable chaos‐based sonar

An acoustic ranging demonstration is presented in a reflective environment using solvable chaos with novel and practical hardware advances. Previously, solvable chaos was optimally detected using ideal alignment, random codes and an 11 element microphone array for environments free of multipath reflections. In this work, ranging experiments were performed with drastically reduced sensor hardware and measurements were extended to include non-ideal effects such as multipath and angle misalignment from reduced line of sight (LOS). Importantly, this new scheme not only reduces the size and power of these systems by over 83%, a novel triggering technique allows for triggering on targeted bit sequences and produces sidelobe levels that are independent of oscillator dynamics. Ranging results from a closed, reflective cavity obtained using a 10 kHz solvable chaotic oscillator are reported. Measured sidelobe levels were typically between −1.80 dB and −15.85 dB. The resulting demonstration shows successful ranging despite drastically reduced hardware and environmental imperfections.

Overview of Arbitrarily High-Order Adjoint Sensitivity and Uncertainty Quantification Methodology for Large-Scale Systems

This work reviews from a unified viewpoint the concepts underlying the “nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Response-Coupled Forward/Adjoint Linear Systems” (nth-CASAM-L) and the “nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems” (nth-CASAM-N) methodologies. The practical application of the nth-CASAM-L methodology is illustrated for an OECD/NEA reactor physics benchmark, while the practical application of the nth-CASAM-N methodology is illustrated for a nonlinear model of reactor dynamics that exhibits periodic and chaotic oscillations. As illustrated both by the general theory and by the examples reviewed in this work, both the nth-CASAM-L and nth-CASAM-N methodologies overcome the curse of dimensionality in sensitivity analysis. The availability of efficiently and exactly computed sensitivities of arbitrarily high order can lead to major advances in all areas that need such high-order sensitivities, including data assimilation, model calibration, uncertainty reduction, and predictive modeling.

Local and Network Dynamics of a Non-Integer Order Resistor–Capacitor Shunted Josephson Junction Oscillators

Spiral waves are an intriguing phenomenon that can be found in a variety of chemical and biological systems. We consider the fractional-order resistor–capacitor shunted Josephson junction chaotic oscillator to investigate the spiral wave pattern. For a preliminary understanding, we look at the dynamics of isolated FJJs and show that infinitely coexisting periodic and chaotic attractors depend on the fractional order. In addition, we perform bifurcation analysis to show the dynamical transition of the attractors as a function of fractional order and basin stability analysis to show the infinitely coexisting attractors. This is followed by the existence of spiral waves which is observed under various intrinsic and extrinsic system parameters. Finally, the impact of noise on SW is also analyzed by dispersing it to the entire stimulation period or defined time-period.

Periodic and chaotic oscillations in mutual-coupled mid-infrared quantum cascade lasers

Dynamic states in mutual-coupled mid-infrared quantum cascade lasers (QCLs) were numerically investigated in the parameter space of injection strength and detuning frequency based on the Lang–Kobayashi equations model. Three types of period-one states were found, with different periods of injection time delay τ inj, 2τ inj, and reciprocal of the detuning frequency. Besides, square-wave, quasi-period, pulse-burst and chaotic oscillations were also observed. It is concluded that external-cavity periodic dynamics and optical modes beating are the mainly periodic dynamics. The interaction of the two periodic dynamics and the high-frequency dynamics stimulated by strong injection induces the dynamic states evolution. This work helps to understand the dynamic behaviors in QCLs and shows a new way to mid-infrared wide-band chaotic laser.

The structure of uni-directional chain for the synchronization of networked chaotic systems

In this paper, we introduce the structure of uni-directional chain (UDC) as a benchmark to probe the synchronizability of networked chaotic system. By studying the relation between the length of UDC and its synchronizability, we identify a region of coupling strength, referred to as strong controllable region (SCR), where a UDC with arbitrary length could realize synchronization that is robust to non-local disturbance. More importantly, it is found that there exists a kind of chaotic oscillators where the region of negative master stability function (MSF) overlaps with the SCR. As MSF implements local stability analysis to provide the necessary condition for the appearance of synchronization, while SCR permits stable synchronization corresponding to non-local disturbance pertinent to sufficient condition, the role of MSF for such kind of oscillator may be equivalent to provide sufficient condition of network synchronization. Besides, we propose methods for the control of network synchronization based on the characteristic of UDC. For networks with fixed structure, by obtaining a minimal spanning tree, each node chooses one of its neighbors and follows its dynamics with uniform coupling strength to realize synchronization. For growing networks, by establishing directed connection from newly added nodes to existing nodes. An interacting structure which is similar to the combination of a group of UDCs is formed and the system is guaranteed to be controlled within the SCR.