Chaotic Regime Research Articles

In this study, we explore the interplay between \mathcal{PT} 𝒫 𝒯 -symmetry and quantum chaos in a non-Hermitian dynamical system. We consider an extension of the standard diagnostics of quantum chaos, namely the complex level spacing ratio and out-of-time-ordered correlators (OTOCs), to study the \mathcal{PT} 𝒫 𝒯 -symmetric quantum kicked rotor model. The kicked rotor has long been regarded as a paradigmatic dynamic system to study classical and quantum chaos. By introducing non-Hermiticity in the quantum kicked rotor, we uncover new phases and transitions that are absent in the Hermitian system. From the study of the complex level spacing ratio, we locate three regimes – one which is integrable and \mathcal{PT} 𝒫 𝒯 -symmetry, another which is chaotic with \mathcal{PT} 𝒫 𝒯 -symmetry and a third which is chaotic but with broken \mathcal{PT} 𝒫 𝒯 -symmetry. We find that the complex level spacing ratio can distinguish between all three phases. Since calculations of the OTOC can be related to those of the classical Lyapunov exponent in the semi-classical limit, we investigate its nature in these regimes and at the phase boundaries. In the phases with \mathcal{PT} 𝒫 𝒯 -symmetry, the OTOC exhibits behaviour akin to what is observed in the Hermitian system in both the integrable and chaotic regimes. Moreover, in the \mathcal{PT} 𝒫 𝒯 -symmetry broken phase, the OTOC demonstrates additional exponential growth stemming from the complex nature of the eigenvalue spectrum at later times. We derive the analytical form of the late-time behaviour of the OTOC. By defining a normalized OTOC to mitigate the effects caused by \mathcal{PT} 𝒫 𝒯 -symmetry breaking, we show that the OTOC exhibits singular behaviour at the transition from the \mathcal{PT} 𝒫 𝒯 -symmetric chaotic phase to the \mathcal{PT} 𝒫 𝒯 -symmetry broken, chaotic phase.

This research shows an analytical and numerical analysis of a nonlinear three-dimensional dynamical system controlled by a unique control parameter, named α. The system exhibits self-excited oscillations through local bifurcations for negative α, including saddle-node and supercritical Hopf bifurcations, leading to periodic orbits and a sequence of period-doubling transitions into chaos. On the other hand, when α is negative and there are no equilibrium points, the system shows long term oscillations that last for a long time through hidden attractors—bounded chaotic dynamics with basins of attraction that are not connected to any equilibrium. Numerical continuation and Lyapunov spectrum analysis confirm the simultaneous existence of periodic, quasiperiodic, and chaotic regimes. The results demonstrate the intricate interplay between local bifurcations and global nonlinear frameworks, emphasizing the distinctive routes to chaos and the emergence of hidden dynamics in systems lacking stability.

Abstract A nonlinear torsional vibration model of precision harmonic drives is developed, incorporating stiffness degradation, static transmission errors, piece-wise backlash, and periodic torque excitation. Dynamic responses are quantified through angular domain solutions of the dimensionless governing equations using the Runge−Kutta method. The research identifies two types of bursting oscillations caused by stiffness and backlash, and reveals their triggering mechanisms. Furthermore, the effects of factors such as the damping coefficient, excitation torque, and speed on the chaotic characteristics are analyzed using bifurcation diagrams, time histories, phase trajectories, and Poincaré maps. The results indicate that with prolonged service time, interface damage-induced stiffness degradation or enlargement of backlash may simultaneously induce bursting oscillations and chaotic motion. Furthermore, the amplitude of the torque fluctuation component critically governs the emergence of bursting oscillations, with these phenomena occurring when Ab′ ≥ To′. The dynamic behavior of harmonic drives exhibits strong dependence on operating conditions, with numerical simulations demonstrating that progressive increases in speed and load torque induce transitions toward increasingly unstable chaotic regimes. Increasing damping can suppress this transition. These findings elucidate the mechanisms underlying undesirable nonlinear oscillations in the system, while establishing theoretical foundations for control strategies in precision harmonic drive applications.

Prey refuge is widely recognized as a critical ecological mechanism that shapes species persistence and ecosystem stability. While most existing studies have focused on lower-dimensional predator–prey models, the stabilizing and destabilizing roles of refuge in higher-dimensional trophic structures remain underexplored. In this study, we investigate the dynamics of a four-dimensional diamond-shaped food web in which refuge is incorporated at the basal prey level. Using isospike diagrams as a novel diagnostic tool, we uncover a spectrum of complex dynamical behaviors, including shrimp-shaped periodic windows, period bubbling, coexistence of multiple attractors, and predator extinction scenarios. These periodic windows form well-structured islands embedded within chaotic regimes, demonstrating that small parameter changes can recover stability from apparent randomness. Our results reveal that moderate refuge levels suppress chaos and promote equilibrium, whereas excessive refuge induces oscillations or top predator extinction, underscoring the double-edged ecological role of this mechanism. These findings extend the classical stabilizing interpretation of prey refuge to higher-dimensional systems and highlight its potential to either buffer or destabilize ecological networks depending on intensity. By integrating nonlinear dynamical analysis with ecological interpretation, our work provides new insights into the resilience and fragility of complex food webs and establishes isospike diagrams as a powerful tool for exploring higher-dimensional ecological models. Overall, the study highlights the dual role of prey refuges: in most cases stabilizing multitrophic dynamics, but occasionally acting as a source of instability in complex ecological networks.

Understanding how neurons respond to weak external signals is crucial for accurate signal transmission and processing in both individual nerve cells and interconnected neuronal networks. One mechanism for the detection of these responses is through resonances. In this paper, we numerically investigate the firing patterns induced in a silent Huber-Braun neuron by a sinusoidal external force. We observe complex resonance patterns, including a sequence of frequency-locking exhibited in a Devil’s Staircase structure. Furthermore, we also explore the emergence of multistability induced by the nonlinear resonance. This multistability manifests as the coexistence of three attractors, such as periodic spiking, chaotic spiking, and subthreshold oscillations. The dynamical behaviors are comprehensively analyzed using time series, bifurcation diagrams, phase portraits, and the basin of attraction. In addition, we compute the maximum Lyapunov exponent to verify chaotic regimes, and estimate the fractal dimension of basin boundaries using the uncertainty exponent. We also analyze the energy consumption of resonance-induced firing patterns and coexisting attractors. The results presented in this paper have important implications for understanding the detection of subthreshold signals and the encoding of stimulus information within a neuron’s firing patterns.

Continuous time crystals (CTCs) are a phase of matter characterized by spontaneous breaking of continuous time-translation symmetry. Recently, CTCs have garnered interest due to breakthroughs in experimental implementation. Here we report the experimental observation of CTCs in noble-gas nuclear spins and uncover previously unexplored dynamical phenomena. We observe that the CTCs manifest as persistent limit cycle oscillations of nuclear spins, with coherence times exceeding hours. Notably, these oscillations are robust against noise perturbations and exhibit random time phases upon repetitive realization, epitomizing continuous time-translation symmetry-breaking intrinsic to CTCs. Additionally, we observe a dynamical phase featuring quasi-periodic oscillations and random time phases, indicating the emergence of the continuous time quasi-crystals proposed by recent theories. By varying the feedback strength and magnetic gradient, we observe complex dynamical phase transitions between time crystal phases and chaotic regimes. This work broadens the catalog of phases of spin gases and unlocks opportunities in precision measurements.

A minimal neural network at the edge of chaos

This paper presents a detailed study of the (3+1)-dimensional Zakharov–Kuznetsov–Burgers equation to investigate shock-wave phenomena in dusty plasmas with quantum effects. The model provides significant physical insight into nonlinear dispersive and dissipative structures arising in charged-dust–ion environments, corresponding to both laboratory and astrophysical plasmas. We then perform a qualitative, numerically assisted dynamical analysis using bifurcation diagrams, multistability checks, return maps, Poincaré sections, and phase portraits. For both the unperturbed and a perturbed system, we identify chaotic, quasi-periodic, and periodic regimes from these numerical diagnostics; accordingly, our dynamical conclusions are qualitative. We also examine frequency-response and time-delay sensitivity, providing a qualitative classification of nonlinear behavior across a broad parameter range. After establishing the global dynamical picture, traveling-wave solutions are obtained using the Paul–Painlevé approach. These solutions represent shock and solitary structures in the plasma system, thereby bridging the analytical and dynamical perspectives. The significance of this study lies in combining a detailed dynamical framework with exact traveling-wave solutions, allowing a deeper understanding of nonlinear shock dynamics in quantum dusty plasmas. These results not only advance theoretical plasma modeling but also hold potential applications in plasma-based devices, wave propagation in optical fibers, and astrophysical plasma environments.

We present a stochastic framework for describing the transition from regular to chaotic dynamics in quantum billiards by incorporating a background scale of fluctuations in the level spacing evolution. Analytic expressions for the nearest-neighbor spacing distribution and power spectral density accurately capture the behavior across integrable, chaotic, and mixed-dynamics regimes. Remarkably, our model predicts a plateau formation in the level repulsion strength as the system approaches chaos-a behavior related to the plasma model of many-body localization and freezing transitions in Gaussian free fields. Comparisons to numerical results for quantum limaçon and mushroom billiards confirm the reliable description of the complex crossover between order and chaos. Our approach provides a complementary perspective to microscopic methods, particularly valuable for understanding the emergence of non-Gaussian statistics and multiscale phenomena in the crossover regime.

We introduce the α-Gauss-Logistic map, a new nonlinear dynamics constructed by composing the logistic and α-Gauss maps. Explicitly, our model is given by x_{t+1}=f_{L}(x_{t})x_{t}^{-α}-⌊f_{L}(x_{t})x_{t}^{-α}⌋, where f_{L}(x_{t})=rx_{t}(1-x_{t}) is the logistic map and ⌊...⌋ is the integer part function. Our investigation reveals a rich phenomenology depending solely on two parameters, r and α. For α<1, the system exhibits multiple period-doubling cascades to chaos as the parameter r is increased, interspersed with stability windows within the chaotic attractor. In contrast, for 1≤α<2, the onset of chaos is abrupt, occurring without any prior bifurcations, and the resulting chaotic attractors emerge without stability windows. For α≥2, the regular behavior is absent. The special case of α=1 allows an analytical treatment, yielding a closed-form formula for the Lyapunov exponent and conditions for an exact uniform invariant density, using the Perron-Frobenius equation. Chaotic regimes for α=1 can exhibit gaps or be gapless. Surprisingly, the golden ratio Φ marks the threshold for the disappearance of the largest gap in the regime diagram. Additionally, at the edge of chaos in the abrupt transition regime, the invariant density approaches a q-Gaussian with q=2, which corresponds to a Cauchy distribution.

Evolution into chaos - Implications of the trade-off between transmissibility and immune evasion.

Krylov complexity, a quantum complexity measure which uniquely characterizes the spread of a quantum state or an operator, has recently been studied in the context of quantum chaos. However, the definitiveness of this measure as a chaos quantifier is in question in light of its strong dependence on the initial condition. This article clarifies the connection between the Krylov complexity dynamics and the initial operator or state. We find that the saturation value of Krylov complexity depends monotonically on the inverse participation ratio (IPR) of the initial condition in the eigenbasis of the Hamiltonian. We explain the reversal of the complexity saturation levels observed in [Phys. Rev. E 107, 024217 (2023)2470-004510.1103/PhysRevE.107.024217] using the initial spread of the operator in the Hamiltonian eigenbasis. IPR dependence is present even in the fully chaotic regime, where popular quantifiers of chaos, such as out-of-time-ordered correlators and entanglement generation, show similar behavior regardless of the initial condition. Krylov complexity averaged over many initial conditions still does not characterize chaos.

The rising computational and energy demands of artificial intelligence systems urge the exploration of alternative software and hardware solutions that exploit physical effects for computation. According to machine learning theory, a neural network-based computational system must exhibit nonlinearity to effectively model complex patterns and relationships. This requirement has driven extensive research into various nonlinear physical systems to enhance the performance of neural networks. In this paper, we propose and theoretically validate a reservoir-computing system based on a single bubble trapped within a bulk of liquid. By applying an external acoustic pressure wave to both encode input information and excite the complex nonlinear dynamics, we showcase the ability of this single-bubble reservoir-computing system to forecast a Hénon benchmarking time series and undertake classification tasks with high accuracy. Specifically, we demonstrate that a chaotic physical regime of bubble oscillation—where tiny differences in initial conditions lead to wildly different outcomes, making the system unpredictable despite following clear rules, yet still suitable for accurate computations—proves to be the most effective for such tasks.

Unstable periodic orbits (UPOs) are believed to be the underlying dynamical structures of spatiotemporal chaos and turbulence. Finding these UPOs is, however, notoriously difficult. Matrix-free loop convergence algorithms deform entire space-time fields (loops) until they satisfy the evolution equations. Initial guesses for these robust variational convergence algorithms are thus periodic space-time fields in a high-dimensional state space, rendering their generation highly challenging. Usually guesses are generated with recurrency methods, which are most suited to shorter and more stable periodic orbits. Here we propose an alternative, data-driven method for generating initial guesses, enabled by the periodic nature of the guesses for loop convergence algorithms: While the dimension of the space used to discretize fluid flows is prohibitively large to construct suitable initial guesses, the dissipative dynamics will collapse onto a chaotic attractor of far lower dimension. We use an autoencoder to obtain a low-dimensional representation of the discretized physical space for the one-dimensional Kuramoto-Sivashinksy equation in chaotic and hyperchaotic regimes. In this low-dimensional latent space, we construct loops based on the latent POD modes with random periodic coefficients, which are then decoded to physical space and used as initial guesses. These loops are found to be realistic initial guesses and, together with variational convergence algorithms, these guesses help us to quickly converge to UPOs. We further attempt to "glue" known UPOs in the latent space to create guesses for longer ones. This gluing procedure is successful and points towards a hierarchy of UPOs where longer UPOs shadow sequences of shorter ones.

Abstract This study presents a novel approximation approach for the Fractional Rössler System using the Variational Iteration Method (VIM). The Fractional Rössler System, an extension of the classical Rössler System, incorporates fractional-order derivatives to capture more intricate dynamical behaviors. VIM is employed due to its efficiency in handling nonlinear fractional differential equations (FDEs) and its novel application to this system. A comparative analysis with the Adams-Bashforth-Moulton method has been conducted, and numerical values are presented to validate the effectiveness of VIM. The graphical results reveal significant chaotic dynamics, including the presence of strange attractors and sensitivity to initial conditions and bifurcation phenomena, indicating transitions between periodic and chaotic regimes. This work provides a new perspective on approximating fractional chaotic systems, demonstrating the potential of VIM in advancing the study of complex dynamical systems.

Abstract The dynamic region of out-of-time-ordered correlators (OTOCs) serves as a powerful indicator of chaos in classical and semiclassical systems, capturing the characteristic exponential growth. However, in spin systems, the dynamic region of OTOCs does not reliably quantify chaos as both integrable and chaotic systems display similar behavior. Instead, we utilize the saturation behavior of OTOCs to differentiate between chaotic and integrable regimes. In integrable systems, the saturation region of OTOCs exhibits oscillatory behavior, while in chaotic systems, it shows a stable saturation. To evaluate this distinction, we investigate a time-dependent Ising spin system subjected to a linearly ramping transverse field, analyzing both integrable (without longitudinal field) and non-integrable (with longitudinal field) scenarios. This setup accelerates system ergodicity due to the introduction of an additional time scale within the system, enhancing chaotic dynamics in the non-integrable regime, and provides a compelling model for studying the interplay between integrability and chaos in quantum systems. To further support our findings, we investigate the level spacing distribution of time-dependent unitary operators, which effectively distinguishes chaotic from regular regions in our system and corroborates the results obtained from the saturation behavior of the OTOC. Additionally, we calculate a metric for the normalized Fourier spectrum of the OTOC which is dependent on the number of frequency components present to gain insights into the observed oscillations and its dependence on the ramping field. Graphical abstract

This paper investigates the complex nonlinear dynamics of an optomechanical system featuring an optical cavity coupled to two mechanical resonators interconnected by a phase-dependent interaction. We specifically explore the role of this phase-dependent phonon hopping as a mechanism for generating synthetic gauge fields without relying on gain-loss or PT-symmetric elements, offering a potentially more robust approach to manipulating mechanical energy transfer. By deriving the semiclassical dynamical equations, we map out the system’s behavior across different parameter regimes. Our findings reveal a rich spectrum of dynamics, including bistability (the coexistence of two steady states) and the emergence of complex attractors such as self-excited oscillations, hidden attractors, and chaos. We demonstrate how controlling system parameters, particularly the mechanical coupling phase and optical drive, allows for tunability between these distinct dynamical states. The presence of tunable bistability and sensitive chaotic regimes offers significant potential for practical applications. Specifically, we discuss how these controlled dynamics could be leveraged for state switching in optical information processing and for enhancing sensitivity in advanced sensor technologies through chaos-based mechanisms. This work deepens our understanding of how synthetic gauge fields, generated via phase-dependent interactions, can sculpt the nonlinear dynamics of optomechanical systems, providing a pathway toward designing robust and tunable devices for signal processing, communication, and sensing.

Abstract The Belousov-Zhabotinsky (BZ) reaction model is known for its rich and chaotic dynamics. It can also exhibit extreme events characterized by significant deviations from typical system behavior. This study numerically examines the critical conditions under which extreme events occur in a reduced BZ model. Statistical analysis tools, including probability distribution functions of events and inter-event intervals, are used to analyze the frequency and nature of extreme events in chaotic regimes. Bifurcation diagrams, threshold values, Lyapunov exponents, and state portraits are used to visualize and characterize system transitions. In addition, inter-event intervals are statistically examined, revealing Poisson-like behavior, typical of uncorrelated extreme events. These results provide new insights into the occurrence of rare, high-impact phenomena in chemical reaction models, thus contributing to a better understanding of nonlinear dynamical systems.

A chain of harmonic oscillators with nonreciprocal coupling exhibits characteristic amplification behavior that serves as a classical analog of the non-Hermitian skin effect (NHSE). We extend this concept of nonreciprocal amplification to nonlinear dynamics by employing double-well Duffing oscillators arranged in ring-structured units. The addition of units induces bifurcations of attractors, driving transitions from limit cycles to tori, chaos, and hyperchaos. Unidirectional couplings between units enable the decomposition of attractors in phase space into projected subspaces corresponding to each unit. In the chaotic regime, amplitude saturation emerges, characterized by monotonically decreasing amplitudes within a unit, in sharp contrast to the increasing profiles seen in the linear NHSE. This work uncovers alternative bifurcation behavior resulting from the intricate interplay between nonreciprocity and nonlinearity.

Abstract Out-of-time-ordered correlators (OTOCs), defined via the squared commutator of a time-evolving and a stationary operator, represent observables that provide useful indicators for chaos and the scrambling of information in complex quantum systems. Here we present a quasiclassical formalism of OTOCs, which is obtained from the semiclassical van Vleck-Gutzwiller propagator through the application of the diagonal approximation. For short evolution times, this quasiclassical approach yields the same result as the Wigner-Moyal formalism, i.e., OTOCs are classically described via the square of the Poisson bracket between the two involved observables, thus giving rise to an exponential growth in a chaotic regime. For long times, for which the semiclassical framework is, in principle, still valid, the diagonal approximation yields an asymptotic saturation value for the quasiclassical OTOC under the assumption of fully developed classical chaos. However, numerical simulations, carried out within chaotic few-site Bose-Hubbard systems in the absence and presence of periodic driving, demonstrate that this saturation value strongly underestimates the actual threshold value of the quantum OTOC, which is normally attained after the Ehrenfest time. This indicates that nondiagonal and hence genuinely quantum contributions, thus exceeding the framework of the quasiclassical description, are primarily responsible for describing OTOCs beyond the short-time regime.