Recent theoretical studies on reservoir computing have shown that when the spectral radius of the adjacency matrix is sufficiently small, the dynamics of a reference system can be embedded in the reservoir space, enabling the reconstruction of dynamical invariants. However, reservoir models often reproduce time series accurately even when the spectral radius is relatively large, where the underlying mechanism is not well understood. In this study, we investigate the reconstruction of dynamical structures from the perspective of Lyapunov analysis using reservoir computing applied to the Hénon map. By comparing the Lyapunov spectrum of the reservoir dynamics with that of the reference system, we show that the reference dynamics are embedded in a low-dimensional inertial manifold in the reservoir space. We further demonstrate that the full Lyapunov spectrum of the reference system can be recovered by restricting the analysis to the tangent space of this manifold, even when the spectral radius is relatively large. These results clarify the geometric mechanism underlying the successful reconstruction of chaotic dynamics by reservoir computing beyond the regime where theoretical guarantees currently exist.

We investigate the ratchet current (RC) in an inertial Brownian particle under the variation of mass, driving parameters, ratchet asymmetry, and temperature, modeled via the noise intensity (Q). Using stochastic simulations combined with parameter-space analysis, we show that the RC is strongly influenced by dynamical regimes, including periodic motion, chaos, bifurcations, and, most important, multistability. Notably, multistability plays a central role in the generation of temperature-induced RCs. The inclusion of Gaussian noise with arbitrarily small values of Q triggers transitions between coexisting deterministic attractors, leading to a preference for specific states. This noise-induced selection mechanism breaks velocity-space symmetry and enables directed transport. Lyapunov spectra, basins of attraction, and trajectory comparisons confirm that multistable regimes are the fundamental source of RC induced by noise. A global temperature analysis further reveals minimal RC at low Q, enhancement at intermediate Q, and suppression at high Q, while positive directed transport persists for small particle masses. These findings highlight the critical role of noise-driven transitions in deterministic multistable systems as a mechanism for controlling stochastic transport.

Abstract We study the quantitative simplicity of the Lyapunov spectrum of ‐dimensional bounded matrix cocycles subjected to additive random perturbations. In dimensions 2 and 3, we establish explicit lower bounds on the gaps between consecutive Lyapunov exponents of the perturbed cocycle, depending only on the scale of the perturbation. In arbitrary dimensions, we show the existence of a universal lower bound on these gaps. A novelty of this work is that the bounds provided are uniform over all choices of the original sequence of matrices. Furthermore, we make no stationarity assumptions on this sequence. Hence, our results apply to random and sequential dynamical systems alike.

Abstract Omnivory, where species feed across multiple trophic levels, is a widespread feature of ecological networks. A key mechanism underlying such complexity is intraguild predation (IGP), in which a top predator consumes both an intermediate predator and a shared resource. Here, we show that Shilnikov homoclinic orbits emerge in a minimal intraguild predation model, triggering a cascade of homoclinic bifurcations near a saddle-focus equilibrium that culminates in chaos. Numerical simulations and Lyapunov spectrum analysis reveal multiple coexistence modes, ranging from regular oscillations to Shilnikov homoclinic orbits and chaos. Our model quantitatively reproduces patterns observed in natural omnivore networks, providing mechanistic insights into complex population fluctuations in ecological systems.

Keywords: Omnivory, intraguild predation, Shilnikov homoclinic orbit, chaos

Lyapunov-exponent-based diagnostics are widely used to quantify deterministic chaos in optically injected semiconductor lasers (OISLs). In most numerical implementations, the optical field is represented either in phasor coordinates (A,ψ,N) or in Cartesian quadrature coordinates (X,Y,N). Although these representations are mathematically related through a smooth coordinate transformation away from vanishing field amplitude, their numerical realizations can exhibit markedly different robustness in variational calculations, directly impacting the reliability of Lyapunov exponent estimation and chaoticity maps. In this work, we present a systematic assessment of the numerical reliability of Lyapunov-based chaos analysis in master-slave optically injected semiconductor lasers using both phasor and quadrature formulations. The full Lyapunov spectrum was computed via a noise-free variational method that integrates the nonlinear dynamics together with the corresponding Jacobian equations using a fourth-order Runge-Kutta scheme combined with periodic QR orthonormalization. High-resolution Lyapunov maps were constructed in the injection strength-frequency detuning parameter space, and the consistency between both formulations was quantitatively evaluated. While both approaches reproduce the overall structure of chaotic and non-chaotic regions, the phasor formulation may generate spurious positive Lyapunov exponents in regimes where the optical field amplitude approaches low values. These discrepancies originate from singular terms proportional to 1/A and 1/A2 in the variational Jacobian of the phasor model, which can lead to numerical amplification and artificial chaotic signatures. The quadrature formulation avoids these singularities and provides numerically stable and physically consistent Lyapunov spectra across the explored parameter space. The results establish practical guidelines for robust chaos quantification in optically injected semiconductor lasers and highlight the importance of representation choice in variational Lyapunov analysis of nonlinear photonic systems.

The Caenorhabditis elegans (C.elegans) nematodes have long been a model organism for quantitative behavioral analysis due to their tractable nervous system and well-characterized genetics. In particular, dynamic diffraction has been a successful method of studying said microorganisms due to its low level of noise and the ability to simultaneously study multiple degrees of freedom of their neuromuscular system through their locomotion. In this study, we estimate the Lyapunov spectrum of C.elegans locomotion, which offers an insight into how volume elements evolve in the phase space of the underlying dynamical system. For that, we used the Sano-Sawada algorithm to estimate the spectra from the trajectories reconstructed using the Takens embedding procedure. In total, two positive and one negative exponents were calculated and verified to be nonspurious through investigations of their stability for different sets of parameters. Those exponents have values of 0.860±0.028s^{-1},0.389±0.014s^{-1}, and -3.451±0.074s^{-1}, respectively. The presence of two positive exponents indicates that C.elegans locomotion is hyperchaotic, while the total sum being negative indicates that the system is dissipative. Those are key observations for the underlying system and will be significant for the potential formulation of future mathematical or computational models.

Signals with flexible control over polarity and frequency provide an essential foundation for reliable and high-speed information transmission. To generate chaotic signals with flexible output characteristics in low-dimensional systems, a novel chaotic system model is proposed by introducing a nonlinear term into the classical Chen chaotic system. Dynamical analysis and MATLAB numerical simulations show that the system is not only highly sensitive to initial conditions but also capable of generating three distinct chaotic attractors. Further simulations confirm that the proposed system demonstrates arbitrary unidirectional and multidirectional offset boosting behaviors, with offset amplitudes in all directions having a wide adjustable range. Furthermore, arbitrary offset constants can effectively control the frequencies of all state variables. This chaotic system, which combines flexible offset control with frequency regulation, is rare in existing research. Additionally, certain parameter ranges in the chaotic regime are relatively narrow. To address this, a method involving control constants to enhance system complexity is proposed, and its effectiveness in increasing system complexity is validated through Lyapunov spectrum and spectral entropy (SE) analysis. Based on the constructed chaotic system, an equivalent circuit model was built using the Multisim 14.0 platform. Experimental results confirm that the system generates chaotic attractors with distinct structures and demonstrates offset boosting behavior in arbitrary directions. Additionally, DSP hardware experiments further validate the physical realizability of the system. To fully exploit the system’s advantages, a synchronization controller was designed for both the drive and response systems, enabling synchronization control of the chaotic system with three offset constants. Based on this, data encryption and transmission experiments were conducted, further establishing the theoretical and experimental foundation for applying the new chaotic system in secure communication.

The dynamics of extended many-body systems are generically chaotic. Classically, a hallmark of chaos is the exponential sensitivity to initial conditions captured by positive Lyapunov exponents. Supplementing chaotic dynamics with stochastic resetting drives a sharp dynamical phase transition: we show that the Lyapunov spectrum, i.e., the complete set of Lyapunov exponents, abruptly collapses to zero above a critical resetting rate. At criticality, we find a sudden loss of analyticity of the velocity-dependent Lyapunov exponent, which we relate to the transition from ballistic scrambling of information to an arrested regime where information becomes exponentially localized over a characteristic length diverging at criticality with an exponent ν=1/2 and a dynamical exponent z=2. We illustrate our analytical results on generic chaotic dynamics by numerical simulations of coupled map lattices.

Abstract We investigate a six-dimensional (6D) dual-oscillator optomechanical system, where two mechanical resonators couple to a single optical mode, and show that \emph{noise} acts as a control parameter that both accelerates the route to chaos and generates \emph{riddled basins}---fractal intermingling of synchronized and anti-synchronized attractors. Combining Poincaré sections, 2D/3D phase-space projections, the full Lyapunov spectrum, Kaplan--Yorke dimension, Melnikov analysis, and Fokker--Planck diagnostics, we obtain a quantitative and self-consistent picture of noise-induced complexity. As the laser drive increases, a Feigenbaum-type period-doubling cascade culminates in chaos and hyperchaos ( $\lambda_1>0$; $\lambda_1,\lambda_2>0$ with $D_{\mathrm{KY}}\!\in\![2,4]$ ), while Melnikov predictions confirm homoclinic intersections underlying global instability. Riddled-basin formation is demonstrated and \emph{quantified} by a stability index $\sigma=0.3\text{--}0.6$, (ii) fractal basin dimension $D_B\approx 1.2\text{--}1.55$, and (iii) basin entropy $S_B\approx 0.3\text{--}0.45$, all peaking in the noise-driven regime. Transverse Lyapunov exponents along the synchronized and anti-synchronized attractors remain negative (transverse stability), while transversely unstable periodic orbits embedded in the attractors provide the dynamical mechanism for riddling. A critical noise threshold $D_c\approx 0.05$ is identified for attractor switching and the smearing of bifurcation structure; largest-exponent maps $\lambda_1(E,D)$ correlate these transitions with parameter space. Our results establish rigorous evidence for noise-modulated (hyper)chaos in high-dimensional optomechanics and offer predictive levers (detuning, coupling, and noise engineering) for controlling instability in microcavity and LIGO-type platforms, as well as enabling applications in chaos-based photonics and sensing.

Triadic percolation turns bond percolation into a dynamical problem governed by an effective one-dimensional unimodal map. We show that the geometry of superstable cycles provides a direct, map-agnostic probe of local nonlinearity: specifically, the distance from the map's maximum to a distinguished next-to-maximum point on the attracting 2^{n} cycle (which coincides with a preimage of the maximum at 2^{n} superstability) scales as |Δp|^{γ}, with γ=1/z, where z is the nonflat order of the maximum. This prediction is verified across canonical unimodal families and heterogeneous triadic ensembles, with Lyapunov spectra corroborating the one-dimensional reduction. A derivative condition on the activation kernel fixes the local nonlinearity order z (and thus, under standard unimodal-map hypotheses, the associated z-logistic universality class) and gives conditions under which z>2 can be realized. The diagnostic operates directly on orbit data under standard regularity assumptions, providing a practical tool to classify universality in higher-order networks.

Abstract This paper proposes an integrated framework that synergizes performance-optimized high-dimensional chaotic systems with dynamic DNA sequence-based encoding, tailored specifically for efficient and secure image encryption in real-time network environments. A novel 6D hyperchaotic generator, derived from coupled conservative and dissipative subsystems, is introduced alongside a dynamic DNA encoding/decoding pipeline, where the coding rules are adaptively selected from chaotic state vectors. This hybrid architecture achieves a robust key space (exceeding 2^256), strong diffusion and confusion characteristics, and throughput suitable for real-time 5G and edge network environments. Mathematical analysis confirms that under typical parameter settings ( λ 1 = 0.85, λ 2 = 0.12, λ 3 = 0.03), the system’s Lyapunov spectrum features three positive exponents, validating its robust chaotic behavior. Empirical tests on 512 × 512 RGB images transmitted over a simulated real-time network with an average packet latency of 10 ms demonstrate that the proposed encryption scheme achieves a Number of Pixels Change Rate (NPCR) of 99.61% (±0.02) and a Unified Average Changing Intensity (UACI) of 33.48% (±0.07). The optimized C implementation (single-threaded) yields an average throughput of 136.4 MB s −1 and an end-to-end latency increase of 4.8 ms (±0.6) per frame compared to unencrypted transmission. Security and performance comparisons demonstrate that the hybrid chaotic-DNA approach provides superior diffusion against differential attacks while meeting practical latency constraints for edge video streams.

Chaotic systems play a crucial role in information security, cryptography, and secure communication systems due to their extreme sensitivity to initial conditions and inherent unpredictability. In this study, the dynamic behavior of the Scaled Zhongtang (SZ) chaotic system, which exhibits rich dynamical characteristics, is analyzed, and a low-cost, high-precision embedded system implementation using the STM32F429 microcontroller is presented. Within the scope of the study, the complexity of the SZ chaotic system is first validated through time series analysis, phase portraits, Lyapunov spectrum, and bifurcation analyses. Following the numerical analyses, the chaotic differential equation set is solved on the embedded system using the fourth-order Runge-Kutta (RK4) algorithm. The obtained chaotic data are converted into analog signals via the microcontroller’s internal 12-bit Digital-to-Analog Converter (DAC) without the need for an external hardware interface. The system performance is evaluated by comparing experimental data acquired from an oscilloscope with MATLAB simulation results. The comparison results demonstrate that the STM32-based implementation exhibits high consistency with theoretical models. This study proposes a flexible and cost-effective alternative for industrial applications of chaotic systems, addressing the stability issues of analog circuits and the high costs of FPGA-based systems.

Steel belt-driven elevators have been increasingly adopted in high-rise buildings due to their compact structure and low noise. Unlike conventional wire ropes, steel belts exhibit strong viscoelasticity, which, together with the time-varying system length and multi-stage speeds operation, introduces a triple-coupling effect that fundamentally alters system dynamics, threatening the normal operation of an elevator. This study develops a nonlinear longitudinal dynamic model of viscoelastic belt-driven time-varying vertical transport systems using the Generalized Hamiltonian Principle, with model accuracy verified through experiments. A reduced-order Duffing oscillator is introduced to capture nonlinear behaviours such as bifurcation, phase trajectory, and Poincaré sections, which are further validated via Lyapunov spectra. Results reveal that viscoelastic damping can suppress vibrations during upward motion but may amplify responses during downward motion, and that mismatched dissipation at high speeds leads to energy accumulation and oscillatory shocks. The triple-coupling mechanism significantly affects both the stability boundaries and energy transfer characteristics of the system. Finally, the influence mechanism of the triple-coupling effect is discussed. This work provides new insights into the dynamic behaviours of viscoelastic traction systems and offers a theoretical foundation for vibration suppression and safe operation of belt-driven elevators, with potential applicability to winches, cranes, and robotic hoists.

This paper presents a comprehensive dynamical systems analysis of a generalized (3 + 1)-dimensional Davey–Stewartson equation with parabolic law nonlinearity. The complex partial differential equation is reduced to a planar dynamical system, enabling detailed bifurcation analysis, equilibrium point characterization, and complete phase portrait construction across different parameter regimes. External periodic forcing transforms the conservative system into a dissipative one, facilitating chaotic dynamics that are confirmed through Lyapunov exponent computation and sensitivity analysis. The complete Lyapunov spectrum provides quantitative evidence of chaotic behavior. Using the new mapping method and the new Kudryashov method, we obtain novel exact solutions, including bright and dark solitons, rogue waves, kink-shaped structures, and periodic patterns. The derivation of these exact solutions addresses the distinctive challenges posed by parabolic law nonlinearity and establishes crucial connections with phase space structures identified through bifurcation analysis. The interplay between analytical solutions and dynamical systems theory reveals fundamental mechanisms governing wave stability, bifurcations, and chaos in higher-dimensional nonlinear systems, with applications in nonlinear optics and fluid dynamics.

We revisit a two-temperature Nosé-Hoover wanderer particle embedded in a two-dimensional periodic 2 × 2 cell with four smooth repulsive corners at (x, y) = (±1, ±1) to explore chaos with anisotropic thermostatting. The model employs separate thermostats in the x and y directions, enabling controlled deviations from equilibrium. By integrating the full six-dimensional equations of motion and computing the complete Lyapunov spectrum, we confirm chaos and quantify phase-space contraction from the fully resolved six-dimensional Lyapunov spectrum. The total contraction rate, interpreted as entropy production, increases nonlinearly with the thermostat anisotropy, deviating from the quadratic dependence expected from linear-response theory, Λ ∝ δ2. We analyze two functional forms for the entropy-production rate, Λ(δ) (with δ = 0.5 - Ty): (i) a quadratic-plus-quartic expansion, consistent with linear-response expectations, and (ii) a power law, Λ ∝ δ2.44. While the latter captures the low-driving regime slightly better, the former more accurately describes the strongly driven regime and remains consistent with linear-response theory near equilibrium. An empirical linear relation between dissipation and phase-space dimensionality loss is also identified, Λ ≈ (DKY - 6)/3, where DKY is the approximate Kaplan-Yorke dimension. Momentum statistics show a significant non-Gaussian behavior under strong driving. Despite its dissipative nature, the model remains strictly time-reversible, offering a pedagogically rich example of microscopic reversibility coexisting with macroscopic entropy production.

The paper is devoted to the properties of a complex matrix “twisted,” otherwise called “spectral,” cocycle, associated with substitution dynamical systems. Following a recent finding of Rajabzadeh and Safaee (2025) of an invariant section for the twisted cocycle, we indicate that this implies presence of a zero Lyapunov exponent. This has consequences for the spectral properties of substitution dynamical systems; in particular, this extends the scope and simplifies the proof of singular spectrum for a large class of substitutions on two symbols. We also obtain some results on positivity of the top exponent. In the appendix we compute the Lebesgue almost everywhere local dimension of spectral measures of some “simple” test functions, for almost every irrational rotation. This sheds some light on the earlier work of Bufetov and the author (2020), relating the local dimension of spectral measures to pointwise Lyapunov exponents of the twisted cocycle. It should be noted that the paper has some (mutually acknowledged) overlap with the article of Rajabzadeh and Safaee (2025).

We study universal aspects of thermalization induced by Trotterization, a procedure routinely used in gate-based quantum computation. We use the reduced-Bardeen-Cooper-Schrieffer model-quantum integrable with a classically integrable mean-field limit-where the effects of Trotter chaos are expected to be particularly stark. The resulting Trotterized chaotic dynamics is characterized by its Lyapunov spectrum and rescaled Kolmogorov-Sinai entropy. The chaos quantifiers depend on the Trotterization time step τ. We observe a Trotter transition at a finite step value τ_{c}≈sqrt[N]. While the dynamics is weakly chaotic for time steps τ≪τ_{c}, the regime of large Trotterization steps is characterized by short temporal correlations. We derive two different scaling laws for the two different regimes by numerically fitting the maximum Lyapunov exponent data. The scaling law of the large τ limit agrees well with the one derived from the kicked top map. Beyond its relevance to current quantum computers, our work opens other directions-such as probing observables like the Loschmidt echo, which lie beyond standard mean-field description--across the Trotter transition we uncover.

Abstract Purpose It is essential for maritime safety, particularly in severe sea conditions, to investigate the nonlinear stability of a ship's pitch-roll motion. It reveals how nonlinear interactions can lead to unexpected instabilities, providing a more accurate design framework. Analysing two-degrees-of-freedom (2DOF) of roll-pitch motion of a vessel explicates complicated coupled dynamics, essential in comprehending nonlinear resonance events and parametric instabilities. Beyond linear approximations, nonlinear stability analysis of coupled pitch-roll ship motion helps capture genuine vessel behavior, particularly in strong sea conditions and large-amplitude waves. Consequently, the current study examines the 2DOF of an excited harmonically pendulum scheme, recognized in the works as an effective model of coupling between pitch and roll motions of a ship. We focus primarily on the dangerous condition of a vessel in which the excitation period is close to the pitch period, and the pitch frequency becomes twice as high as the roll frequency. Method The existing methodology is based mainly on a non-perturbative approach (NPA), which facilitates a unique analysis that is independent of Taylor expansion. He’s frequency formula (HFF) represents the principal tool employed in constructing NPA. The principal purpose of NPA is to transform weakly oscillating nonlinear ordinary differential equations (ODEs) into linear ones. The quick evaluation of frequency-amplitude correlation is essential in acquiring successive approximations of responses to parametric nonlinear variations. The inspiration for some criticisms on the stability of steady states is examined. A chaotic analysis of specified models is conducted using bifurcation diagrams, phase portraits, Poincaré maps, and Lyapunov spectrum. This strategy enables us to identify and distinguish distinct forms of motion exhibited by every system.

This paper examines the nonlinear behavior of the generalized stochastic intermediate dispersive velocity (SIdV) equation, which has been widely analyzed in a non-noise deterministic framework but has yet to be studied in any depth in the presence of varying forcing strength and noise types, in particular how it switches between periodic, quasi-periodic, and chaotic regimes. A stochastic wave transformation reduces the equation to simpler ordinary differential equations to make soliton overlap analysis feasible to analyze soliton robustness under deterministic and stochastic conditions. Lyapunov exponents, power spectra, recurrence quantification, correlation dimension, entropy measures, return maps, and basin stability are then used to measure the effect of white, Brownian, and colored noise on attractor formation, system stability, and spectral correlations. Order-chaos transitions as well as noise-induced complexity are more effectively described by bifurcation diagrams and by Lyapunov spectra. The results of this experiment improve the theoretical knowledge of stochastic nonlinear waves and offer information that will be useful in the fields of control engineering, energy harvesting, optical communications, and signal processing applications.

The formation of complex chaotic layers and the study of chaotic transport in stochastic magnetic fields in tokamaks represent two important challenges in fusion research and fusion engineering design. Hamiltonian dynamics, stability, and instability manifolds of hyperbolic periodic points and mapping techniques are considered plausible and successful tools to study physical properties in magnetically confined plasmas in tokamaks. Recently, a new cylindrical model has been introduced to study the chaotic behavior, dynamics, and statistical properties of magnetic fields by developing the dynamic divertor ergodic (DED) map. The DED map offers several motivating properties due to its symmetric structure representing the chaotic trajectories of magnetic field lines in tokamaks. However, it was observed in various fusion sciences that intricate filaments with fractal patterns arise in twist and nontwist maps, and that in the chaotic region, lines escape to the tokamak wall through a fractal-like structure. Motivated by the relevance of fractals in Hamiltonian dynamics and nuclear fusion, we generalize in this study the DED map in fractal dimensions, with the construction of a DED fractal map (DED FM). The DED FM was studied for monotonic and nonmonotonic profiles in order to describe the magnetic field lines in the toroidal plane of the TEXTOR torus in the presence of a magnetic perturbation. The Poincaré sections, the Lyapunov exponent, and the bifurcation diagrams were analyzed. We studied the parameter spaces of the DED FM, revealing a number of interesting chaotic and fractal structures that are also seen in the Lyapunov spectrum. We interpreted the results using the Poincaré sections. All of the DED FM maps displayed motivating behavior that described the fractal transport of the magnetic field lines in tokamaks. The DED FM exhibited a mixed phase space, with Kolmogorov-Arnold-Moser (KAM) islands, KAM barriers, and chaotic regions filled with intricate filaments having fractal patterns. Each map had its advantages and weaknesses and exhibited typical properties of nontwist maps, like twin island chains, shearless invariant curves, and separatrix reconnection. A number of features and properties were obtained and analyzed accordingly.