Dissipative Chaotic Systems Research Articles

Operator dynamics in Lindbladian SYK: a Krylov complexity perspective

We use Krylov complexity to study operator growth in the q-body dissipative Sachdev-Ye-Kitaev (SYK) model, where the dissipation is modeled by linear and random p-body Lindblad operators. In the large q limit, we analytically establish the linear growth of two sets of coefficients for any generic jump operators. We numerically verify this by implementing the bi-Lanczos algorithm, which transforms the Lindbladian into a pure tridiagonal form. We find that the Krylov complexity saturates inversely with the dissipation strength, while the dissipative timescale grows logarithmically. This is akin to the behavior of other \U0001d52e-complexity measures, namely out-of-time-order correlator (OTOC) and operator size, which we also demonstrate. We connect these observations to continuous quantum measurement processes. We further investigate the pole structure of a generic auto-correlation and the high-frequency behavior of the spectral function in the presence of dissipation, thereby revealing a general principle for operator growth in dissipative quantum chaotic systems.

Modeling method of a class of 5D Hamiltonian conservative hyperchaotic systems with adjustable signal amplitude

Compared to dissipative chaotic systems, conservative chaotic systems have gained attention because they can avoid reconstruction attacks due to the absence of attractors. This paper reports a general method for constructing 5D Hamiltonian conservative hyperchaotic systems, mainly by coupling three 5D sub-rigid bodies with two identical axes to obtain 5D Euler equations, and then combining Hamiltonian energy and Casimir energy analysis to obtain a 5D conservative hyperchaotic system. This method is general and convenient, and the constructed conservative hyperchaotic system has good performance. In addition, this paper investigates the impact of parameters and initial values on system performance using energy analysis and proposes a simple signal amplitude adjustment method. This method has no restrictions on the mathematical models of chaotic systems, can quickly adjust signal amplitudes, and enhances the hyperchaotic characteristics of the system based on this method. Finally, the correctness of the theoretical and simulation analysis is verified using a DSP hardware platform.

Characteristic analysis of 5D symmetric Hamiltonian conservative hyperchaotic system with hidden multiple stability

Conservative chaotic systems have unique advantages over dissipative chaotic systems in the fields of secure communication and pseudo-random number generator because they do not have attractors but possess good traversal and pseudo-randomness. In this work, a novel five-dimensional (5D) Hamiltonian conservative hyperchaotic system is proposed based on the 5D Euler equation. The proposed system can have different types of coordinate transformations and time reversal symmetries. In this work, Hamilton energy and Casimir energy are analyzed firstly, and it is proved that the new system satisfies Hamilton energy conservation and can generate chaos. Then, the complex dynamic characteristics of the system are demonstrated and the conservatism and chaos characteristics of the system are verified through the correlation analysis methods such as phase diagram, equilibrium point, Lyapunov exponent, bifurcation diagram, and SE complexity. In addition, a detailed analysis of the multistable characteristics of the system reveals that many energy-related coexisting orbits exist. Based on the infinite number of center-type and saddle-type equilibrium points, the dynamic characteristics of the hidden multistability of the system are revealed. Then, the National Institute of Standards and Technology (NIST) test of the new system shows that the chaotic sequence generated by the system has strong pseudo-random. Finally, the circuit simulation and hardware circuit experiment of the system are carried out with Multisim simulation software and digital signal processor (DSP) respectively. The experimental results confirm that the new system has good ergodicity and realizability.

Detection of Weak Fault Signals in Power Grids Based on Single-Trap Resonance and Dissipative Chaotic Systems

Aiming to solve the problem that the performance of classical time–frequency domain signal detection methods is severely degraded in highly noisy environments, a single-trap approximate model of the stochastic resonance of bistable systems is studied in this paper. This method improves the defects of the classical bistable stochastic resonance model that cause it to be inapplicable during non-periodic signal detection. Combining this method with the particle swarm optimization algorithm based on an attenuation factor and cross-correlation detection technology, detection experiments determining the impulse voltage fluctuation signals, motor speed fluctuation signals and low-frequency oscillation signals of a power system are conducted. The results show that the single-trap resonance model has good phase matching performance and noise cancellation abilities. Furthermore, combining it with two kinds of dissipative chaotic systems, a comprehensive frequency and amplitude detection experiment was carried out for multiple harmonic aliasing signals. The results show that the single-trap resonance model can achieve error-free detection of each harmonic frequency and high-precision detection of each harmonic amplitude in highly noisy environments. The research results will provide new ideas for the detection of various types of weak fault signals in power systems.

A four-dimensional chaotic system with coexisting attractors and its backstepping control and synchronization

This paper constructs a new four-dimensional dissipative chaotic system with coexisting attractors. The system's dynamic behaviors are analyzed through the numerical simulation of the phase portrait, Lyapunov exponent graph, bifurcation diagram, etc. Numerical simulation results indicate that the system has abundant dynamic characteristics, and the polarity of the chaotic signal can be flexibly changed by introducing offset boosting control. By studying the spectral entropy (SE) complexity of the two different initial conditions, an initial condition with higher complexity is selected as the initial state when the system is synchronized. Finally, a backstepping controller is designed to implement the synchronization of the chaotic system. Multisim is used to simulate the system's circuit and field programmable gate arrays (FPGA) are used to implement the system on actual hardware.

LQR Chaos Synchronization for a Novel Memristor-Based Hyperchaotic Oscillator

In a three-dimensional dissipative chaotic system circuit, by superimposing a cubic magnetron-type memristor and connecting a feedback circuit, a new four-dimensional synchronous controlling system is established. The control parameters have a significant impact on the system, and the system displays rich dynamic features such as hyperchaos, chaos, and period states. At the same time, the synchronization scheme for the chaotic system is designed based on the linear quadratic regulator (LQR), which effectively improves the system response speed and reduces the complexity of the synchronous controlling system. Further, numerical verification is carried out. Finally, a detailed verification of the chaotic system’s dynamic characteristics is performed by hardware simulation. Simulation results and performance analysis show that the proposed method has synchronous controlling performance. Compared to some existing synchronous controlling schemes, this method is more widely applicable.

A new 5D fractional-order conservative hyperchaos system

At present, most of the encryption algorithms based on chaotic systems use dissipative chaotic systems. However, the dissipative chaotic systems have attractors and are easy to reconstruct, which leads to potential security risks in the process of data transmission. Therefore, a novel five-dimensional conservative hyperchaotic system is proposed in this paper, and the integer order system is transformed into a fractional-order system based on the Adomian decomposition method(ADM). The dynamic characteristics of the system are discussed by using classical analysis methods such as Lyapunov exponent spectrum(LEs), bifurcation diagram, phase diagram, and timing diagram. By changing the system parameters and the differential order q, we found a wealth of dynamic phenomena, such as quasi-periodic flow, chaotic flow, and hyperchaotic flow. When the initial value is used as a variable, it is found that the system has initial offset boosting behavior, multiple stability, and special transient behavior. In addition, we use the spectral entropy algorithm to analyze the complexity of the system. Finally, hardware experiments are also carried out using digital signal processor (DSP) to verify the correctness of the numerical simulation, and also to prove the physical realizability of the system, to create conditions for its subsequent engineering applications.

A Simple Method for Constructing a Family of Hamiltonian Conservative Chaotic Systems

Conservative chaotic systems (CCSs) have unique advantages over dissipative chaotic systems (DCSs) in the fields of secure communication and pseudo-random number generators (PRNGs), etc. However, there are relatively fewer reports on CCSs than DCSs. To this end, this paper proposes an effective method for constructing a family of Hamiltonian conservative chaotic systems (HCCSs) by letting any three of the four sub-bodies denoted by 4D generalized Euler equations share a rotation axis. From theoretical analysis to experimental verification, one of the proposed HCCSs is studied thoroughly to demonstrate the effectiveness of this method. The example system has an infinite number of equilibrium points, which belong to either centers or saddles, resulting in hidden chaos. Besides, through the bifurcation diagram, parametric chaotic set, and Lyapunov exponent, richly dynamic behaviors related to parameters are displayed. Moreover, the 3D phase portraits verify that the Hamiltonian energy is conservative. Numerous energy-related coexisting orbits are discovered in this system, such as the coexistence of quasi-periodic orbits, chaotic orbits, and chaotic quasi-periodic orbits. Furthermore, the breadboard-based circuit is implemented to illustrate the HCCS’s physical feasibility. Finally, the PRNG based on the HCCS has excellent randomness in terms of NIST and TESTU01 test results.

A novel conservative system with hidden flows evolved from the simplest memristive circuit.

Over the past few decades, the research of dissipative chaotic systems has yielded many achievements in both theory and application. However, attractors in dissipative systems are easily reconstructed by the attacker, which leads to information security problems. Compared with dissipative systems, conservative ones can effectively avoid these reconstructing attacks due to the absence of attractors. Therefore, conservative systems have advantages in chaos-based applications. Currently, there are still relatively few studies on conservative systems. For this purpose, based on the simplest memristor circuit in this paper, a non-Hamiltonian 3D conservative system without equilibria is proposed. The phase volume conservatism is analyzed by calculating the divergence of the system. Furthermore, a Kolmogorov-type transformation suggests that the Hamiltonian energy is not conservative. The most prominent property in the conservative system is that it exhibits quasi-periodic 3D tori with heterogeneous coexisting and different amplitude rescaling trajectories triggered by initial values. In addition, the results of Spectral Entropy analysis and NIST test show that the system can produce pseudo-random numbers with high randomness. To the best of our knowledge, there is no 3D conservative system with such complex dynamics, especially in a memristive conservative system. Finally, the analog circuit of the system is designed and implemented to test its feasibility as well.

Spectral Statistics of Non-Hermitian Matrices and Dissipative Quantum Chaos.

We propose a measure, which we call the dissipative spectral form factor (DSFF), to characterize the spectral statistics of non-Hermitian (and nonunitary) matrices. We show that DSFF successfully diagnoses dissipative quantum chaos and reveals correlations between real and imaginary parts of the complex eigenvalues up to arbitrary energy scale (and timescale). Specifically, we provide the exact solution of DSFF for the complex Ginibre ensemble (GinUE) and for a Poissonian random spectrum (Poisson) as minimal models of dissipative quantum chaotic and integrable systems, respectively. For dissipative quantum chaotic systems, we show that the DSFF exhibits an exact rotational symmetry in its complex time argument τ. Analogous to the spectral form factor (SFF) behavior for Gaussian unitary ensemble, the DSFF for GinUE shows a "dip-ramp-plateau" behavior in |τ|: the DSFF initially decreases, increases at intermediate timescales, and saturates after a generalized Heisenberg time, which scales as the inverse mean level spacing. Remarkably, for large matrix size, the "ramp" of the DSFF for GinUE increases quadratically in |τ|, in contrast to the linear ramp in the SFF for Hermitian ensembles. For dissipative quantum integrable systems, we show that the DSFF takes a constant value, except for a region in complex time whose size and behavior depend on the eigenvalue density. Numerically, we verify the above claims and additionally show that the DSFF for real and quaternion real Ginibre ensembles coincides with the GinUE behavior, except for a region in the complex time plane of measure zero in the limit of large matrix size. As a physical example, we consider the quantum kicked top model with dissipation and show that it falls under the Ginibre universality class and Poisson as the "kick" is switched on or off. Lastly, we study spectral statistics of ensembles of random classical stochastic matrices or Markov chains and show that these models again fall under the Ginibre universality class.

Classification of Chaotic Behaviors in Jerky Dynamical Systems

Differential equations are widely used to model systems that change over time, some of which exhibit chaotic behaviors. This paper proposes two new methods to classify these behaviors that are utilized by a supervised machine learning algorithm. Dissipative chaotic systems, in contrast to conservative chaotic systems, seem to follow a certain visual pattern. Also, the machine learning program written in the Wolfram Language is utilized to classify chaotic behavior with an accuracy around 99.1±1.1%.

Dynamical analysis of a new chaotic system: asymmetric multistability, offset boosting control and circuit realization

In this paper, a new four-dimensional dissipative chaotic system which can produce multiple asymmetric attractors is designed and its dynamical behaviors are analyzed. The basin of attraction reveals the asymmetric multistability of the system. In addition, it is very interesting to observe different types of asymmetric coexisting attractors as the bifurcation parameters change. The spectral entropy complexity chaotic diagrams are used to observe the changes in the sequence complexity when the two bifurcation parameters change simultaneously. Moreover, the difference of the system complexity between the two different initial values is analyzed. In order to facilitate engineering applications, the offset boosting control is introduced to the state variable, and the numerical simulation shows that the offset boosting control scheme can flexibly change the polarity of the chaotic signal. Finally, an analog circuit and a digital circuit were designed to verify the new chaotic system. The new research results will enrich the theoretical basis of multistability, offset boosting control and circuit implementation of chaos.

A novel image encryption scheme based on conservative hyperchaotic system and closed-loop diffusion between blocks

Dissipative chaotic systems have been widely used in digital image encryption schemes in the past 20 years. However, compared to conservative chaotic systems, the dissipative chaotic systems have attractors thus attacker can reconstruct the chaotic systems by reconstructing the attractors. Therefore, the conservative chaotic systems are more suitable in chaos-based encryption system because they have no attractors thus can avoid the reconstructing attacks. Based on this, an image encryption scheme based on conservative hyper-chaotic system and closed-loop diffusion between blocks is proposed in this paper. On the one hand, the conservative hyperchaotic system has strong pseudo-randomness and anti-reconstruction attack property. On the other hand, different from the existing closed-loop diffusion schemes which change pixel values one by one until the last pixel value is changed, the proposed closed-loop diffusion method cannot only generate the first ciphertext and other ciphertext blocks but also update the first ciphertext block using the other generated ciphertext block information. In addition, the key streams are related to plaintext and ciphertext. Consequently, the key, plaintext and ciphertext form an organic whole to ensure the sensitivity of the encryption system. Moreover, simulation results and analysis show that the encryption scheme has strong security and excellent performance.

Linear response for macroscopic observables in high-dimensional systems.

The long-term average response of observables of chaotic systems to dynamical perturbations can often be predicted using linear response theory, but not all chaotic systems possess a linear response. Macroscopic observables of complex dissipative chaotic systems, however, are widely assumed to have a linear response even if the microscopic variables do not, but the mechanism for this is not well-understood. We present a comprehensive picture for the linear response of macroscopic observables in high-dimensional coupled deterministic dynamical systems, where the coupling is via a mean field and the microscopic subsystems may or may not obey linear response theory. We derive stochastic reductions of the dynamics of these observables from statistics of the microscopic system and provide conditions for linear response theory to hold in finite dimensional systems and in the thermodynamic limit. In particular, we show that for large systems of finite size, the linear response is induced via self-generated noise. We present examples in the thermodynamic limit where the macroscopic observable satisfies linear response theory (LRT), although the microscopic subsystems individually violate LRT, as well as a converse example where the macroscopic observable does not satisfy LRT despite all microscopic subsystems satisfying LRT when uncoupled. This latter, maybe surprising, example is associated with emergent nontrivial dynamics of the macroscopic observable. We provide numerical evidence for our results on linear response as well as some analytical intuition.

Modeling of a Hamiltonian conservative chaotic system and its mechanism routes from periodic to quasiperiodic, chaos and strong chaos

In this paper, the important role of 3D Euler equation playing in forced-dissipative chaotic systems is reviewed. In mathematics, rigid-body dynamics, the structure of symplectic manifold, and fluid dynamics, building a four-dimensional (4D) Euler equation is essential. A 4D Euler equation is proposed by combining two generalized Euler equations of 3D rigid bodies with two common axes. In chaos-based secure communications, generating a Hamiltonian conservative chaotic system is significant for its advantage over the dissipative chaotic system in terms of ergodicity, distribution of probability, and fractional dimensions. Based on the proposed 4D Euler equation, a 4D Hamiltonian chaotic system is proposed. Through proof, only center and saddle equilibrium lines exist, hence it is not possible to produce asymptotical attractor generated from the proposed conservative system. An analytic form of Casimir power demonstrates that the breaking of Casimir energy conservation is the key factor that the system produces the aperiodic orbits: quasiperiodic orbit and chaos. The system has strong pseudo-randomness with a large positive Lyapunov exponent (more than 10 K), and a large state amplitude and energy. The bandwidth for the power spectral density of the system is 500 times that of both existing dissipative and conservative systems. The mechanism routes from quasiperiodic orbits to chaos is studied using the Hamiltonian energy bifurcation and Poincaré map. A circuit is implemented to verify the existence of the conservative chaos.

Universality classes of quantum chaotic dissipative systems

We study the ensemble of complex symmetric matrices. The ensemble is useful in the study of the effect of dissipation on systems with time-reversal invariance. We consider the nearest-neighbor spacing distribution and spacing ratio to investigate the fluctuation statistics and show that these statistics are similar to that of dissipative chaotic systems with time-reversal invariance. We show that, unlike cubic repulsion in eigenvalues of Ginibre matrices, this ensemble exhibits a weaker repulsion. The nearest-neighbor spacing distribution exhibits for small spacings. We verify our results for quantum kicked rotor with time-reversal invariance. We show that the rotor exhibits similar spacing distribution in dissipative regime. We also discuss a random matrix model for transition from the time-reversal invariant to the broken case.

Adaptive Synchronization Strategy between Two Autonomous Dissipative Chaotic Systems Using Fractional-Order Mittag-Leffler Stability.

Compared with fractional-order chaotic systems with a large number of dimensions, three-dimensional or integer-order chaotic systems exhibit low complexity. In this paper, two novel four-dimensional, continuous, fractional-order, autonomous, and dissipative chaotic system models with higher complexity are revised. Numerical simulation of the two systems was used to verify that the two new fractional-order chaotic systems exhibit very rich dynamic behavior. Moreover, the synchronization method for fractional-order chaotic systems is also an issue that demands attention. In order to apply the Lyapunov stability theory, it is often necessary to design complicated functions to achieve the synchronization of fractional-order systems. Based on the fractional Mittag–Leffler stability theory, an adaptive, large-scale, and asymptotic synchronization control method is studied in this paper. The proposed scheme realizes the synchronization of two different fractional-order chaotic systems under the conditions of determined parameters and uncertain parameters. The synchronization theory and its proof are given in this paper. Finally, the model simulation results prove that the designed adaptive controller has good reliability, which contributes to the theoretical research into, and practical engineering applications of, chaos.

A new class of Hamiltonian conservative chaotic systems with multistability and design of pseudo-random number generator

• A new modeling method is employed to develop a class of high dimensional Hamiltonian conservative chaotic systems. • A new conservative hyperchaotic system is proposed and exhibits some behaviour previously unobserved. • Multistability including infinite-orbit and infinite-scroll is observed in the proposed systems. • For encryption, pseudo-random digital signal generators based on conservative chaotic systems are developed. Compared to dissipative chaotic system, conservative chaotic systems have some attractive features valuable for engineering applications, such as security communication. This paper proposes a method for constructing new high dimensional conservative chaotic systems, for instance 4 dimensional conservative chaotic systems and 5 dimensional conservative hyper-chaotic systems. The dynamic properties of the constructed systems are verified by the dynamical evolution map, equilibrium points, Lyapunov dimension, the volume of flow and the Hamiltonian energy. An interesting co-existence of quasi-period, chaos and hyper-chaos flows are observed in the developed conservative chaotic systems. With trigonometric transformation, parallel conservative chaotic systems are achieved with infinitely many scrolls. To accommodate the engineering application requirement, the security communication is considered as a case study. The proposed conservative chaotic systems passed the validation National Institute of Standard Statistic Test criteria for the pseudo-random property , which is the critical factor of security communication. Furthermore, the pseudo-random signal generator is developed using the Field Programmable Gate Array technique. Due to the complicated topology structure, such digital implementation was not commonly reported in literatures, instead most conservative chaotic systems were validated only by analog circuit simulation.

A complete analytical study on the dynamics of simple chaotic systems

We report in this paper a complete analytical study on the bifurcations and chaotic phenomena observed in certain second-order, non-autonomous, dissipative chaotic systems. One-parameter bifurcation diagrams obtained from the analytical solutions proving the numerically observed chaotic phenomena such as antimonotonicity, period-doubling sequences, Feignbaum remerging have been presented. Further, the analytical solutions are used to obtain the basins of attraction, phase-portraits and Poincare maps for different chaotic systems. Experimentally observed chaotic attractors in some of the systems are presented to confirm the analytical results. The bifurcations and chaotic phenomena studied through explicit analytical solutions is reported in the literature for the first time.

Some New Dissipative Chaotic Systems with Cyclic Symmetry

Designing new chaotic system with specific features is an interesting field in nonlinear dynamics. In this paper, some new chaotic systems with cyclic symmetry are proposed. In order to understand the overall behavior of such systems, the dynamical analyses such as stability analysis, bifurcation and Lyapunov exponent analysis are done. The accurate examination of bifurcation plot represents that these systems are multistable which makes them more interesting. Also, the basin of attraction of these systems is investigated to detect the type of attractors of these systems which are self-excited. Finally, the circuit implementation is carried out to show their feasibility.