N-dimensional System Research Articles

Quantum algorithm for dynamic mode decomposition integrated with a quantum differential equation solver

We present a quantum algorithm that analyzes time series data simulated by a quantum differential equation solver. The proposed algorithm is a quantum version of a dynamic mode decomposition algorithm used in diverse fields such as fluid dynamics, molecular dynamics, and epidemiology. Our quantum algorithm can also compute matrix eigenvalues and eigenvectors by analyzing the corresponding linear dynamical system. Our algorithm handles a broad range of matrices, particularly those with complex eigenvalues. The complexity of our quantum algorithm is O(polylogN) for an N-dimensional system. This is an exponential speedup over known classical algorithms with at least O(N) complexity. Thus, our quantum algorithm is expected to enable high-dimensional dynamical systems analysis and large matrix eigenvalue decomposition, intractable for classical computers. Published by the American Physical Society 2024

Second-Order Modified Nonstandard Explicit Euler and Explicit Runge–Kutta Methods for n-Dimensional Autonomous Differential Equations

Nonstandard finite-difference (NSFD) methods, pioneered by R. E. Mickens, offer accurate and efficient solutions to various differential equation models in science and engineering. NSFD methods avoid numerical instabilities for large time steps, while numerically preserving important properties of exact solutions. However, most NSFD methods are only first-order accurate. This paper introduces two new classes of explicit second-order modified NSFD methods for solving n-dimensional autonomous dynamical systems. These explicit methods extend previous work by incorporating novel denominator functions to ensure both elementary stability and second-order accuracy. This paper also provides a detailed mathematical analysis and validates the methods through numerical simulations on various biological systems.

A fast numerical algorithm for finding all real solutions to a system of N nonlinear equations in a finite domain

AbstractA highly recurrent traditional bottleneck in applied mathematics, for which the most popular codes (Mathematica, Matlab, and Python as examples) do not offer a solution, is to find all the real solutions of a system of n nonlinear equations in a certain finite domain of the n-dimensional space of variables. We present two similar algorithms of minimum length and computational weight to solve this problem, in which one resembles a graphical tool of edge detection in an image extended to n dimensions. To do this, we discretize the n-dimensional space sector in which the solutions are sought. Once the discretized hypersurfaces (edges) defined by each nonlinear equation of the n-dimensional system have been identified in a single, simultaneous step, the coincidence of the hypersurfaces in each n-dimensional tile or cell containing at least one solution marks the approximate locations of all the hyperpoints that constitute the solutions. This makes the final Newton-Raphson step rapidly convergent to all the existent solutions in the predefined space sector with the desired degree of accuracy.

Fixed/predefined-time stability and designing of terminal sliding mode control of impulsive dynamical systems

The contribution of the present work is two-fold. Firstly, some results on the predefined and fixed-time stability of non-linear impulsive systems are derived. Impulse-dependent sufficient conditions are derived for predefined-time estimation of settling-time under the influence of stabilising impulses. In contrast, a fixed-time estimate of the settling-time is obtained for destabilising impulses. Secondly, terminal sliding mode (TSM) control is designed in the presence of hybrid (continuous and impulsive) matched disturbances for n-dimensional systems. It shows that the trajectories can reach the sliding surfaces in fixed-time and thus will stay on it after that under the influence of the proposed TSM. Thus, the system states will converge to the origin in a fixed-time. Finally, three examples, which also consist of the stability of Cohen-Grossberg BAM neural networks, are provided to justify the proposed results numerically.

A secure fractal compression scheme based on irregular Latin square, Julia and 2D-FCICM

With the proliferation of digital image data, ensuring its secure and efficient storage and transmission has become important. One-dimensional (1D) chaotic maps come with their set of limitations, such as chaotic interval discontinuity and simple chaotic behavior. To increase the complexity of chaotic systems, this paper proposed a novel 2D fractal coupled model (2D-FCM) to generate chaotic systems and extend it to n-dimensional chaotic systems. Meanwhile, the newly generated 2D-FCICM is employed in secure fractal compression. Compared to existing chaotic systems, the pseudo-random sequences generated by the 2D-FCICM exhibit better randomness and complexity. To enhance the plaintext sensitivity of the algorithm, a localized magnified fractal image is produced using plaintext information. Control parameters for the chaotic system are then derived from both the fractal and the original images. Given the compressed data's I × 5 size, this study proposed a technique for generating irregular Latin squares and integrated it into the scrambling phase. The experimental findings indicate that this scheme boasts high security levels, strong plaintext sensitivity and key sensitivity, and is resilient against various statistical attacks.

Stability Analysis of a Class of Nonlinear Fractional Differential Systems With Riemann-Liouville Derivative

This paper investigates the stability of n-dimensional nonlinear fractional differential systems with Riemann-Liouville derivative. By using the Mittag-Leffler function, Laplace transform and the Gronwall-Bellman lemma, one sufficient condition is attained for the asymptotical stability of a class of nonlinear fractional differential systems whose order lies in (0, 2). According to this theory, if the nonlinear term satisfies some conditions, then the stability condition for nonlinear fractional differential systems is the same as the ones for corresponding linear systems. Several examples are provided to illustrate the applications of our result.

Multidimensional Model of Information Struggle with Impulse Perturbation in Terms of Levy Approximation

The focus of this research was on building a decision support system for a model that characterizes the conflict interaction of n-dimensional complex systems with non-trivial internal structures. The interpretation of the new model was focused on information warfare as the impact of rare events that quickly change certain perceptions of a large number of people. Consequently, the support for various ideas experiences stochastic jumps, a phenomenon observable through a non-classical Levy approximation scheme. The essence of our decision support system lies in its ability to navigate the complex dynamics of conflict interaction among multifaceted systems. Through the utilization of advanced modeling techniques, our aim is to illuminate the complicated interplay of factors influencing information warfare and its cascading effects on societal perceptions and behaviors. Key components of our decision support system encompass model development, simulation capabilities, data integration, and visualization tools. The significance of our work lies in its potential to inform policy formulation, conflict resolution strategies, and societal resilience in the face of information warfare. By providing decision-makers with actionable intelligence and foresight into emerging threats and opportunities, our decision support system serves as a valuable tool for navigating the complexities of modern conflict dynamics. In conclusion, developing a decision support system for modeling conflict interaction in complex systems represents an essential step toward enhancing our understanding of information warfare and its consequences. Through interdisciplinary collaboration and innovative modeling techniques, we aim to provide stakeholders with the insights and capabilities needed to navigate the developing landscape of conflict and ensure the stability and resilience of society.

N-Dimensional Lattice Integrable Systems and Their bi-Hamiltonian Structure on the Time Scale Using the R-Matrix Approach

A time scale is a special measure chain that can unify continuous and discrete spaces, enabling the construction of integrable equations. In this paper, with the Lax operator generated by the displacement operator, N-dimensional lattice integrable systems on the time scale are given by the R-matrix approach. The recursion operators of the lattice systems are derived on the time scale. Finally, two integrable hierarchies of the discrete chain with a bi-Hamiltonian structure are obtained. In particular, we give the structure of two-field and four-field systems.

Predefined-time sliding mode control of chaotic systems based on disturbance observer.

In this paper, in order to realize the predefined-time control of n-dimensional chaotic systems with disturbance and uncertainty, a disturbance observer and sliding mode control method were presented. A sliding manifold was designed for ensuring that when the error system runs on it, the tracking error was stable within a predefined time. A sliding mode controller was developed which enabled the dynamical system to reach the sliding surface within a predefined time. The total expected convergence time can be acquired through presetting two predefined-time parameters. The results demonstrated the feasibility of the proposed control method.

A universal method for constructing n-dimensional polynomial hyperchaotic systems with any desired positive Lyapunov exponents

Most existing chaotic maps have many defects in engineering applications, such as discontinuous parameter range, uneven output of chaotic sequences and dynamic degradation. Based on this, a generalized n-dimensional polynomial chaotic map is proposed in this paper. By setting the coefficient of the linear term and the order of the highest order term of the polynomial, a series of n-dimensional polynomial chaotic maps of specific Lyapunov exponents can be obtained. The system solves the defects of the above system well, in addition, one can get the desired number of positive Lyapunov exponents, and one can get the desired value of positive Lyapunov exponents. Then, the effectiveness of the map is verified by a specific numerical example, and its dynamic analysis shows that the map has complex dynamic behavior. Finally, the map is applied to secure communication technology. Compared with other chaotic maps of the same dimension, the maps can obtain a smaller bit error rate, indicating that the chaotic map is more suitable for chaotic secure communication applications.

N-dimensional polynomial hyperchaotic systems with synchronization application

n-dimensional polynomial hyperchaotic systems with synchronization application

A decoupled approach for determination of the joint probability density function of a high-dimensional nonlinear stochastic dynamical system via the probability density evolution method

The joint probability density function (PDF) of multiple response processes of a system is a crucial topic in the fields of science and engineering. It is adopted to describe the dependent uncertainty propagation in complex stochastic dynamical systems, including the displacement and velocity of one degree of freedom (DOF), and the nonlinear probabilistic dependencies between multiple variables. However, the partial differential equations (PDEs) governing the time evolution of joint PDFs for multiple stochastic processes are often high-dimensional and coupled, making them analytically or numerically intractable for systems with high dimensionality. The probability density evolution method (PDEM) has provided a new perspective from the standpoint of random events description of the principle of preservation of probability. For an n-dimensional system, if merely the joint PDF of m processes is needed, where m is not larger than n, then it is sufficient to solve the generalized density evolution equation (GDEE), i.e., the Li–Chen equation, in m dimensions. This approach is particularly convenient for cases where n is large while m is small. However, for cases where m is not less than two, it is still challenging to avoid solving high-dimensional PDEs. This paper proposes a new method for the determination of joint PDF of multiple responses based on the principle of preservation of probability. To capture the joint PDF of m processes, the m-dimensional PDE can be transformed into m one-dimensional PDEs, significantly reducing the computational complexity of numerical implementation. The proposed method gives an enhanced version to the PDEM, enabling it to not only determine the individual probabilistic response, but also efficiently determine the dependent probabilistic distribution of multiple responses of high-dimensional stochastic dynamical systems without solving the coupling PDE. It reduces the computational cost from exponential growth of m powers to linear growth. Some numerical examples are presented to verify the effectiveness of the proposed method. Finally, problems for future investigations are discussed.

Adaptive finite-time anti-disturbance fuzzy control for uncertain time-delay nonlinear systems and its application to SVC

The adaptive finite-time anti-disturbance fuzzy control method for n-dimensional uncertain nonlinear systems with input delay is proposed in this paper. For unknown nonlinear terms of the controlled system, the fuzzy logic systems (FLSs) are utilized to approximate them. To further improve the robustness of the controlled system, the disturbance observer is established to estimate external disturbances, facilitating the designed controller to compensate for the influence of external disturbances. Moreover, the Pade approximation method is used to transform the time-delay system model, which is convenient for controller design. With the aid of the finite-time command filtered backstepping technique, the adaptive finite-time disturbance rejection controller with low computational complexity is designed, which can compensate for the errors caused by the filters and assure that system state variables are stable and bounded in a finite time. In the end, the simulation for the single-machine infinite bus (SMIB) system with static var compensator (SVC) is presented to illustrate the effectiveness of the proposed control approach.

The Reduction, First Integral and Kam Tori for n-Dimensional Volume-Preserving Systems

The Reduction, First Integral and Kam Tori for n-Dimensional Volume-Preserving Systems

An n-dimensional modulo chaotic system with expected Lyapunov exponents and its application in image encryption

One-dimensional (1D) chaotic maps have many shortcomings, including simple structures, uncomplicated chaotic behaviors, and narrow chaotic ranges. Multi-dimensional (MD) chaotic maps certainly have more complex constructions and more abundant chaotic phenomena. However, it is still challenging to artificially customize its Lyapunov exponents (LEs), which are characterization indicators of the chaotic state. This paper proposed a novel n-dimensional (nD) modulo chaotic system (nD-MCS) based on preceding works that can generate nD chaotic maps by using existing 1D chaotic maps as seed chaotic maps. The proposed nD-MCS can customize its LEs by setting system parameters. Three 2D and three 3D chaotic maps are presented as examples. Performance evaluations demonstrated that they both exhibit hyperchaotic behaviors, broad chaotic ranges, and large complexities. We present an encryption scheme to confirm the application of the developed chaotic maps to image encryption. Finally, a hardware experiment utilizing the field-programmable gate array (FPGA) is adapted to verify the proposed cryptosystem.

Li-Yorke chaos in weak topology of the n-dimensional linear systems

The paper deals with the complicated dynamics of the linear systems on the n-dimensional Euclidean space with the weak topology. A weak topology defined by a family of semi-norms is found to systematically investigate the chaotic dynamics in weak topology. It is rigorously proved that the one-dimensional linear systems are not Li-Yorke chaotic in weak topology and the n-dimensional linear systems with n>1 exhibit weak Li-Yorke chaos in weak topology. Moreover, some necessary and sufficient conditions for weak Li-Yorke chaos in weak topology of n-dimensional linear systems with n≥2 are established by proving the existence of a semi-irregular or an irregular vector in weak topology. As an application, the dynamics in weak topology for the three-dimensional linear systems with real distinct, real repeated and complex conjugate eigenvalues are classified.

Clifford group is not a semidirect product in dimensions N divisible by four

The paper is devoted to projective Clifford groups of quantum N-dimensional systems (with configuration space ). Clearly, Clifford gates allow only the simplest quantum computations which can be simulated on a classical computer (Gottesmann–Knill theorem). However, it may serve as a cornerstone of full quantum computation. As to its group structure it is well-known that—in N-dimensional quantum mechanics—the Clifford group is a natural semidirect product provided the dimension N is an odd number. For even N special results on the Clifford groups are scattered in the mathematical literature, but they mostly do not concern the semidirect structure. Using appropriate group presentation of it is proved that for even N the projective Clifford groups are not natural semidirect products if and only if N is divisible by four.

On global dynamics of type-K competitive Kolmogorov differential systems

This paper deals with global asymptotic behaviour of the dynamics for N-dimensional type-K competitive Kolmogorov systems of differential equations defined in the first orthant. It is known that the backward dynamics of such systems is type-K monotone. Assuming the system is dissipative and the origin is a repeller, it is proved that there exists a compact invariant set Σ which separates the basin of repulsion of the origin and the basin of repulsion of infinity and attracts all the non-trivial orbits. There are two closed sets S H and S V , their restriction to the interior of the first orthant are -dimensional hypersurfaces, such that the asymptotic dynamics of the type-K system in the first orthant can be described by a system on either S H or S V : each trajectory in the interior of the first orthant is asymptotic to one in S H and one in S V . Geometric and asymptotic features of the global attractor Σ are investigated. It is proved that the partition holds such that and . Thus, Σ0 contains all the ω-limit sets for all interior trajectories of any type-K subsystems and the closure as a subset of Σ is invariant and the upper boundary of the basin of repulsion of the origin. This Σ has the same asymptotic feature as the modified carrying simplex for a competitive system: every nontrivial trajectory below Σ is asymptotic to one in Σ and the ω-limit set is in Σ for every other nontrivial trajectory.

Invariant manifold growth formula in cylindrical coordinates and its application for magnetically confined fusion

For three-dimensional vector fields, the governing formula of invariant manifolds grown from a hyperbolic cycle is given in cylindrical coordinates. The initial growth directions depend on the Jacobians of Poincaré map on that cycle, for which an evolution formula is deduced to reveal the relationship among Jacobians of different Poincaré sections. The evolution formula also applies to cycles in arbitrary finite n-dimensional autonomous continuous-time dynamical systems. Non-Möbiusian/Möbiusian saddle cycles and a dummy X-cycle are constructed analytically as demonstration. A real-world numeric example of analyzing a magnetic field timeslice on EAST is presented.

Laplace Homotopy Perturbation Method (Lhpm) For Solving Systems Of N-Dimensional Non-Linear Partial Differential Equation

In this research, we proposed coupling the Laplace transform method and the homotopy Perturbation Method (LHPM). We employed the fusion of the Laplace method to make up for the shortcomings of other semi-analytical approaches like the homotopy perturbation method, variation iteration method, and the Adomian decomposition method. We aim to obtain an approximate and semi-analytic solution of the n-dimensional system of nonlinear partial differential equations. N-dimensional partial differential equations with nonlinear terms are known as nonlinear partial differential equations. They have been used to solve mathematical problems like the Poincaré conjecture and the Calabi-Yau conjecture and describe physical systems, from gravity to fluid dynamics. Therefore, we proffer a semi-analytic solution in the form of a Taylor multivariate series of displacements x, y, and time t using the proposed method. A side-by-side comparison was carried out to compare the exact solution with the new solution using 3-dimensional graphs, and thus the graph analysis followed. Results show excellent agreement, and the emergence of this method as a viable alternative demonstrates its viability by requiring fewer computations and being much easier and more convenient than others, making it suitable for widespread use in engineering as well.