Five-dimensional Systems Research Articles

3D medical model encryption based on five-dimensional hyperchaotic systems with 3D Arnold transform and selectable multiple spiral arrangements

3D medical model encryption based on five-dimensional hyperchaotic systems with 3D Arnold transform and selectable multiple spiral arrangements

High Schmidt Number Concentration in Quantum Bound Entangled States.

A deep understanding of quantum entanglement is vital for advancing quantum technologies. The strength of entanglement can be quantified by counting the degrees of freedom that are entangled, which results in a quantity called the Schmidt number. A particular challenge is to identify the strength of entanglement in quantum states that remain positive under partial transpose (PPT), otherwise recognized as undistillable states. Finding PPT states with high Schmidt numbers has become a mathematical and computational challenge. In this Letter, we introduce efficient analytical tools for calculating the Schmidt number for a class of bipartite states called grid states. Our methods improve the best-known bounds for PPT states with high Schmidt numbers. Most notably, we construct a Schmidt number 3 PPT state in five-dimensional systems and a family of states with a Schmidt number of (d+1)/2 for odd d-dimensional systems, representing the best-known scaling of the Schmidt number in a local dimension. Additionally, these states possess intriguing geometrical properties, which we utilize to construct indecomposable entanglement witnesses.

Gauge Field Induced Chiral Zero Mode in Five-Dimensional Yang Monopole Metamaterials.

Owing to the chirality of Weyl nodes characterized by the first Chern number, a Weyl system supports one-way chiral zero modes under a magnetic field, which underlies the celebrated chiral anomaly. As a generalization of Weyl nodes from three-dimensional to five-dimensional physical systems, Yang monopoles are topological singularities carrying nonzero second-order Chern numbers c_{2}=±1. Here, we couple a Yang monopole with an external gauge field using an inhomogeneous Yang monopole metamaterial and experimentally demonstrate the existence of a gapless chiral zero mode, where the judiciously designed metallic helical structures and the corresponding effective antisymmetric bianisotropic terms provide the means for controlling gauge fields in a synthetic five-dimensional space. This zeroth mode is found to originate from the coupling between the second Chern singularity and a generalized 4-form gauge field-the wedge product of the magnetic field with itself. This generalization reveals intrinsic connections between physical systems of different dimensions, while a higher-dimensional system exhibits much richer supersymmetric structures in Landau level degeneracy due to the internal degrees of freedom. Our study offers the possibility of controlling electromagnetic waves by leveraging the concept of higher-order and higher-dimensional topological phenomena.

A class of 5D Hamiltonian conservative hyperchaotic systems with symmetry and multistability

Conservative chaos systems have been investigated owing to their special advantages. Taking symmetry as a starting point, this study proposes a class of five-dimensional(5D) conservative hyperchaotic systems by constructing a generalized Hamiltonian conservative system. The proposed systems can have different types of coordinate-transformation and time-reversal symmetries. Also, the constructed systems are conservative in both volume and energy. The constructed systems are analyzed, and their conservative and chaotic properties are verified by relevant analysis methods, including the equilibrium points, phase diagram, Lyapunov exponent diagram, bifurcation diagram, and two-parameter Lyapunov exponent diagram. An interesting phenomenon, namely that the proposed systems have multistable features when the initial values are changed, is observed. Furthermore, a detailed multistable characteristic analysis of two systems is performed, and it is found that the two systems have different numbers of coexisting orbits under the same energy. And, this type of system can also exhibit the coexistence of infinite orbits of different energies. Finally, the National Institute of Standards and Technology tests confirmed that the proposed systems can produce sequences with strong pseudo-randomness, and the simulation circuit is built in Multisim software to verify the simulation results of some dynamic characteristics of the system.

Multistability and bifurcations in a 5D segmented disc dynamo with a curve of equilibria

Multistability, i.e., coexisting attractors, is one of the most exciting phenomena in dynamical systems. This paper presents a new category of coexisting hidden attractor: five-dimensional (5D) systems with a curve of equilibria. Based on the segmented disc dynamo, a new 5D hyperchaotic system is proposed. The paper studies not only coexisting self-excited attractors but also coexisting hidden attractors in the new system with four types of equilibria: a curve of equilibria, a line equilibrium, a stable equilibrium, and no equilibria. Furthermore, the paper proves that the degenerate Hopf and pitchfork bifurcations occur in the system. Numerical simulations demonstrate the emergence of the two bifurcations.

Autonomous memristor chaotic systems of infinite chaotic attractors and circuitry realization

Memristor chaotic system has been attracted by many researchers because of the rich dynamical behaviors. However, some existed memristor chaotic systems have finite numbers of chaotic attractors. In this paper, a simple, effective method is given for designing the autonomous memristor chaotic systems of infinite chaotic attractors. Autonomous memristor chaotic systems are proposed from the start of memristor chaotic system counterparts. Three-dimensional, four-dimensional, and five-dimensional memristor chaotic systems are given in standard form with sine functions and tangent functions to prove the effectiveness of this method. Eventually, an analog circuit of three-dimensional memristor chaotic system is designed and implemented to prove its feasibility.

Wavelet characterization of hyper-chaotic time series

A wavelet scaling numerical characterization of time series based on the variance of the wavelet coefficients is used for three well-known four-dimensional and one five-dimensional hyper-chaotic systems. We report several scaling behaviors for the states of these hyper-chaotic systems.

Complete Coefficient Criteria for Five-Dimensional Hopf Bifurcations, with an Application to Economic Dynamics

Paper presents a complete mathematical characterization of coefficient criteria for five-dimensional Hopf bifurcations and an example of the application of these criteria to a model of economic dynamics. The application illustrates that the proposed criteria are practical and useful in determining the existence or nonexistence of Hopf bifurcations of five-dimensional dynamical systems in entire ranges of the system’s parameters.

Adaptive Modified Function Projective Lag Synchronization of Uncertain Hyperchaotic Dynamical Systems with the Same or Different Dimension and Structure

Modified function projective lag synchronization (MFPLS) of uncertain hyperchaotic dynamical systems with the same or different dimensions and structures is studied. Based on Lyapunov stability theory, a general theorem for controller designing, parameter update rule designing, and control gain strength adapt law designing is introduced by using adaptive control method. Furthermore, the scheme is applied to four typical examples: MFPLS between two five-dimensional hyperchaotic systems with the same structures, MFPLS between two four-dimensional hyperchaotic systems with different structures, MFPLS between a four-dimensional hyperchaotic system and a three-dimensional chaotic system and MFPLS between a novel three-dimensional chaotic system, and a five-dimensional hyperchaotic system. And the system parameters are all uncertain. Corresponding numerical simulations are performed to verify and illustrate the analytical results.

Magic-State Distillation in All Prime Dimensions Using Quantum Reed-Muller Codes

We propose families of protocols for magic state distillation -- important components of fault tolerance schemes --- for systems of odd prime dimension. Our protocols utilize quantum Reed-Muller codes with transversal non-Clifford gates. We find that, in higher dimensions, small and effective codes can be used that have no direct analogue in qubit (two-dimensional) systems. We present several concrete protocols, including schemes for three-dimensional (qutrit) and five-dimensional (ququint) systems. The five-dimensional protocol is, by many measures, the best magic state distillation scheme yet discovered. It excels both in terms of error threshold with respect to depolarising noise (36.3%) and the efficiency measure know as "yield", where, for a large region of parameters, it outperforms its qubit counterpart by many orders of magnitude.

On the existence and stability of cycles in five-dimensional models of gene networks

Sufficient conditions for the existence and stability of closed trajectories in five-dimensional nonlinear dynamical systems that model gene networks with negative feedback are obtained.

Second Hopf map and supersymmetric mechanics with a Yang monopole

We propose to use the second Hopf map for the reduction [via $SU(2)$ group action] of the eight-dimensional $\mathcal{N}=8$ supersymmetric mechanics to five-dimensional supersymmetric systems specified by the presence of an $SU(2)$ Yang monopole. For our purpose we develop the relevant reduction procedure. The reduced system is characterized by its invariance under the $\mathcal{N}=5$ or $\mathcal{N}=4$ supersymmetry generators (with or without an additional conserved Becchi-Rouet-Stora-Tyutin charge operator) which commute with the $su(2)$ generators.

Bell inequalities for three particles

We present tight Bell inequalities expressed by probabilities for three four- and five-dimensional systems. A tight structure of Bell inequalities for three $d$-dimensional systems (qudits) is proposed. Some interesting Bell inequalities of three qubits reduced from those of three qudits are also studied.

Hyperchaos–chaos–Hyperchaos Transition in a Class of On–Off Intermittent Systems Driven by a Family of Generalized Lorenz Systems

Blowout bifurcation in nonlinear systems occurs when a chaotic attractor lying in some symmetric subspace becomes transversely unstable. A class of five-dimensional continuous autonomous systems is considered, in which a two-dimensional subsystem is driven by a family of generalized Lorenz systems. The systems have some common dynamical characters. As the coupling parameter changes, blowout bifurcations occur in these systems and brings on change of the systems' dynamics. After the bifurcation the phenomenon of on–off intermittency appears. It is observed that the systems undergo a symmetric hyperchaos–chaos–hyperchaos transition via or after blowout bifurcations. An example of the systems is given, in which the drive system is the Chen system. We investigate the dynamical behaviour before and after the blowout bifurcation in the systems and make an analysis of the transition process. It is shown that in such coupled chaotic continuous systems, blowout bifurcation leads to a transition from chaos to hyperchaos for the whole systems, which provides a route to hyperchaos.

Blowout bifurcation and chaos–hyperchaos transition in five-dimensional continuous autonomous systems

Blowout bifurcation in chaotic systems occurs when a chaotic attractor lying in some symmetric subspace, becomes transversely unstable. There has been previous reports of chaos–hyperchaos transition via blowout bifurcation in synchronization of identical chaotic systems. In this paper, two five-dimensional continuous autonomous systems are considered, in which a two-dimensional subsystem is driven by a chaotic system. As a system parameter changes, blowout bifurcations occur in these systems and bring on changes of the systems’ dynamics. It is observed that one system undergoes a symmetric hyperchaos–chaos–hyperchaos transition via blowout bifurcations, while the other system does not transit to hyperchaos after the bifurcations. We investigate the dynamical behaviours before and after the blowout bifurcation in the systems and make an analysis of the transition process. It is shown that in such coupled chaotic continuous systems, blowout bifurcation may indicate a transition from chaos to hyperchaos for the whole systems, which provides a possible route to hyperchaos.

Generation of on–off intermittency based on Rössler chaotic system

In this paper, one-state on–off intermittency and two-state on–off intermittency are generated in two five-dimensional continuum systems respectively. In each system, a two-dimensional subsystem is driven by the Rossler chaotic system. The parameter conditions under which the on–off intermittency occurs are discussed in detail. The statistical property of the intermittency is investigated. It is shown that the distribution of the laminar phase duration time follows a power law with an exponent of −3/2, which is a signature of on–off intermittency. Moreover, the phenomenon of intermingled basins is observed when attractors in the two symmetric invariant subspaces are stable. We provide an effective way to generate on–off intermittency based on a chaotic system, which is important for application and theoretical study.

On–off intermittency in continuum systems driven by Lorenz system

Previous studies of on–off intermittency in continuum systems are generally in the synchronization of identical chaotic oscillators or in the nonlinear systems driven by the Duffing oscillator. In this paper, one-state on–off intermittency and two-state on–off intermittency are observed in two five-dimensional continuum systems, respectively. The systems have skew product structure in which a two-dimensional subsystem is driven by the well-known Lorenz chaotic system. Moreover, the phenomenon of intermingled basins is observed below the blowout bifurcation. The statistical properties of the intermittency in the systems are investigated. It is shown that the distribution of the laminar phase duration time follows a - 3 2 power law, and that of the burst phase amplitude shows a −1 power law, which coincide with the basic statistical characteristics of on–off intermittency.

Quasi-Lagrangian systems of Newton equations

Systems of Newton equations of the form q̈=−12A−1(q)∇k with an integral of motion quadratic in velocities are studied. These equations generalize the potential case (when A=I, the identity matrix) and they admit a curious quasi-Lagrangian formulation which differs from the standard Lagrange equations by the plus sign between terms. A theory of such quasi-Lagrangian Newton (qLN) systems having two functionally independent integrals of motion is developed with focus on two-dimensional systems. Such systems admit a bi-Hamiltonian formulation and are proved to be completely integrable by embedding into five-dimensional integrable systems. They are characterized by a linear, second-order partial differential equation PDE which we call the fundamental equation. Fundamental equations are classified through linear pencils of matrices associated with qLN systems. The theory is illustrated by two classes of systems: separable potential systems and driven systems. New separation variables for driven systems are found. These variables are based on sets of nonconfocal conics. An effective criterion for existence of a qLN formulation of a given system is formulated and applied to dynamical systems of the Hénon–Heiles type.

A Fast Algorithm for Backbones

A matching algorithm for the identification of backbones in percolation problems is introduced. Using this procedure, percolation backbones are studied in two- to five-dimensional systems containing 1.7×107 sites, two orders of magnitude larger than was previously possible using burning algorithms.

Development of a computer algorithm based on a conjugate gradient approach for optimization of fed-batch fermentations

The problem of optimization of fed-batch fermentations using the substrate feed rate as the control variable is singular in nature. Previous approaches, including the boundary condition iteration method and transformation to a nonsingular problem using a different control variable, do not work well for solving optimization of systems governed by more than four differential equations. The applicability of a first-order conjugate gradient algorithm for optimizing fed-batch fermentations was tested for systems of varing complexity. This approach does not need any variable transformation ora priori knowledge of the control arc sequence. Constraints on the feed rate are handled in a simple and direct manner. The algorithm worked very well for three, four, and five-dimensional singular systems. The correctness of the optimal profile was judged by observing the variation in the sign of the gradient of the Hamiltonian. The gradient was found to be zero during the singular period and had the appropriate sign on the boundary arcs. The optimization method based on conjugated gradient approach can be complementary to the boundary condition iteration method for determination of the exact optimum profile.