4D System Research Articles

Development and implementation of a dynamic and 4D GIS based on semantic location model

ABSTRACTIn the big data era, robust solutions are obliged to be proposed to integrate and represent data from different formats and with different contents to assist the decision-making. Current cartographic and geographic information systems have limited capabilities for solving these problems. This paper describes an automatic and comprehensive system that conducts data fusion from all potentially related sources. In this system, a new Semantic Location Model (SemLM) is established to present the semantic concepts and location feature and demonstrate how locations are interrelated. In the SemLM, various types of location descriptors in different application scenarios can be analyzed and understood. Additionally, considering the challenges involved in data-intensive computation and visualization, this paper implements a Place-based Pan-Information System (P2S) as an innovative 4D system that dynamically associates and visualizes place-based information, using public security as the case study.

Entropy of Reissner-Nordström 3D Black Hole in Roegenian Economics.

The subject of this paper is to analyse the Mathematical Principia of Economic 3D Black Holes in Roegenian economics. In detail, we study two main problems: (i) mathematical origin of economic 3D black holes; and (ii) entropy and internal political stability depending on national income and the total investment, for economic Reissner–Nordström (RN) 3D black hole. To solve these problems, it was necessary to jump from macroeconomic side to microeconomic side (a substantial approach as they are so different), to complete the thermodynamics–economics dictionary with new entities, and to introduce the flow between two macroeconomic systems. The main contribution is about introducing and studying the Schwarzschild-type metric on an economic 4D system, together with Rindler coordinates, Einstein 4D partial differential equations (PDEs), and economic RN 3D black holes. In addition, we introduce some economic Ricci type flows or waves, for further research.

Transit and non-transit 3D EPID dosimetry versus detector arrays for patient specific QA.

PurposeDespite their availability and simplicity of use, Electronic Portal Imaging Devices (EPIDs) have not yet replaced detector arrays for patient specific QA in 3D. The purpose of this study is to perform a large scale dosimetric evaluation of transit and non‐transit EPID dosimetry against absolute dose measurements in 3D.MethodsAfter evaluating basic dosimetric characteristics of the EPID and two detector arrays (Octavius 1500 and Octavius 1000SRS), 3D dose distributions for 68 VMAT arcs, and 10 IMRT plans were reconstructed within the same phantom geometry using transit EPID dosimetry, non‐transit EPID dosimetry, and the Octavius 4D system. The reconstructed 3D dose distributions were directly compared by γ‐analysis (2L2 = 2% local/2 mm and 3G2 = 3% global/2 mm, 50% isodose) and by the percentage difference in median dose to the high dose volume (%∆HDVD 50).ResultsRegarding dose rate dependency, dose linearity, and field size dependence, the agreement between EPID dosimetry and the two detector arrays was found to be within 1.0%. In the 2L2 γ‐comparison with Octavius 4D dose distributions, the average γ‐pass rate value was 92.2 ± 5.2%(1SD) and 94.1 ± 4.3%(1SD) for transit and non‐transit EPID dosimetry, respectively. 3G2 γ‐pass rate values were higher than 95% in 150/156 cases. %∆HDVD 50 values were within 2% in 134/156 cases and within 3% in 155/156 cases. With regard to the clinical classification of alerts, 97.5% of the treatments were equally classified by EPID dosimetry and Octavius 4D.ConclusionTransit and non‐transit EPID dosimetry are equivalent in dosimetric terms to conventional detector arrays for patient specific QA. Non‐transit 3D EPID dosimetry can be readily used for pre‐treatment patient specific QA of IMRT and VMAT, eliminating the need of phantom positioning.

Efficacy of the use of the McKenzie and Vojta methods to treat discopathy-associated syndromes in the pediatric population.

This paper evaluates the efficacy of using the McKenzie and Vojta methods for patients with low back pain and the use of the DIERS Formetric 4D system as an objective diagnostic tool. The study enrolled 28 patients aged 15-17 years old. The patients were hospitalized at the Department of Orthopedics and Traumatology of the Świętokrzyskie Center for Pediatrics in Kielce with a diagnosis of back pain associated with a discopathy. The patients were rehabilitated according to the McKenzie and Vojta methods. Assessment by means of the DIERS Formetric system had taken place before the first therapy session and on the day that pain was eliminated to evaluate trunk inclination, angle of thoracic kyphosis, angle of lumbar lordosis, lateral deviation, trunk torsion, surface rotation and pelvic obliquity. Pain intensity and change in pain intensity over time were assessed by means of a numerical rating scale. Pain intensity was reduced to 0 over 3-12 days. The study participants demonstrated reduction in anterior trunk inclination of the mean value at 1.83°. The angle of thoracic kyphosis was also reduced by 7.95°. The angle of lordosis increased by 7.6°. The lateral spinal curvature was reduced by 8.92 mm. There was a reduction of 4.64° in trunk torsion. Surface rotation was reduced by 1.61° and pelvic obliquity was reduced by 3.78°. In discopathic patients, postural parameters comprising trunk inclination, angle of thoracic kyphosis, angle of lumbar lordosis, lateral deviation, trunk torsion, vertebral rotation and pelvic obliquity fail to reach Hartzmann's physiological reference ranges. A therapeutic intervention based on the Vojta and McKenzie methods may normalize the posture to physiological reference ranges and is effective in the treatment of patients with back pain. The DIERS system is an objective tool for tracing the effects of therapy in patients with back pain. Int J Occup Med Environ Health. 2019;32(1):33-41.

Investigation of Internal Target Volumes Using Device and Deviceless Four-dimensional Respiratory Monitoring Systems for Moving Targets in Four-dimensional Computed Tomography Acquisition.

Aims and Objectives:The influence of target motion on the reconstructed internal target volume (ITV) for device-based (DB) external surrogate system and Smart deviceless (DL) 4-dimensional (4D) system were compared in a controlled phantom experiment. The volumetric changes in reconstructed ITVs from the average intensity projection (AveIP) images using DB method (Anzai Respiratory Gating System, ANZAI MEDICAL CO., LTD, Japan) and DL method (Smart deviceless 4D system by GE Medical Systems (Chicago, USA)) with the theoretical true volume (ITVth) for moving target with the increasing target motion in anterior-posterior (A-P), lateral (left-right [L-R]) and inferior-superior (S-I) directions were assessed.Materials and Methods:4D computed tomography (4DCT) of CIRS dynamic phantom (Computerized Imaging Reference Systems Inc., Norfolk, VA, USA) with 2.5 cm diameter spherical target of volume 8.2 cc programmed to move in a cos4(x) motion pattern placed in the lung volume were acquired for various target motion pattern using DB and DL method of gating. AveIP images of 10 phase binned image sets were generated and ITVs were delineated.Results:The maximum absolute percent differences between ITVave and ITVth for DL and DB methods were 15.91% and 4.94 % respectively for target motion of 5 mm in AP with 15 mm S-I direction. When the S-I motion was decreased to 10 mm, the observed % difference of the ITVs were also decreased to 12.5% and 0.3% for DL and DB method. When the lateral [L-R] motion was varied from 0 mm to 5 mm for S-I motion of 5 mm to 15 mm, the differences in the ITVs were significant (P = 0.004) with the maximum absolute percent difference of 18.61% and 4.94 % for DL and DB gating. With the simultaneous motion of the target in all the 3 directions, the difference in the reconstructed ITVs were statistically significant for DL method (P = 0.0002) and insignificant for DB method (P = 0.06) with an average increase of 10% in ITVDL against 2% in the ITVDB. The difference in ITVDL was significant for the target motion above 3 mm in A-P and L-R directions for S-I movement of above 10 mm (P = 0.0002). However, for low excursions of the target movement, no significant difference in the ITVs were observed (P > 0.06). In general, ITVDBs were closer to the ITVth (within 7.8%) than ITVDL (18.61%).Conclusion:The results showed that the DL method is an effective way of image sorting in 4D acquisition for smaller target excursion. When the target motion exceeds 3 mm in A-P and L-R directions with S-I more than 10 mm, DB method is the choice due to its accuracy in reproducing the absolute target volume.

Two-dimensional dose reconstruction using scatter correction of portal images.

Electronic portal imaging devices (EPIDs) could potentially be useful for patient setup verification and are also increasingly used for dosimetric verification. The accuracy of EPID for dose verification is dependent on the dose-response characteristics, and without a comprehensive evaluation of dose-response characteristics, EPIDs should not be used clinically. A scatter correction method is presented which is based on experimental data of a two-dimensional (2D) ion chamber array. An accurate algorithm for 2D dose reconstruction at midplane using portal images for in vivo dose verification has been developed. The procedure of scatter correction and dose reconstruction was based on the application of several corrections for beam attenuation, and off-axis factors, measured using a 2D ion chamber array. 2D dose was reconstructed in slab phantom, OCTAVIUS 4D system, and patient, by back projection of transit dose map at EPID-sensitive layer using percentage depth dose data and inverse square. Verification of the developed algorithm was performed by comparing dose values reconstructed in OCTAVIUS 4D system and with that provided by a treatment planning system. The gamma analysis for dose planes within the OCTAVIUS 4D system showed 98% ±1% passing rate, using a 3%/3 mm pass criteria. Applying the algorithm for dose reconstruction in patient pelvic plans showed gamma passing rate of 96% ±2% using the same pass criteria. An accurate empirical algorithm for 2D patient dose reconstruction has been developed. The algorithm was applied to phantom and patient data sets and is able to calculate dose in the midplane. Results indicate that the EPID dose reconstruction algorithm presented in this work is suitable for clinical implementation.

203. Patient specific pre-treatment Quality Assurance in multiple lesions lung SBRT VMAT plans

Purpose SBRT of lung multiple lesions is a frequent clinical request. VMAT is preferentially used because it is possible to treat multiple lesions with single arc and single isocenter in a relatively short time. This treatment technique frequently make use of MLC interdigitation and the peripheral part of the treatment field, challenging conditions for both TPS calculation and LINAC delivery. The aim of this study is to evaluate the reliability of this irradiation technique in our department through different methods of patient specific pre-treatment Quality Assurance (QA). Methods VMAT plans have been calculated with the Monaco TPS for an Elekta Synergy Linac mounting the Agility MLC (80 leaf pair, 5 mm width, interdigitation). PTW Octavius 4D system and Octavius detector have been used to measure 3D dose distributions. Each plan has also been recalculated on the Tomotherapy Cheese phantom in order to measure absorbed dose at the center of each lesion with an Exradin A1SL ionization chamber. EBT3 gafchromic films have also been exposed in the Cheese phantom in coronal, sagittal or transversal planes to measure absorbed dose distributions. Results Even with good gamma passing rate (>90%, 3%, 3 mm, local), measurements with PTW Octavius 4D are sometime affected by poor spatial resolution and/or limited acquisition volume, and occasional failures in the high dose region have also been found. On the contrary, very good agreement ( Conclusions By using different independent methods of pre-treatment QA we came to the conclusion that lung SBRT VMAT plans of multiple lesion with single isocenter can be accurately delivered with the technology in use in our department.

Evaluation of jaws-only intensity modulated radiation therapy treatment plans using Octavius 4D system

Abstract Introduction: Jaws-Only Intensity modulated radiation therapy (JO-IMRT) is a technique uses the collimator jaws of the linear accelerator (LINAC) to delivery of complex intensity patterns. In previous studies, pretreatment patient specific quality assurance for those JO-IMRT were also performed using ionization chamber, MapCHECK2, and Octavius 4D and good agreements were shown. The aim of this study is to further verify JO-IMRT plans in 2 different cases: one with the gantry angle set equal to beam angle as in the plans and the other with gantry angle set to zero degree. Materials and Methods: Twenty-five JO-IMRT, previously verified, were executed twice for each plan. The first one used a real gantry angle, and the second one used a 0° gantry angle. Measurements were performed using Octavius 4D 1500. Results: The results were analyzed using Verisoft software. The results show that the Gamma average was 97.32 ± 2.21% for IMRT with a 0° gantry angle and 94.72 ± 2.67% for IMRT with a true gantry angle. Conclusion: In both cases, gamma index of more than 90% were found for all of our 25 JO-IMRT treatment plans.

Imperfect chimeras in a ring of four-dimensional simplified Lorenz systems

Analyzing the dynamics of networks of coupled oscillators possessing chimera states has attracted considerable attention in recent years. Here we study a ring of coupled 4D simplified Lorenz systems whose prototype was first studied by Li and Sprott [1]. An interesting feature of this system is the coexistence of a limit cycle and two symmetric strange attractors for some specific range of parameters. Investigating a network of this type under long range interactions, we find a multitude of very interesting chimera multi-chimera states, whose appearance we relate to initial conditions chosen within the basins of attraction of the limit cycle and strange attractor of the individual 4D system. Thus, for sufficiently low coupling, selecting more and more initial conditions from the strange attractor basins leads to one or more “rebel” particles departing from the synchronous group and producing fascinating “imperfect” chimera states. For larger coupling these “rebels” multiply and the system develops what one might call “perfect” chimera states with multiple “heads”.

Improvement of off-axis SABR plan verification results by using adapted dose reconstruction algorithms for the Octavius 4D system.

Stereotactic ablative body radiotherapy (SABR) for lung patients can be performed with volumetric-modulated arc therapy (VMAT) plans using off-axis target geometry to allow treatment in their CBCT verified position. For patient-specific quality assurance measurements using the PTW Octavius 4D phantom (PTW, Freiburg, Germany) (OCT4D) in conjunction with an Octavius 1000SRS array (OCT1000) (PTW, Freiburg, Germany), repositioning the phantom off-axis is required to ensure the measurement area coincides with the tumor. The aim of this work is to quantify delivery errors using an array repositioned off-axis and evaluate new software which incorporates corrections for off-axis phantom measurements. Dynamic conformal arcs and 25 lung SABR plans were created with the isocenter at the patient midline and the target volume off-axis. Measurements were acquired with an OCT4D phantom in conjunction with a 729 array (PTW, Freiburg, Germany) (OCT729) placed at isocenter. These plans were recalculated and delivered to both the OCT729 and OCT1000 arrays repositioned so that the high-dose region was at the center of the phantom. Comparisons were made using VeriSoft v7.0 (PTW, Freiburg, Germany) and the newly implemented version 7.1 with 2%/2mm gamma criterion (10% threshold) and results correlated with off-axis distance to the tumor. Average pass rates for VeriSoft v7.0 significantly reduced from 92.7±2.4% to 84.9±4.1% when the phantom was repositioned compared to the isocenter setup for the OCT729. The gamma pass rates significantly decreased the further the phantom was moved off-axis. Significantly higher pass rates were observed for the OCT1000 of 95.7±3.6% and a significant decrease in gamma pass rate with off-axis phantom distance was again observed. In contrast, even with phantom repositioning, the pass rates for analysis with VeriSoft v7.1 were 93.7±2.1% and 99.4±1.1% for OCT729 and OCT1000, respectively. No significant difference in gamma pass rate was observed with off-axis phantom position irrespective of array type with the new software. The errors in QA phantom measurements due to dose reconstruction at off-axis target geometry have been demonstrated for conformal arcs and clinical VMAT SABR plans. A novel software solution implemented by the vendor to allow accurate pass rates has been tested. This solution enables high-resolution arrays with small active detection areas to be used for quality assurance of SABR treatment plans in the off-axis treatment position.

Generalized projective series synchronization between chaotic systems and its application

The existence of generalized projective series synchronization scheme between different chaotic systems is investigated. The synchronization scheme consists of three 3D systems and one 4D system to show the generalized projective series synchronization. The synchronization can be well implemented with non-linear adaptive coupling. Sufficient conditions for synchronization are also derived according to Lyapunov stabilization theory. Numerical simulations show the effectiveness of the analytical result. The application of generalized projective series synchronization scheme is implemented to image encryption. The correlation between adjacent pixels in original image and encrypted image is analyzed to show the effectiveness of the synchronization scheme.

Coexistence of hidden attractors, 2-torus and 3-torus in a new simple 4-D chaotic system with hyperbolic cosine nonlinearity

Recently, the study of dynamic systems with hidden attractors has become one of the most followed research avenues owing to its theoretical and practical relevance. A new 4D system with hyperbolic cosine nonlinearity [instead of quadratic nonlinearity as in Jay and Roy (Int J Dyn Control, 2017) https://doi.org/10.1007/s40435-017-0332-8 ] which can exhibit complex and varied dynamics including hidden attractors and multiple attractors is proposed. The proposed system is studied using numerical simulations and nonlinear analysis methods in order to characterize its dynamics with respect to its parameters. A particular feature of the new system is the simplicity of the corresponding electronic analogue (no analog multiplier chip is employed). Although the system has no equilibrium points, it can display rich behaviors such as chaos, 2-torus, and 3-torus behavior. Moreover, the phenomenon of multistability showing the coexistence of hidden attractors, 2-torus and 3-torus is also observed. In addition, the feasibility of the system is demonstrated through PSpice simulations based on an appropriate electronic analog of the model.

Photonic topological boundary pumping as a probe of 4D quantum Hall physics.

When a two-dimensional (2D) electron gas is placed in a perpendicular magnetic field, its in-plane transverse conductance becomes quantized; this is known as the quantum Hall effect. It arises from the non-trivial topology of the electronic band structure of the system, where an integer topological invariant (the first Chern number) leads to quantized Hall conductance. It has been shown theoretically that the quantum Hall effect can be generalized to four spatial dimensions, but so far this has not been realized experimentally because experimental systems are limited to three spatial dimensions. Here we use tunable 2D arrays of photonic waveguides to realize a dynamically generated four-dimensional (4D) quantum Hall system experimentally. The inter-waveguide separation in the array is constructed in such a way that the propagation of light through the device samples over momenta in two additional synthetic dimensions, thus realizing a 2D topological pump. As a result, the band structure has 4D topological invariants (known as second Chern numbers) that support a quantized bulk Hall response with 4D symmetry. In a finite-sized system, the 4D topological bulk response is carried by localized edge modes that cross the sample when the synthetic momenta are modulated. We observe this crossing directly through photon pumping of our system from edge to edge and corner to corner. These crossings are equivalent to charge pumping across a 4D system from one three-dimensional hypersurface to the spatially opposite one and from one 2D hyperedge to another. Our results provide a platform for the study of higher-dimensional topological physics.

Analysis of VMAT dosimetric verifications with different techniques

Objective To investigate the gamma (γ) passing rates for volumetric-modulated arc therapy (VMAT) dosimetric verification with different techniques. Methods A total of 12 VMAT plans for the treatment of different anatomical sites in cancer patients were chosen. The Octavius 4D system was used to measure the dose distributions in two different settings: the gantry was rotating (three-dimensional (3D) and 2D γ-analysis) and the gantry was fixed at 0°(2D γ-analysis). The γ passing rates were analyzed with 3%/3 mm and 2%/2 mm criteria, using the paired t test or Wilcoxon signed-rank test. The 2D γ passing rates for different irradiation methods were calculated. Results For the 3D and 2D dose distributions obtained at a rotating gantry angle as well as the 2D dose distribution obtained at zero gantry angle, the average γ passing rates were 96.03%, 96.98%, and 98.90% for 3%/3 mm (P= 0.227, P= 0.000, P= 0.003); 82.08%, 84.04%, and 90.90% for 2%/2 mm (P= 0.379, P= 0.000, P= 0.000). For the 2D dose distributions obtained with different irradiation methods, the average γ passing rate was 98.99% for 3%/3 mm and 93.68% for 2%/2 mm. Conclusions The VMAT dosimetric verification based on a 3D volumetric dosimeter at a rotating gantry position can be clinically useful for delivery quality assurance (QA), and can achieve the most reliable dose calculation for VMAT, which has more referential values. Key words: Volumetric intensity modulated arc therapy; Dosimetric verification; Octavius 4D system; γ passing rates

Application of detector array in treatment planning system modeling adjustment

Objective To investigate the feasibility of detector array in Monaco modeling for MLC parameters adjustment. Methods One parameter was fixed, and then the other parameter was changed. The γ pass rates of the test beams, namely 3ABUT, 7SegA, and FOUR L, were assessed to determine the values of leaf transmission and leaf offset. A total of 12 tumor cases from different anatomical sites were randomly selected. Two-dimensional dose verification (rack angle zero) of Step& Shot and dMLC plans as well as three-dimensional dose validation of VMAT plan were performed using Octavius 4D system. The γ pass rates were analyzed at a standard of 3%/3 mm. Meanwhile, the point dose verification for these three plans was analyzed to obtain the dose deviations. Results The values of leaf transmission and leaf offset were 0.0105 and-0.08 mm, respectively. The average γ pass rates (%) of Step& Shot, dMLC, and VMAT plans were 88.59±2.94, 87.81±3.28, and 87.45±2.24 before adjustment and 98.45±1.23, 98.9±1.01, and 96.03±1.66 after adjustment. In addition, the average dose deviations (%) according to the point dose verification were 0.85±0.75, 0.95±0.39, and 0.98±0.40 before adjustment and 0.97±0.57, 1.08±0.76, and 0.86±0.45 after adjustment. Conclusions Octavius detector 729 ionization chamber array is a feasible and reliable device in Monaco modeling for MLC parameters adjustment. Key words: Post modeling adjustment; Detector array; γpass rates; Leaf transmission; Leaf offset

Non-linear vibrating systems excited by a nonideal energy source with a large slope characteristic

This paper revisits the problem of an unbalanced motor attached to a fixed frame by means of a nonlinear spring and a linear damper. The excitation provided by the motor is, in general, nonideal, which means it is affected by the vibratory response. Since the system behaviour is highly dependent on the order of magnitude of the motor characteristic slope, the case of large slope is considered herein. Some Perturbation Methods are applied to the system of equations, which allows transforming the original 4D system into a much simpler 2D system. The fixed points of this reduced system and their stability are carefully studied. We find the existence of a Hopf bifurcation which, to the authors’ knowledge, has not been addressed before in the literature. These analytical results are supported by numerical simulations. We also compare our approach and results with those published by other authors.

Virtual patient 3D dose reconstruction using in air EPID measurements and a back-projection algorithm for IMRT and VMAT treatments

PurposeAt our institute, a transit back-projection algorithm is used clinically to reconstruct in vivo patient and in phantom 3D dose distributions using EPID measurements behind a patient or a polystyrene slab phantom, respectively. In this study, an extension to this algorithm is presented whereby in air EPID measurements are used in combination with CT data to reconstruct ‘virtual’ 3D dose distributions. By combining virtual and in vivo patient verification data for the same treatment, patient-related errors can be separated from machine, planning and model errors. Methods and materialsThe virtual back-projection algorithm is described and verified against the transit algorithm with measurements made behind a slab phantom, against dose measurements made with an ionization chamber and with the OCTAVIUS 4D system, as well as against TPS patient data. Virtual and in vivo patient dose verification results are also compared. ResultsVirtual dose reconstructions agree within 1% with ionization chamber measurements. The average γ-pass rate values (3% global dose/3mm) in the 3D dose comparison with the OCTAVIUS 4D system and the TPS patient data are 98.5±1.9%(1SD) and 97.1±2.9%(1SD), respectively. For virtual patient dose reconstructions, the differences with the TPS in median dose to the PTV remain within 4%. ConclusionsVirtual patient dose reconstruction makes pre-treatment verification based on deviations of DVH parameters feasible and eliminates the need for phantom positioning and re-planning. Virtual patient dose reconstructions have additional value in the inspection of in vivo deviations, particularly in situations where CBCT data is not available (or not conclusive).

Complex novel 4D memristor hyperchaotic system and its synchronization using adaptive sliding mode control

First, this paper announces a flux controlled novel 4-D chaotic system displaying chaotic attractor modified from a 4D hyperchaotic system. The phase portraits of the novel chaotic system are displayed and the qualitative properties are discussed. Next we derive the complex model of the proposed 4D system. We investigate the dynamic properties of the complex systems. Adaptive sliding mode control is used to achieve non-identical synchronization of the complex systems. Numerical simulations are conducted to validate the results.

Improving the Global Analysis of Mechanical Systems via Parallel Computation of Basins of Attraction

Numerical integrations represent a time-consuming element in the long-term dynamics analysis of mechanical systems. This limits the resolution of the computations and the dimension of the system to be investigated numerically. In fact, even pushing memory resources to their thresholds, only few tools can deal with higher-dimensional systems. This work illustrates, in a preliminary manner, the results that can be obtained reducing the aforementioned constraints thanks to the implementation of algorithms based on a parallel computing approach. In particular, by focusing on basins of attraction, four applications are discussed. i) The full domain of attraction for a four-dimensional (4D) system describing a linear oscillator coupled with a nonlinear absorber is calculated. ii) The variation of a safe basin with respect to the system dimension is then analyzed. It is highlighted how 4D and 3D analyses provide more confident results with respect to 2D analyses. iii) The parametric variation of a 2D system with a reduced step is performed by building a 3D representation which allows to highlight a smooth transition between the states. iv) A convergence study of a basin of attraction resolution is carried out. The integrity factor is used as a comparison measure.

Improving the Complexity of the Lorenz Dynamics

A new four-dimensional, hyperchaotic dynamic system, based on Lorenz dynamics, is presented. Besides, the most representative dynamics which may be found in this new system are located in the phase space and are analyzed here. The new system is especially designed to improve the complexity of Lorenz dynamics, which, despite being a paradigm to understand the chaotic dissipative flows, is a very simple example and shows great vulnerability when used in secure communications. Here, we demonstrate the vulnerability of the Lorenz system in a general way. The proposed 4D system increases the complexity of the Lorenz dynamics. The trajectories of the novel system include structures going from chaos to hyperchaos and chaotic-transient solutions. The symmetry and the stability of the proposed system are also studied. First return maps, Poincaré sections, and bifurcation diagrams allow characterizing the global system behavior and locating some coexisting structures. Numerical results about the first return maps, Poincaré cross sections, Lyapunov spectrum, and Kaplan-Yorke dimension demonstrate the complexity of the proposed equations.