THE ALPHA GROUP DYNAMIC MAPPING

THE ALPHA GROUP DYNAMIC MAPPING Cleber Souza Corrêa Instituto de Aeronáutica e Espaço, São José dos Campos, SP, Brazil E-mail: cleb[email protected] Thiago Braido Nogueira de Melo Instituto de Aeronáutica e Espaço, São José dos Campos, SP, Brazil E-mail: [email protected] July 25, 2025 Abstract This paper investigates the dynamical behavior of a system of ordinary differential equations (ODEs) governed by a matrix that represents the division in the algebra of the Alpha group. As the system evolves, the matrix induces topological transitions in geometric spaces, controlled by a rotational parameter. Numerical simulations are performed using a fourth-order Runge-Kutta method implemented in Python. The results reveal the emergence of topological nodes, the existence of critical points at which the rotation between dividing planes transitions from 0 to π/2radians. Near zero radians, the system exhibits a Euclidean geometric structure, while rotations close to π/2define an Alpha Group space. At these nodes, the matrix-driven ODE system undergoes qualitative dynamic changes, reflecting distinct topological behaviors. The Alpha Group matrix is interpreted as a generator of symmetry transformations, potentially analogous to gauge fields under local or global symmetries. This work provides a computational framework for exploring dynamic topologies, attractors at infinity, and internal coherence in hyper-complex vector spaces. Keywords: Alpha Group, Asymptotic Compactification, Hyperboloid, Manifolds, Hypercomplex Topology. 1 Introduction The use of ordinary differential equation (ODE) systems is a common method in science to study the dynamics of complex systems. This methodology is well-established in the literature, where ODE simulations are used to observe 1 arXiv:2507.18303v1 [math.DG] 24 Jul 2025 resulting behavior and map properties by changing initial conditions. The Alpha Group defines a structure based on a division-based operation. Corrêa et al. (2022) proposed the Alpha group, using group theory, formed by the transformation of two infinite planes interacting through a change in the division operation between them. This interaction creates a third element with morphism and preserves the operations in both planes. Infinity is associated with a geometric representation that induces topological deformations and generates an attractor-like structure in R4. This situation resulted from a division operation associated with the rotation between the planes that form the Alpha group. Corrêa et al. (2024) demonstrated that the metrics of infinitesimal distance from Riemannian and Euclidean space are special cases of the Alpha group’s metrics. The Alpha group satisfies the properties defined by group theory. These findings establish a consistent structure that geometrically and topologically characterizes a hypercomplex numerical group, possessing inherent properties that define transformations between surfaces. These definitions enable the development of new mathematical research. The geometry of the alpha group is based on George Cantor’s theories, which explored the nature of different types of infinity (Corrêa and de Melo, 2025). This allows for the interpretation of the existence of numerous and varied types of infinity and their connection to geometry and topology. This structure has a trigonometric representation associated with the hypercomplex plane, resulting in a 4×4antisymmetric matrix with a system that incorporates tangent and cotangent functions between its elements. The matrix’s properties must be analyzed dynamically. In such a scenario, ODE systems enable changes in initialization conditions and observation of the resulting dynamic behavior. A significant feature of the Alpha Group is that when zero radian rotation occurs in the formation of the numeric group, it is associated with Euclidean topology. However, when the rotation is π/2radians, maximum deformation occurs, generating a tangent plane at infinity. In this aspect, it converges asymptotically and may be a manifestation of asymptotic symmetry, where the field tends to a stable configuration at infinity. This stable structure is associated with a type of global attractor in four dimensions and is defined by an imaginary number (µ) that has geometric and topological representations in the context of hyperbolic topology. In a geometric context, the global attractor can represent an asymptotic symmetry, where the field tends to a stable or homogeneous configuration as it approaches infinity. This topological change along the parameter θ, from the local Euclidean space to the infinite tangent plane, reflects a fundamental concept that can be associated with Gauge theory. The idea is that local symmetries can extend to describe global properties, converging asymptotically into a stable asymptotic convergence (closed hyperbolic attractor). Thus, this analysis can effectively characterize how classical geometry can integrate with Gauge theory, accurately describing both local interactions and global transformations. Because the resulting matrix model exhibits properties that can change the topology, transitioning from a Euclidean topology to an R4topology in the Alpha group space, this work aims to map the dynamic characteristics of the matrix resulting from group theory, which leads to the Alpha group, using 2 different initializations. 2 Methodology 2.1 Mathematical Model The Alpha Group can be constructed through a tensor-based division operation between two matrices formulated via the De Moivre identity, even if the quaternion in the Alpha Group can be made of two complex planes. The construction of the division approach is analogous to the Kronecker product, as demonstrated in Graham’s (2018) equation (I). The result of equation (II) emerged naturally from the relationship between the operators; however, there may be other possible ways of representing this operation. Looking at the trigonometric relationship, the ratio between sin θand cos θrepresents the tangent (tan θ), and the cotangent (cot θ) is the inverse function of the tangent. The tangent of an angle can also be defined by the ratio between the measure of the opposite side and the measure of the adjacent side to the angle θin a right triangle. It can replace trigonometric mathematical relations and possibly division operations. The matrix A(II) resulting from the division of two complex planes generates a transformation into a matrix of a hypercomplex space. Matrix Ahas some interesting properties: •Antisymmetry: The matrix is not symmetric, meaning that A=AT, where ATis the transpose of A. This implies distinct properties compared to symmetric matrices. •Non-zero determinant: The determinant of Ais non-zero, meaning the matrix is non-singular. This implies that the matrix has an inverse. •Non-zero diagonal elements: The diagonal elements of Aare nonzero; the matrix has non-zero diagonal elements, but is not diagonal. •Elements off the main diagonal: Elements off the main diagonal have a specific relationship based on angle θ, indicating a non-trivial structure. •Parameter dependency: Matrix Adepends on the parameter θ, which implies that its properties can vary based on this parameter. •Transformation properties: Matrix Acan represent a linear transformation in a vector space, with specific properties related to the application of that transformation. Accordingly, the antisymmetric matrix Aexhibits dynamic properties as the rotation between the planes occurs to form the Alpha Group (Corrêa et al., 2022 and 2024). One way to analyze this matrix Ais to associate it with a system of ordinary differential equations (ODEs). An ordinary differential equation can be written as: d dtx=A·x(I) 3 An improvement to matrix (II) can be achieved by inserting the parameter µ along the main diagonal. This modification characterizes the operation defined within the domain of the Alpha Group, taking into account both the effects of topological deformation between planes due to rotations in the interval [0,2π] and the numerical construction principles underlying the Alpha Group. The inclusion of µon the main diagonal ensures that the matrix is nonsingular (i.e., its determinant is non-zero), while also preserving its antisymmetric structure. This property is essential for maintaining the coherence of the internal transformations governed by the group. The matrix M(θ)is then defined as the product: M(θ) = A(θ)·B(µ), where A(θ)is an angular matrix explicitly dependent on the rotational parameter θ, and B(µ)is a phase amplitude matrix incorporating the deformation parameter µ. The resulting matrix M(θ)is a non-Hermitian structure that acts as a generator of internal vectorial variations. In the context of the Alpha Group algebra, this composition formalizes a fundamental mechanism of internal dynamics, coupling angular and amplitude contributions into a unified framework. M(θ) =     1−cot θ−tan θ1 cot θ1−1−tan θ tan θ−1 1 −cot θ 1 tan θcot θ1     ·     1 1 1 1 1i1 1 1 1 µ1 1 1 1 iµ     (II) The system of ODEs is represented as: d dt     x1 x2 x3 x4     =     1−cot θ−tan θ1 cot θ1−1−tan θ tan θ−1 1 −cot θ 1 tan θcot θ1         x1 x2 x3 x4     (III) The ODE system (III) can express each row as a differential equation. Let’s use variables x1,x2,x3, and x4for the dependent variables and assume they are functions of some independent variable, usually denoted as t(time). Within the ODE system with matrix A,dx/dt is initialized as a vector x0= (1,1,1,1) with the same size as x. The four differential equations are then defined based on the specific model of the system: The derivative of x1for time is defined by the equation x1−cot θ·x2−tan θ·x3+x4, and the other three derivatives (x2,x3,x4) are defined similarly. Essentially, this is a system of first-order ODEs with four differential equations, each representing the rate of change of a system variable over time. This system is then solved numerically using the fourth-order Runge-Kutta method. A script was built to numerically simulate the ODE system, written in Python, and the simulation environment was the website mycompiler.io (https://www.mycompiler.io/pt). In this script, the 4 fourth-order Runge-Kutta function was used. The discretization factor hwas 0.001 and integrated over time at 1.5. The value of µwas also replaced by 1. The Python script was programmed to generate the Poincaré map associated with each rotation angle of the matrix Asystem. The NumPy Python package is the fundamental package for scientific computing in Python. It is a Python library that provides a multidimensional array object, various derived objects (such as masked arrays and matrices), and an assortment of routines for fast operations on arrays, including mathematical, logical, shape manipulation, sorting, selecting, discrete Fourier transforms, basic linear algebra, basic statistical operations, random simulation, and much more. 2.2 The Poincaré Map The Poincaré map is a fundamental technique in the study of dynamical systems, providing a way to analyze the behavior of differential equations by projecting trajectories onto a lower-dimensional plane. This method simplifies the analysis by reducing the continuous dynamics to a discrete system, allowing the identification of periodic orbits, bifurcations, and attractors. The central idea behind the Poincaré map is to define a specific subset of the system’s phase space, known as the Poincaré section, and to observe the intersection points of the trajectories with this section. Typically, this is done by fixing one of the system’s variables and recording the points at which the trajectory crosses the section transversally. These intersection points generate a two-dimensional map—called the Poincaré map—which captures the essence of the system’s dynamics in a discrete form. It is particularly useful in the analysis of complex or chaotic systems where continuous observation is difficult or uninformative. In computational implementations, such as in Python, a custom function (e.g., poincare_map(x_values)) can be defined to generate the Poincaré map numerically. This function typically extracts the coordinates of the points where the trajectories intersect the chosen section, using tools from numerical integration and array handling (e.g., NumPy). This approach allows for an effective visualization and classification of dynamic behavior, serving as a bridge between continuous differential systems and discrete dynamics. 2.3 Phase Diagram The Python script also generated the phase diagram, which graphically represents the system’s trajectories within the state space defined by a system of four differential equations. These trajectories illustrate how the state variables evolve and interact dynamically. In a phase diagram, the evolution of time is visualized through projections in selected pairs of state variables, allowing a two-dimensional analysis of the system’s behavior. This method provides valuable insight into the structure and stability of the system. 5 The interpretation of the phase diagram depends on the specific characteristics of the dynamic system to be studied. Among the general features that may appear in such diagrams are: •Trajectories and their global behavior over time; •Equilibrium points (fixed points); •Attractors and repellors; •Stability regions and their boundaries; •Limit cycles and periodic orbits; •Saddle points and separatrices; •Emergent chaotic behavior. Phase diagrams are essential tools for identifying and classifying dynamic regimes, especially in systems where analytical solutions are difficult or impossible. When combined with Lyapunov analysis and Poincaré maps, they offer a powerful framework for understanding both local and global stability properties. 2.4 Lyapunov Function To identify attractors at infinity, it is often necessary to employ specific numerical techniques capable of analyzing the long-term behavior of a dynamical system. In this context, the Lyapunov method was used to assess both the stability and the presence of attractors located at infinity. The key idea is that if a system possesses an attractor at infinity, the state variables tend to grow without bound, often at an exponential rate. This growth can be quantified using a Lyapunov function evaluated along the solution trajectories of the system. Numerically, the Lyapunov values were computed using the following function: lyapunov_values = np.apply_along_axis(lyapunov_function, 1, ode_solution) where ode_solution is obtained by numerical integration of the system using: from scipy.integrate import odeint The SciPy library is a collection of mathematical algorithms and convenience functions built on top of NumPy. It adds substantial computational power to Python by offering high-level routines and data manipulation tools, making it suitable for tasks such as the numerical analysis of dynamical systems, including the detection of asymptotic behaviors and attractors. This approach enables the classification of system stability through the sign and magnitude of the Lyapunov values, especially when investigating structures that diverge toward infinity or stabilize on non-trivial asymptotic manifolds. 6 2.5 Bifurcation Diagram The bifurcation diagram of the differential equation system defined by matrix A was generated using the Python libraries NumPy and matplotlib.pyplot. The NumPy library plays a fundamental role in scientific computing with Python, providing efficient support for multidimensional arrays and high-performance mathematical operations. To solve the system, a custom implementation of the classical fourth-order Runge–Kutta method was employed. This method numerically integrates the system’s dynamics over a range of initial conditions or parameters, focusing on the variation concerning the angular parameter θ. The function calls: bifurcation_data.append(solution_rk4[-1]) was used to collect the final value of the state variables from each Runge–Kutta integration. These terminal values were then plotted to construct the bifurcation diagram, which reveals changes in the qualitative behavior of the system as θvaries. The resulting diagram provides insight into the structure of the system’s dynamics, including the appearance of fixed points, periodic solutions, branching behavior, and transitions that may indicate bifurcations or the onset of chaotic regimes. 2.6 Jacobian Matrix The change in the ODE system of the matrix Ais associated with the θangle. If the maximum deformation occurs at π/2radians and the formation of an Alpha group number space results in a significant change in the system’s behavior. This may indicate a critical point or singularity in the system’s configuration space. The eigenvalues of the Jacobian matrix of the ODE system of matrix Awere also calculated close to π/2to observe whether the majority of the eigenvalues were composed of complex numbers with a negative real part. This suggests that the system has asymptotically stable behavior. The Jacobian matrix of a system of differential equations is a matrix where each element Jij represents the partial derivative of the function fito the variable xj. The formula for the Jacobian matrix of the ODE system is given by: J4x4=      ∂f1 ∂x1 ∂f1 ∂x2 ∂f1 ∂x3 ∂f1 ∂x4 ∂f2 ∂x1 ∂f2 ∂x2 ∂f2 ∂x3 ∂f2 ∂x4 ∂f3 ∂x1 ∂f3 ∂x2 ∂f3 ∂x3 ∂f3 ∂x4 ∂f4 ∂x1 ∂f4 ∂x2 ∂f4 ∂x3 ∂f4 ∂x4      (IV) where fiis the i-th function of the ODE system of matrix Aand xjis the j-th state variable. The ‘sympy‘ library was used to calculate the eigenvalues of the Jacobian matrix. 7 3 Results The results were important because they show the dynamic behavior of the system of ordinary differential equations associated with the matrix A. Figure 1 illustrates the system dynamics under zero-radian rotation, representing the Euclidean regime. In this aspect, its topology can be associated with an Euclidean space. The system converges to a stable zero-equilibrium state when the rotation approaches 0 + nπ radians. In a certain aspect, it generates a type of dynamically stable and convergent behavior. Figure 1: The dynamics of the ODE system and Poincaré map associated with simulation near zero radian rotation or 0 + nπ radians. An analysis of the Poincaré map can provide valuable information about the dynamics of a dynamical system. Figure 1 shows a distribution whose dynamic system has a stable equilibrium point. It can be seen that the points in the Poincaré figure converge to this point. The trajectories in the phase diagram represent the solutions of the system over time. Each point on the trajectory corresponds to the state of the system at a given moment. As shown in Figure 2, trajectory convergence behavior is shown for a specific region at zero value. The topology of Euclidean space is characterized by zero rotation between planes. It was also calculated for the angle of 0.2 degrees (Euclidean Space); however, for the zero-degree angle, the ODE system exhibits trajectories converging toward the origin, and the Lyapunov function does not grow exponentially over time but instead approaches a constant value. Figure 3 shows the situation when the rotation is π/2 + nπ radians (90 degrees). In this case, the ODE system shows that the system grows infinitely, tending to infinity asymptotically. In this case, the maximum topological deformation and the formation of the Alpha group space are associated. In this case, the ODE with matrix Acan be a system behavior generator in R4, where in the context of Gauge’s theory, there is a relationship between local behavior 8 Figure 2: The phase diagram and the Lyapunov function for the Matrix A ODE system for the angle 0.02 degrees close to 0 radians. (zero radians) and global behavior (like the asymptotic attractor (π/2radians)), which is similar to the idea of its dynamical system growing and forming complex geometric structures. The alpha group and its matrix M(θ)could be related to the way gauge fields are transformed under local (or global) symmetries defined by the group. The matrix would then represent a generator of this specific symmetry and asymmetry. The deformations are associated with the canonical vector imaginary number µ(Corrêa et al., 2022 and 2024). The vector element µ, considered as a canonical vector and topological invariant, transcends the traditional role of an imaginary unit. Within the algebraic formulation of the Alpha Group, it µacquires an internal vectorial condition, functioning as a generator of symmetry that organizes and structures the internal algebraic relations. Importantly, the element µremains unaffected by fluctuations in the angular parameter θ, internal rotations governed by M(θ), or by any external perturbations. It thus becomes an absolute topological reference, serving as an invariant anchor within a complex and topologically structured vector space. This framework supports a transition of topological domains from a collapsed 3-sphere (S3), which corresponds to a real regime, to a bundled 4-sphere (S4), representing a complex regime. In this transition, a hidden degree of freedom begins to emerge through the imaginary component of the structure. The dynamic behavior of the complex eigenvalues of M(θ)plays a central role: when Re(λ)→0and Im(λ)→ ∞, phenomena such as torsion and internal curvature become prominent. 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