Multiscale topology and dynamic internal vectorial geometry - Alpha group

1 Studies in Engineering and Exact Sciences, Curitiba, v.6, n.1, p.01-23, 2025 Multiscale topology and dynamic internal vectorial geometry - alpha group Topologia multiescala e geometria dinâmica vetorial interna - grupo alpha Topología multiescala y geometría vectorial dinámica interna - grupo alpha DOI: 10.54021/seesv6n1-042 Originals received: 5/20/2025 Acceptance for publication: 6/13/2025 Cleber Souza Corrêa PhD in Water Resources Institution: Institute of Aeronautics and Space Address: São Jose dos Campos, São Paulo, Brazil E-mail: [email protected] Thiago Braido Nogueira de Melo Master in Aeronautical and Mechanical Engineering Institution: Institute of Aeronautics and Space Address: São Jose dos Campos, São Paulo, Brazil E-mail: [email protected] ABSTRACT This work sought to apply the Continuous Wavelet Transform (CWT) to the multiscale time-frequency analysis of non-stationary signals associated with the temporal evolution behavior of the complex eigenvalues of the matrix M(θ(t)), with θ=2π radians. The results of CWT analysis have proven to be particularly effective in identifying oscillatory patterns, transient regimes and resonant features that are not evident through conventional spectral analysis, such as that which occurs using Fourier Transform. The results show that the CWT scalogram clearly reveals a persistent multiscale structure associated with an internal dynamic vector geometry, which exhibits a complex foliated topology. Thus, the use of CWT enabled a refined characterization of the system's spectral dynamics, serving as a key tool for identifying topological phenomena, internal resonances, and recurrent structures intrinsic to the vectorial geometry of the Alpha Group. Keywords: Spectral Dynamics. Multiscale Analysis. Complex Eigenvalues. Topological Structures. RESUMO Este trabalho buscou aplicar a Transformada Wavelet Contínua (CWT) à análise multiescala tempo-frequência de sinais não estacionários associados ao comportamento de evolução temporal dos autovalores complexos da matriz 2 Studies in Engineering and Exact Sciences, Curitiba, v.6, n.1, p.01-23, 2025 M(θ(t)), com θ=2π radianos. Os resultados da análise da CWT se mostraram particularmente eficazes na identificação de padrões oscilatórios, regimes transitórios e características ressonantes que não são evidentes por meio da análise espectral convencional, como a que ocorre utilizando a Transformada de Fourier. Os resultados mostram que o escalograma da CWT revela claramente uma estrutura multiescala persistente associada a uma geometria vetorial dinâmica interna, que exibe uma topologia foliada complexa. Assim, o uso da CWT permitiu uma caracterização refinada da dinâmica espectral do sistema, servindo como uma ferramenta-chave para a identificação de fenômenos topológicos, ressonâncias internas e estruturas recorrentes intrínsecas à geometria vetorial do Grupo Alpha. Palavras-chave: Dinâmica Espectral. Análise Multiescala. Autovalores Complexos. Estruturas Topológicas. RESUMEN Este trabajo tuvo como objetivo aplicar la Transformada Wavelet Continua (CWT) al análisis multiescala tiempo-frecuencia de señales no estacionarias asociadas con el comportamiento de evolución temporal de los autovalores complejos de la matriz M(θ(t)), con θ=2π radianes. Los resultados del análisis CWT demostraron ser particularmente efectivos en la identificación de patrones oscilatorios, regímenes transitorios y características resonantes que no son evidentes a través del análisis espectral convencional, como el que ocurre utilizando la Transformada de Fourier. Los resultados muestran que el escalograma CWT revela claramente una estructura multiescala persistente asociada con una geometría vectorial dinámica interna, que exhibe una topología foliada compleja. Así, el uso de la CWT permitió una caracterización refinada de la dinámica espectral del sistema, sirviendo como una herramienta fundamental para la identificación de fenómenos topológicos, resonancias internas y estructuras recurrentes intrínsecas a la geometría vectorial del Grupo Alpha. Palabra clave: Dinámica Espectral. Análisis Multiescala. Valores Propios Complejos. Estructuras Topológicas. 1 INTRODUCTION In the Alpha group algebra, its central operation that defines this group is the division operation, which defines a rotation between complex planes by an angle of π/2 radians, implying symmetries with global characteristics and defining a dynamic and foliated topology (Correa et al., 2022). The group contains familiar numerical subgroups (real and complex) and provides a way to visualize infinity that, in a projective way, geometrically will be on the edge of the domain by the process of asymptotically compactification, and defining, by geometric projection, 3 Studies in Engineering and Exact Sciences, Curitiba, v.6, n.1, p.01-23, 2025 an imaginary number μ, which can be useful in physics and mathematics, particularly in understanding symmetries and structures that vary in scale. Then, the Alpha Group is a group that can be defined as by a radial division operation according to which a division ring is characterized by two complex planes. The hypercomplex geometry is composed of four fundamental elements: 1, i, μ and iμ. These elements in R4 define a tensor metric, which generalizes the Riemann and Euclid metrics, Corrêa et al. (2024). Therefore, this geometry defines aspects of great topological relevance in the matrix M(θ), the main one being the characterization of an internal dynamic vector geometry, associated with a complex foliated topology. This behavior is evidenced by the analysis of its eigenvalues and eigenvectors of the matrix M(θ), and when the angle is θ = π/2 radians, as demonstrated by Corrêa and Melo (2025). Its tangent term existing in the matrix grows rapidly, giving rise to an asymptotic compactification process, whose limit tends to form global symmetries. At this geometric point of maximum deformation and projective projection, on the edge of the geometric domain, the imaginary number μ emerges, which defines a canonical vector of the numerical space proposed by the Alpha Group. This internal vectorial dynamic geometry that forms with the emergence of a complex foliated topology would have a highfrequency dynamics associated with the eigenvalues with magnitudes of the imaginary part of 1016 calculated from the matrix M(ɵ) in π/2 radians. Therefore, it is extremely important to map these dynamics and observe what its structure would be and whether there would be a multiscale nature in the space-time domain. In the literature, the use of the Continuous Wavelet Transform (CWT) has been consolidated as a powerful tool for the analysis of continuous time series in the joint time and frequency (or scale) domain. Unlike Fourier Transform, which represents data only in the global frequency domain, the CWT allows the capture of local and multiscale phenomena, being particularly effective in the detection of transient patterns, singularities and hierarchical structures. Applications in several areas, such as meteorology, physics, geophysics, engineering and dynamic systems, show that CWT provides a more faithful and rich representation of nonstationary signals, with the possibility of associating scales with characteristic frequencies over time, Stark (1992); Torrence and Compo (1998); Grinsted et al. (2004); and Addison (2017). 4 Studies in Engineering and Exact Sciences, Curitiba, v.6, n.1, p.01-23, 2025 Such tools become particularly powerful in the analysis of eigenvalues of matrices that encode internal geometries, as is the case of the matrix M(θ) in the Alpha Group. The scalogram resulting from the CWT applied to the eigenvalues of this matrix, it would allow the identification of regions where internal topological changes occur, characterized by spectral singularities located at certain scales. Thus, the use of CWT in multiscale geometric contexts not only enriches the quantitative analysis of signals and spectra but also provides a bridge between analytical and geometric interpretations, being fundamental for the understanding of mathematical systems that incorporate dynamical symmetries, emergent topologies and non-trivial internal structure. In this context, this work aims to use CWT to search for possible coherence between dynamical structures at multiple scales and regions. It will be a technique relevant for models with foliated topologies and internal vector geometries. In these models, local variations of curvature and internal symmetry can be reflected in the wavelet scalograms, revealing topological 2 METHODOLOGY 2.1 ALPHA GROUP The Division as Radial Vector Relation – Alpha Group by Corrêa and Melo (2025), it was presented as the conformal matrix M(θ), being a geometric central nucleus that characterizes the Alpha Group. This matrix has the angle θ, which defines a continuous topological transition between a local Euclidean geometry (θ = 0) and a non-Euclidean global symmetry structure associated with the Alpha group (θ = π/2). In the latter case, the matrix M(π/2) has pairs of complex conjugate eigenvalues with modulus of the order of 1016, revealing a dynamic vectorial internal geometry, interpreted as a complex foliated topology. The construction is conceptually close to Weyl's gauge theory (1929), suggesting an analogy with internal transformations that preserve dynamic symmetries of the system. 5 Studies in Engineering and Exact Sciences, Curitiba, v.6, n.1, p.01-23, 2025 M(θ) = (1 −cotg θ −tg θ 1 cotg θ 1 −1 −tg θ tg θ −1 1 −cotg θ 1tg θ cotg θ 1) (1) 2.2 CONTINUOUS WAVELET TRANSFORM (CWT) CWT is a signal analysis tool that decomposes a time-domain signal into different frequency or scale components (Sadowsky, 1996; Daubechies, 2002; Wee et al., 2008; Debnath, 2012; and Pachori, 2023). Technically, it is defined as the convolution of the input signal x(t) with a family of wavelet functions ψa,b(t), which are scaled and translated versions of a single "wavelet" ψ(t). The mathematical definition is: 𝐶𝑊𝑇(𝑎,𝑏)=1 √|𝑎|∫𝑥 ∞ −∞ (𝑡)𝜓(𝑡−𝑏 𝑎)𝑑𝑡 (2) The CWT(a,b) are the wavelet coefficients, which represent the similarity between the signal and the wavelet at a given scale and position; The term a is the scale (or dilation) factor, which controls the width of the wavelet. Small scales (a small value of a) correspond to compressed wavelets (or high frequencies), ideal for capturing fast and transient details. However, large scales (a large value of a) correspond to stretched wavelets (low frequencies), which are ideal for capturing trends and long-term features; The term b is the translation (or displacement) factor, which moves the wavelet over time, allowing analysis of the signal at different time points; The term ψ(t) is the mother wavelet, an oscillatory function with zero mean and localized in both time and frequency; the Morlet wavelet is the one being used (Goupillaud et al., 1984; Büssow, 2007); and the term ψ is the complex conjugate of the mother wavelet. In essence, CWT provides a representation of the signal in a two-dimensional time-frequency (or time-scale) space, allowing us to analyze how the different frequency components of the signal evolve over time. This is particularly useful for non-stationary signals, in which the frequency characteristics change over time, such as the "bursts" and multiscales that we seek to observe in the eigenvalues of the matrix M(θ) near the value π/2 radians. 6 Studies in Engineering and Exact Sciences, Curitiba, v.6, n.1, p.01-23, 2025 The data series that CWT analyzes is the eigenvalue module, and the behavior of these modules is directly constructed and determined by the variation of θ over time. The fundamental parameter (θ), in the matrix M(θ) depends directly on the angle θ. All the resonance dynamics, explosions and reorganizations that we observe in the eigenvalues and eigenvectors are caused by the properties of tan(θ) and cot(θ) when θ approaches multiples of π/2 rad. Its generation of the variability of the angle θ over time. It was carried out in the development of a Python script and it was run on Google Colab. In this script, the variation of θ is parameterized by time t through the sinusoidal function: theta_t = A np.sin(omega t). This means that as time t progresses, theta_t oscillates, exploring the critical values where the matrix M(θ) exhibits its resonant behavior. The series generated for CWT shows that this series would represent how the magnitude of one of the specific eigenvalue changes. Since the values are calculated for each theta_t (which in turn is generated for each t), the time series created is a time series (indexed by t_vals), but whose dynamics and characteristics are intrinsically linked to the values of θ. Therefore, although CWT operates on a series indexed by time (t_vals), the "information" it is analyzing (the peaks and multiscales) is a direct consequence of the behavior of the matrix M(θ) when θ assumes certain values. In other words, the series is built on the variation of θ, which is the engine of the system dynamics. Even though the series we look at using the Continuous Wavelet Transform (CWT) is tracked over time, the patterns and sudden changes we notice are mainly influenced by the internal structure of the matrix M(θ). This means that as time progresses, it is the specific values and properties within M(θ) that actually dictate the dynamics observed in the eigenvalue modules. Part of the Python script: # Parameters A = np.pi / 2 # Adjusted so that theta_vals includes pi/2 and -pi/2 omega = 2 * np.pi #The sampling performed of t_vals was 1000 elements. t_vals = np.linspace(0, 2, 1000) # Matrix M(θ) 7 Studies in Engineering and Exact Sciences, Curitiba, v.6, n.1, p.01-23, 2025 def M(theta): #Defines the matrix M(θ) as specified. Handles cases where tan(theta) or cot(theta) may be undefined. epsilon = 1e-16 tan_theta = np.tan(theta) cot_theta = 1 / (tan_theta + epsilon) return np.array([ [1, -cot_theta, -tan_theta, 1], [cot_theta, 1, -1, -tan_theta], [tan_theta, -1, 1, -cot_theta], [1, tan_theta, cot_theta, 1] ], dtype=complex) # Calculate eigenvalues and eigenvectors eigenvalues = [] eigenvectors = [] theta_vals = [] for t in t_vals: theta_t = A * np.sin(omega * t) theta_vals.append(theta_t) M_t = M(theta_t, mu) w, v = np.linalg.eig(M_t) idx = np.argsort(np.real(w)) w_sorted = w[idx] v_sorted = v[:, idx] eigenvalues.append(w_sorted) eigenvectors.append(v_sorted.T) eigenvalues = np.array(eigenvalues) # ... (rest of code for other analyses and plots) ... # ... Wavelet Analysis for Eigenvalue Modules ... # ... (definition of 'widths' and 'for i in range(4)' loop) ... signal_to_analyze = np.abs(eigenvalues[:, i]) signal_to_analyze[np.isinf(signal_to_analyze)] = 0 signal_to_analyze[np.isnan(signal_to_analyze)] = 0 8 Studies in Engineering and Exact Sciences, Curitiba, v.6, n.1, p.01-23, 2025 # ... (CWT call and scalogram plot) … 2.3 DYNAMIC SPECTRAL ENTROPY H(t) - SHANNON ENTROPY Shannon entropy can be used as a quantitative measure of multiscale order/disorder over time (Shannon, 1948; Cincotta & Simó, 1999; Nagaraj & Balasubramanian, 2017; Cincotta et al., 2021) i.e., how concentrated or dispersed the energy is between the "leaves" revealed by the scalogram. The information from the scalogram S(t,f) (CWT module) can be processed so that at each instant t, we can normalize it with the following probability distribution over the frequencies. 𝑃(𝑓∨𝑡)=𝑆(𝑡,𝑓)² ∑𝑆 𝑓′ (𝑡,𝑓′)² (3) where: S(t,f) is the value of the scalogram (CWT module), S(t,f)2 represents the power density at time t and frequency f, the denominator is the instantaneous total energy at time t. P(f∣t) is therefore a normalized probability distribution over frequencies for each time t. The frequency f is the fixed frequency for which we want to calculate P(f∣t) and f’ is a sum variable, running through all possible frequencies in the denominator, to ensure the normalization of the distribution. Dynamic Spectral Entropy can be calculated using the following equation for spectral entropy at time t: 𝐻(𝑡)=−∑𝑃 𝑓(𝑓∨𝑡)𝑙𝑜𝑔𝑃(𝑓∨𝑡) (4) This entropy H(t) quantifies the degree of spectral dispersion of energy at time t. Low values indicate coherent concentration, and high values indicate diffuse or chaotic distribution. It can classify the regions of order/chaos using thresholds to classify regimes: H(t) < 3: structural coherence and H(t) > 4: multiscale dispersion, (Cover, 1999 and Mackay, 2003). 9 Studies in Engineering and Exact Sciences, Curitiba, v.6, n.1, p.01-23, 2025 2.4 LYAPUNOV EXPONENTS OF M(θ) To complement the Shannon entropy analysis, we calculate the local Lyapunov exponent (Wolf et al., 1985; Brown et al., 1991; Parlitz, 2016; Caligiuri et al., 2021) which is based on measuring how much an eigenvalue of the matrix M(θ) moves away from itself between two close instants in time, reflecting an instantaneous rate of expansion or contraction of the system in a direction associated with that eigenvalue. Therefore, the local Lyapunov exponent would measure the rate of divergence or convergence of close trajectories in a dynamic system, but in a finite and specific time interval (a "sliding window"), instead of a global average over infinite time. Methodologically, it is calculated by evaluating the exponential growth rate of the distance between two infinitesimally close trajectories within a small-time window. As this window moves along the time series of the system, a different value is obtained for each point, revealing the local sensitivity of the system to small perturbations and identifying regions of transient instability (positive peaks) or temporary stability (negative valleys) that would be masked by a global average. Discrete computational expression is giving a time vector of eigenvalues λ₀, 𝜆₁, 𝜆₂, .....𝜆ₙ (one per time), We have: 𝜒𝑖=1 𝛥𝑡𝑙𝑜𝑔(|𝜆𝑖+1| |𝜆𝑖|) for i = 0, 1,..., n−1 The eigenvalues of M(θ(t)) reflect the natural modes of the vector dynamic geometry. The local Lyapunov exponent can allow us to obtain information about how much each mode grows or decays with time, if there is internal resonance (positive values, indicating growth), and if there is coherent stability (zero or negative values, indicating topological containment). An analysis of the nature of the bands, their composition in real and imaginary parts, band length, and their phase will also be carried out. 16 Studies in Engineering and Exact Sciences, Curitiba, v.6, n.1, p.01-23, 2025 in these characteristics we have the results obtained from the average (global) Lyapunov Exponents calculated by eigenvalue: λ0: -3.538731, λ1: -3.973244, λ2: - 2.753547, and λ3: -2.329518. Figure 5 shows the evolution of the local Lyapunov exponents for the four eigenvalues over time. Which presents an intense and synchronized oscillatory behavior: The four curves exhibit a periodic and highly synchronized oscillation pattern. This means that fluctuations in the sensitivity to the initial condition occur for all modes in a coordinated manner in time. It also presents positive peaks (interpreted as Local Instability). There are sharp positive peaks, reaching values that can be approximately 5 or more. These peaks indicate moments of local exponential divergence of the trajectories, meaning that at these specific points in time, the system becomes highly sensitive to small disturbances, exhibiting a "quasi-chaos" behavior or transient instability. Showing deep negative valleys (interpreted as local stability/dissipation), the curves also present deep negative valleys, falling to approximately of the order of more than -200. These valleys represent moments of local convergence, in which the system is less sensitive to disturbances and its trajectories tend to approach each other. This may indicate phases of strong dissipation or a return to a more stable state after a period of high energy/instability. The amplitude of the oscillations is significant, ranging from approximately -200 to +5. This large variation reiterates that the dynamics of the system are not uniformly stable or unstable but constantly transition between these regimes. Period and repetition, the oscillation pattern appears to be periodic and repetitive over time. Figure 5 shows that the Lyapunov exponent cyclically crosses both the real and imaginary parts of the eigenvalues, such dynamic behavior. It is an indication that they can lead to certain bifurcation regimes, such as fixed point (stationary), that occur when a real part of an eigenvalue (real or complex) crosses zero. If a real eigenvalue crosses zero, it can lead to bifurcations such as saddle-node (or fold), (Lu et al., 1995; Kuznetsov et al., 2004; Kadar et al., 2025) the equilibrium point appears/disappears. Transcritical bifurcation, the equilibrium points exchange stability, (Rasband, 2015 and Strogatz, 2024). At a pitchfork equilibrium point, one splits into three (or vice versa), (Kuznetsov et al., 2004; Lóczi, 2015; Bissell, 2022). However, if the pair of complex conjugate eigenvalues crosses the imaginary axis (i.e., their real parts cross zero), this leads to a Hopf bifurcation, (Hopf, 1942; Hassard et al., 1981; 17 Studies in Engineering and Exact Sciences, Curitiba, v.6, n.1, p.01-23, 2025 Marsden & Mccracken, 2012; Liang & Meng, 2023) . The Hopf bifurcation is an equilibrium point that loses (or gains) stability and can give rise to a type of periodic orbit (limit cycle). The imaginary part of the eigenvalues determines the frequency of this oscillation, and as the magnitude of the eigenvalue part is of the order of 1010, its frequency is very high, as an example, Theta=-6.28067, Eigenvalue=(0.00002 + 397.80946i), Type: Hopf bifurcation (complex conjugate pair crossing) Theta=6.28319, Eigenvalue=(-9999999997.94002 + 24492.93598i), Type: Hopf bifurcation (complex conjugate pair crossing). As the matrix M(θ) can generate 4 eigenvalues, and these can be complex and can cross the complex zero axis, one of the direct consequences of this 4×4 structure of the matrix M(θ) with singular angular dependence (via tan(θ), cot(θ)) and non-Hermitian coupling would be the emergence of eigenvalue bands that can collide or simultaneously cross the complex axis, resulting in multiple coupled bifurcations, such as two Hopf occurring together or in close sequence. Such a system could enter into double resonance, and this could manifest as compound oscillatory patterns (e.g., beats, modulations). It could give rise to quasi-periodic or chaotic behaviors in extended dynamical systems. As well as two coupled saddle-nodes colliding on the real axis or even forming more complex structures such as torus bifurcation if the pairs interact with different internal frequencies. This suggests that the parameter θ, which is modulating the matrix M(θ) and, consequently, the dynamics of the eigenvalues, is also varying periodically, which is consistent with the sinusoidal function used to generate θ(t). This figure 5 is a crucial visual representation of the resonance and bifurcation, the local Lyapunov peaks (divergence) are the direct signature of the intense resonance of the system and the bifurcations that occur dynamically. They show the moments when the internal geometry reconfigures itself, leading to high sensitivity. The coexistence of high peaks of magnitude of the eigenvalues (resonance) with these local Lyapunov peaks may reinforce this interpretation. Mean stability compared to local dynamics, figure 5 perfectly illustrates the concept that the mean dynamics can be globally stable (with global Lyapunov exponents close to zero or slightly negative), but with a highly complex local dynamics alternating between instability and stability (four-band). This dynamic behavior of the eigenvalues emerges from the vectorial internal dynamic geometry of complex four-band topology, affected by the imaginary number μ 18 Studies in Engineering and Exact Sciences, Curitiba, v.6, n.1, p.01-23, 2025 idempotence and it has an interpretation like as a canonical vector of the Alpha group geometry and also a global gauge field. In short, Figure 5 is clear evidence of the complexity and dynamic richness of the system described and revealing in the fluctuations and transitions that are fundamental to generate a field associated with the Alpha Group. 4 CONCLUSIONS The geometry associated with the Alpha Group describes and reveals the inner nature of geometric dynamics as a multi-scale web, in which resonances, phases, topology and curvature are deeply intertwined. These are not treated as external or independent attributes, but as emergent manifestations of a unified geometric inner structure: the Dynamic Internal Vector Geometry (DIVG). This dynamic geometry defines a complex, foliated topology, regulated by non-trivial symmetries, local phase planes, and conformal radial transformations. Across all scales, new oscillatory modes emerge, weaving the local and global, the discrete and continuous, into a coherent, rotational, and resonant framework. In this view, every topological aspect of space is a distinct expression of a single internal geometric vibration, distributed and synchronized across multiple layers of scale. A scalogram of the continuous wavelet transform (CWT) of the first eigenvalue of the matrix M(θ), evaluated at θ=π/2 rad, reveals this rich multiscale structure, confirming the presence of complex internal dynamics unfolding over a broad range of time scales. The spectral content demonstrates that each eigenvalue encodes self-organized internal modulations, which reflect the topological foliation intrinsic to the Alpha Group's vectorial geometry. Moreover, the spectral density across scales uncovers internal resonances potentially linked to nonlinear global symmetries induced by M(θ) at θ = n.π/2 rad. The CWT scalogram thus captures the dynamics of internal geometry, revealing a fundamental phenomenon at its multiscale: the coherent and massive conservation of energy. Unlike conventional systems, where energy conservation is limited by scale, this multiscale structure imposes conservation over vast orders of magnitude, reaching 1015 to 1020. This coherence is not merely quantitative; it is a structural stabilizer, ensuring the integrity of the internal dynamics, associated 19 Studies in Engineering and Exact Sciences, Curitiba, v.6, n.1, p.01-23, 2025 with the geometric nature of the Alpha group in 4 dimensions and the canonical vector μ is idempotent and acts in a projective invariant way. The matrix M(θ) acts as a geometric operator, modulating the evolution of energy and imposing its rigorous and synchronized distribution among the topological sheets. It imposes deep geometric and topological constraints necessary for maintaining system stability, even in the presence of bifurcations and local nonlinear dynamics driven by the rotational and spectral modulation of the internal geometry. This nonlinear regime manifests through sensitive dependence on initial conditions, abrupt fluctuations, and irregular spectral folding, all emergent from the evolving topological curvature. The presence of resonant couplings mediated by M(θ(t)) intensifies the structural sensitivity of the system, creating conditions conducive to the emergence of dynamic bifurcations in the internal sheets of the vector topology. This relationship between resonance and bifurcation defines critical points where the internal geometry is qualitatively reorganized. Crucially, θ(t) and H(t) operate in phase: variations in θ, associated with rotations and high-frequency modulations (with eigenvalue magnitudes around 1016), directly drive changes in spectral entropy. This interdependence translates the system’s internal complexity into a topological curvature, one that both emerges from and sculpts the internal vectorial geometry. It simultaneously defines a spatial information field embedded within the foliated topology. The internal vectorial dynamic geometry structured on a complex 4-band topology manifests a global coherence that sustains the stability of the system even in the face of local fluctuations. These fluctuations are revealed by the Lyapunov exponents, which, although predominantly negative, indicating containment and internal organization, also present positive peaks cyclically. The peaks observed in Figure 4 reflect a dynamic of internal resonance and local bifurcation, in which the bands exhibit a dynamic and interacting behavior in a nontrivial manner, redistributing the energy and modulating the phase structure of the system. The existence of cyclic resonance processes within a stable and coherent regime may reveal the ability of the 4-band dynamical system to oscillate between distinct topological modes without losing its global integrity and, in this context, highlight the critical role of the matrix M(θ) as a regulator of the topological 20 Studies in Engineering and Exact Sciences, Curitiba, v.6, n.1, p.01-23, 2025 dynamics and vector coherence of the bands. The matrix M(θ) defines a dynamic internal vector field structured in four components, which encodes the topological torsion and holonomy of the underlying geometry. This internal structure gives rise to massive coherent resonance, not through conventional external forces, but as an emergent phenomenon of pure topological organization. The dynamics governed by M(θ) operate within a vectorial framework that maintains coherence and energy conservation through nontrivial geometric constraints, enabling stable and self-organized patterns across multiple scales. Despite its local nonlinear behavior, the internal metric induced by the DIVG guarantees global stability. The coherence is not signaled by local smoothness, but by structural invariance across scales. In essence, the geometric framework M(θ) provides the core mechanism for this multiscale integrity. It is the fundamental pillar sustaining the emergence and persistence of stable, self-organized patterns, even under energy fluctuations and topological complexity. The key result of this research is that the interaction of two Hopfs bifurcations can show how oscillatory modes happen and interact, which can come together to create stable and dynamic fields. 21 Studies in Engineering and Exact Sciences, Curitiba, v.6, n.1, p.01-23, 2025 REFERENCE ADDISON, Paul S. 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