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Continuous Exceptional Manifolds in Purely Real-Parameter Passive Systems

Submitted:

23 July 2026

Posted:

24 July 2026

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Abstract
Research on exceptional points (EPs) as singularities, where eigenvalues and eigenvectors coalesce, has sparked a revolution in non-Hermitian physics [1–7], offering unprecedented sensitivity and wave manipulation. Previous studies have predominantly focused on isolated points in the complex plane [8–12], often relying on nonphysical complex parameters or active gain–loss modulation. However, such approaches introduce significant system complexity and hinder scalability, leaving the realization of continuous EP structures in purely passive, real-world systems an open challenge [13–18]. To address this challenge, this study reports the discovery of an EP surface within a purely passive, real-parameter, canonical two-degree-of-freedom (2DOF) damped system. A hidden physical landscape was unveiled, termed the L-surface, representing clusters of loci of innumerable EPs. Leveraging the L-surface derived from exact analytical solutions, this study identified and validated fundamental topological phenomena, including topological jump, imprint, and nucleation, all of which were previously obscured by numerical noise [19–23]. The comprehensive analysis and precise identification of this manifold required a physical parameter precision of 100 decimal places. This regime has been conventionally dismissed as mere numerical artifacts in standard 64-bit double-precision floating-point formats. The L-surface provides a robust foundation for next-generation ultrasensitive sensing and energy dissipation across frontiers, ranging from quantum computing [24–26] to advanced biosensing [27, 28], extending its transformative impact to the broader realms of electronics [29, 30] and optics [31–33].
Keywords: 
exceptional point
;  
critical damping
;  
analytical exact solution
;  
purely real parameter
;  
passive EP
;  
innumerable loci
;  
non-Hermitian physics
;  
isomorphism
Subject: 
Physical Sciences  -   Condensed Matter Physics

Main

Non-Hermitian physics has long been predominantly focused on the complex-parameter plane [34,35,36] primarily because existing frameworks face inherent challenges to identify the exact conditions for exceptional points (EPs) within a real-parameter space. Traditionally, EPs in passive mechanics have been dismissed as isolated, unstable singularities (needle-in-a-haystack points) that vanish under slight environmental noise or manufacturing tolerance [37,38], which is a direct consequence of underestimating the intricate linear coupling between oscillators [39]. This fundamental bottleneck and the consequent heavy reliance on active complex-parameter modulation [8,40] arise not from physical impossibility but from resolution and topological limitations in prior research [22,23], leaving the realization of robust, passive EP structures a persistent challenge [15,41].
This reliance is challenged by explicitly defining a linear coupling manifold, in which the stiffness, damping, and mass ratios are precisely intertwined. The regions of EPs are rigorously analyzed through the fundamental relationship between the characteristic equation of a two-degree-of-freedom (2DOF) damped system and the coupling dependencies [42]. Therefore, the linear characteristics of the system are governed by the equation D X = 0 , where D represents the system characteristic matrix, and the amplitude vector X is defined as X = X 1 , X 2 T , where X 1 and X 2 denote the physical amplitudes of the primary and secondary systems, respectively. The characteristic matrix D is explicitly expressed as follows:
D = λ 2 + 2 ζ 1 + β ω n 1 λ + 1 + α ω n 1 2 − ( β ω n 1 λ + α ω n 1 2 ) − β ω n 1 λ + α ω n 1 2 μ 0 λ 2 + β ω n 1 λ + α ω n 1 2
In Eq. (1), λ is eigenvalue, and ζ 1 and ω n 1 are the damping ratio and natural frequency of the primary system, respectively. The linear coupling manifold α ( μ 0 , ζ 1 ) and β μ 0 , ζ 1 can be defined as a function of the constant mass ratio ( μ 0 = m 2 / m 1 ) and damping ratio ζ 1 of the primary system. By leveraging the structural isomorphism between the complex Hamiltonian H 0 and mechanical characteristic matrix D , a profound physical similitude is highlighted that transcends disciplinary boundaries. While H 0 typically encapsulates non-Hermitian dynamics through complex gain–loss modulation [43], matrix D achieves the same physical essence by intertwining the real-parameter manifolds of mass, stiffness, and damping. The inherent scalability of the proposed manifold extends far beyond conventional vibration analysis, offering a universal framework for any system governed by physical similitude, ranging from fundamental regimes, such as quantum mechanics and condensed matter physics, to advanced frontiers, such as semiconductor technology, aerospace engineering, and bioengineering.
Traditional studies of EPs in optical and quantum systems are often hindered by the inherent physical interdependence between the real and imaginary components of complex parameters, which complicates precise individual tuning. In contrast, the mechanical system in this study utilizes mass, stiffness, and damping as physically decoupled independent parameters [44]. Crucially, although the system relies on specific physical coupling to manifest EP characteristics, the variables governing this interaction can be tuned independently without unintended interference. This independent tunability resolves the design uncertainties prevalent in complex-parameter systems and provides a robust engineering framework for precisely projecting the parametric correlations of EP formation onto tangible physical architectures [45].
The L-surface is defined as a manifold where EPs exist infinitely and is identified by utilizing real parameters to map EP regions that were previously elusive under complex-parameter frameworks. This conceptualization represents a paradigm shift, extending the manifestation of exceptional phenomena from isolated singular points to a continuous and expansive design domain. Consequently, a high degree of engineering design freedom is secured, enabling the free selection and implementation of the desired EP states through the direct modulation of physical entities, namely, mass, stiffness, and damping, without the need for complex active control. To elucidate the geometric origin of this freedom, the interaction between the primary and secondary systems is established using a linear coupling manifold. This manifold bridges the physical domain to the non-Hermitian state space, as analytically defined by the constituent parameters of the damped 2DOF model in Figure 1a as follows:
α μ 0 , ζ 1 = μ 0 1 + μ 0 2 1 ± ζ 1 μ 0 1 + μ 0 − ζ 1 2 2 β μ 0 , ζ 1 = 2 μ 0 1 + μ 0 2 ζ 1 1 − μ 0 ∓ 1 + μ 0 − 2 ζ 1 2 μ 0 1 + μ 0 − ζ 1 2
The linear coupling manifolds, α and β , defined in Eq. (2), encapsulate the physical interaction within the 2DOF model shown in Figure 1a and constitute the mathematical foundation of the L-surface, as shown in Figure 1b. By defining the properties of the secondary system relative to the primary system and maintaining the constant mass ratio ( μ 0 ), the real-parameter space is effectively projected into two dimensions (2D). This strategic reduction allows the non-Hermitian dynamics of the system to be governed primarily by the stiffness ( k 2 ) and damping ( c 2 ) of the secondary system, thereby providing a robust analytical framework for precise system optimization. Consequently, the linear coupling manifold allows the parameters of the secondary system to be explicitly defined as k 2 = α k 1 and c 2 = β m 1 k 1 . In addition, the parameter ζ 2 μ 0 , ζ 1 = β / 2 α μ 0 can be explicitly defined in terms of the mass ratio and the linear coupling manifolds. In the limit where μ 0 → 0 , the relationship converges to ζ 1 = ζ 2 , whereas for μ 0 → ∞ , the value of ζ 2 remains bounded within the interval − 1 < ζ 2 < 1 .
In the ζ 1 - ζ 2 plane of Figure 1b, the topological phase-transition boundaries represent the upper and lower limits of the L-surface, defining the regions of complex degenerate roots where both eigenvalues and eigenvectors coalesce. This phenomenon constitutes the definitive condition for EPs. From Eq. (1), a quartic polynomial for the dimensionless roots ( λ ¯ = λ / ω n 1 ) can be defined as follows:
λ ¯ 4 + a 1 λ ¯ 3 + a 2 λ ¯ 2 + a 3 λ ¯ + a 4 = 0
In this Eq. (3), the coefficients a 1 , a 2 , a 3 , and a 4 represent 1 + μ 0 β + 2 μ 0 ζ 1 / μ 0 , 1 + μ 0 α + μ 0 + 2 β ζ 1 / μ 0 , β + 2 α ζ 1 / μ 0 , and α / μ 0 , respectively. The L-surface shown in Figure 1b constitutes the topological manifold of exceptional point clusters. This surface hosts a quasi-continuous density of EPs, with its structural influence extending asymptotically toward ± ∞ along the ζ 1 -axis. μ 0 is defined in terms of ζ 2 and substituted it into Eq. (3) to eliminate μ 0 . From the resultant method, followed by sequentially eliminating linear coupling manifolds from the characteristic equation, the characteristic equations that dictate the upper and lower topological phase-transition boundaries ( Γ I ) in Figure 1b are derived as follows:
2 ζ 2 2 − 1 2 ζ 1 2 − 2 4 ζ 2 3 − 3 ζ 2 ζ 1 + 5 ζ 2 2 − 4 = 0
This analytical approach identifies a unique geometric architecture underlying the observed dynamics. Specifically, at ζ 1 = 0, the parameter ζ 2 takes a value of ± 4 / 5 , and when ζ 2 = ± 1, ζ 1 becomes ζ 1 = ± 1. Furthermore, when ζ 2 = 0, ζ 1 becomes ζ 1 = ± 2 . The locus where ζ 1 = ζ 2 is represented by a dotted line ( Γ I I I ), which serves as the axis of antisymmetry for the values along the ζ 1 -axis. Within this region of the L-surface, complex and purely imaginary degenerate roots coexist. To identify purely imaginary degenerate roots in Eq. (3), the depressed quartic equation λ ¯ 0 4 + p 1 λ ¯ 0 2 + p 2 λ ¯ 0 + p 3 = 0 is considered by imposing the condition that the cubic coefficient in Eq. (3) vanishes. Under this framework, purely imaginary degenerate roots are rigorously characterized by the constraints p 1 2 − 4 p 3 = 0, p 2 = 0, and a 1 = 0. These continuous points, which represent purely imaginary degenerate roots, are explicitly denoted as Γ I I in Figure 1b. The trajectory that determines the nature of the complex degenerate roots is defined by the following equation:
ζ 1 2 1 − 2 ζ 2 2 2 − ζ 2 2 = 0
The procedure for defining this equation is consistent with the steps taken for Eq. (4); however, it is derived directly without the need for a resultant method. Among the four roots of the quartic equation, the specific solutions satisfying the conditions ζ 1 2 / ζ 2 2 > 1 and ζ 1 ζ 2 < 0 constitute the curve ζ 2 = 1 − 1 + 8 ζ 1 2 / 4 ζ 1 in Figure 1b. This analytical correspondence further validates the structural integrity of the L-surface in the parameter space. The curve represents the locus of purely imaginary degenerate roots. Crucially, this curve serves as a topological boundary: the region above the curve corresponds to complex degenerate roots with positive real parts, whereas the region below signifies those with negative real parts. Furthermore, in Eqs. (4) and (5), the parameter ζ 1 asymptotically approaches the constant value of ζ 2 = ± 2 / 2 as it tends toward ∞ . This asymptotic convergence suggests that the stability boundaries of the 2DOF damped system remain bounded within a predictable regime, even under extreme parameter variations.
The L-surface represents a paradigm shift in non-Hermitian engineering by transforming isolated unstable EPs into a continuous and deterministic manifold within a real-parameter space (Figure 1b) [46]. The antisymmetry axis defined by the ζ 1 = ζ 2 condition serves as a topological hallmark that ensures the structural stability and multiple invariances of the EP clusters (Figure 2). In addition, the locus of purely imaginary degenerate roots effectively linearizes the complex interactions of mass, stiffness, and damping, providing an explicit boundary that distinguishes regions of extreme signal amplification (structural gain) and perfect energy dissipation (structural loss) without the need for active complex-variable modulation (Figure 1).
Intriguingly, any attempt at parametric control along the EP manifold fails to preserve the specific output characteristics of the system because each EP remains an island. Despite being geometrically aligned on the L-surface, these infinite singularities possess zero correlation, representing a paradox in which global geometric order coexists with local functional independence [47]. Nevertheless, by strategically modulating the system stiffness, it is feasible to maintain, guide, and exploit the specific output characteristics of each individual EP.
For the topological jump and imprint (Figure 3), the frequency of topological jumps varies depending on changes in real parameters, such as stiffness, damping, and mass ratios (Figure 4). Furthermore, owing to topological imprint, the presence of EPs within the L-surface induces significantly larger signal amplification compared with general systems, even when the real parameters and roots are truncated to their integer parts. Notably, the capability of fixing four key parameters of the primary and secondary systems as integers, leaving only two as tunable variables, significantly enhances the operational control efficiency and design simplicity [48].
The topological nucleation process observed in this study exhibits contrasting dynamic characteristics depending on the mass ratio (Figure 5) [39]. Crucially, the evolutionary pathways of this nucleation vary distinctly with changes in both the stiffness and damping ratios (Figure 6). The emergence of these nuclei is inherently scale-dependent, and their true nature often remains obscured in macroscopic regimes where the mass ratio is large. For example, at a unit mass ratio, the overwhelming inertia of the system prevents the subtle topological phase-transition boundaries of the EP clusters from manifesting, leading to a perceived stochasticity that veils the underlying orderly patterns.
Finally, the locus of EPs generated by a constant μ 0 within the L-surface manifold coincides with the critical damping locus on the hyperplane where infinite critical damping points cluster (LCC-surface), ultimately merging into a single unified locus (Figure 7a). In a purely real-parameter 2DOF damped system, because the complex degenerate roots of the EPs are characterized by real parts corresponding to either pure dissipation or effective gain, their locus remains independent of the critical damping locus on the plane defined by pairs of negative and positive real double roots (Figure 7b). Surprisingly, these complex non-Hermitian boundary lines do not merely form arbitrary trajectories; instead, they converge into a mathematically rigorous single ellipse overlapping the critical damping limit, revealing a topologically protected geometric order.
In conclusion, the proposed L-surface realizes a self-balancing architecture that inherently satisfies the mathematical conditions for EPs through its passive structure, thereby maintaining robust stability without active external intervention. In this framework, the integration of active control is not a prerequisite for system survival or stability; rather, it functions as a strategic booster designed to exponentially amplify the performance over established passive robustness. This paradigm shift underscores the harmonious coexistence of topological resilience, design simplicity, and extreme scalability, paving the way for a new era of high-precision sensors and energy-harvesting systems in non-Hermitian engineering.

Methods

To capture the topological characteristics of EPs within the L-surface, a variable-precision arithmetic (VPA) framework was implemented using MATLAB R2021b software, securing an ultrahigh numerical precision of over 256 significant figures. This framework enabled a resolution in which the difference between the imaginary parts of the two roots vanished to over 100 decimal places, a prerequisite for verifying perfect degeneracy, which, in turn, allowed for the rigorous analysis of real parameters with over 100 decimal digits. Furthermore, such a high-precision approach is essential for overcoming extreme sensitivity at complex degenerate root boundaries and preventing infinitesimal errors from amplifying into significant numerical artifacts. Consequently, it was confirmed that the exponential amplification remained linear from a macroscopic perspective on a logarithmic scale across a vast numerical range. In addition, the sorting order of the roots was determined based on their underlying physical characteristics. The complex degenerate roots (loci of EPs) were arranged in ascending order of absolute values. For absolute reproducibility, two sets of purely real parameters yielding exact EP coalescence on the L-surface are provided in the Supplementary Information.

Author Contributions

Jung Woo Lee conceived the study and conceptualized the L-surface framework. Jung Woo Lee and Jin Kim jointly performed formal theoretical analyses to identify and characterize the properties of EPs on the L-surface. Jin Kim conducted the systematic investigation and performed the high-precision numerical computations. Jung Woo Lee wrote the manuscript and supervised the project. All authors discussed the results and commented on the final version of the manuscript.

Conflicts of Interest

Jung Woo Lee and Jin Kim are the inventors of the patent application filed by Kyonggi University related to this work (Korean Patent Application No. 10-2026-0124097).

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Figure 1. L-surface: Unveiling the hidden global topology of exceptional loci clusters within the 2D parameter plane. a. Schematic of the 2DOF mechanical system in which k 2 and c 2 of the secondary system are substituted by m 1 , k 1 , and c 1 of the primary system using linear coupled manifolds α and β . b. The L-surface is presented as a continuous EP manifold, a clustering site where isolated singularities are mapped between topological boundaries within the real-parameter space of ζ 1 and ζ 2 . The curves labeled as Γ I represent the topological phase transition boundaries and the single curve denoted as Γ I I represents the locus of purely imaginary degenerate roots, marking the specific parametric limit where ζ 2 = ± 2 / 2 at the limit of ζ 1 → ∓ ∞ . The dotted line assigned as Γ I I I represents the specific condition of ζ 1 = ζ 2 , serving as the axis of antisymmetry for the EP manifold. Furthermore, the upper portion of Γ I I represents the purely negative real part of the complex degenerate roots (structural loss), whereas the lower portion corresponds to the purely positive real part (structural gain).
Figure 1. L-surface: Unveiling the hidden global topology of exceptional loci clusters within the 2D parameter plane. a. Schematic of the 2DOF mechanical system in which k 2 and c 2 of the secondary system are substituted by m 1 , k 1 , and c 1 of the primary system using linear coupled manifolds α and β . b. The L-surface is presented as a continuous EP manifold, a clustering site where isolated singularities are mapped between topological boundaries within the real-parameter space of ζ 1 and ζ 2 . The curves labeled as Γ I represent the topological phase transition boundaries and the single curve denoted as Γ I I represents the locus of purely imaginary degenerate roots, marking the specific parametric limit where ζ 2 = ± 2 / 2 at the limit of ζ 1 → ∓ ∞ . The dotted line assigned as Γ I I I represents the specific condition of ζ 1 = ζ 2 , serving as the axis of antisymmetry for the EP manifold. Furthermore, the upper portion of Γ I I represents the purely negative real part of the complex degenerate roots (structural loss), whereas the lower portion corresponds to the purely positive real part (structural gain).
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Figure 2. Symmetric pair of EPs. a. To demonstrate that the locus of ζ 1 = ζ 2 acts as the axis of antisymmetry for the EPs, the results represented by star and circle marks were comparatively analyzed. b. The amplitudes on a logarithmic scale with a damping ratio of ζ 1 = ± 0.5 at t =0. c, d. Time-domain signals at the EPs for ζ 1 = ± 0.5 , with an initial displacement x 0 applied to m 1 . These were obtained under conditions where the decimal places of both the real parameters and roots are fully truncated. A comparison between panel c and the magnification in panel d reveals that the amplification ratios of both results are in exact agreement at t =0. The observation that the difference between the two points decreases in one case and is exponentially amplified in the other signifies a symmetric correspondence between the dissipating (structural loss) and growing (structural gain) states of the system. Moreover, maintaining the unique features of countless EPs along the two trajectories through stiffness tuning confirms that the system possesses multi-invariance.
Figure 2. Symmetric pair of EPs. a. To demonstrate that the locus of ζ 1 = ζ 2 acts as the axis of antisymmetry for the EPs, the results represented by star and circle marks were comparatively analyzed. b. The amplitudes on a logarithmic scale with a damping ratio of ζ 1 = ± 0.5 at t =0. c, d. Time-domain signals at the EPs for ζ 1 = ± 0.5 , with an initial displacement x 0 applied to m 1 . These were obtained under conditions where the decimal places of both the real parameters and roots are fully truncated. A comparison between panel c and the magnification in panel d reveals that the amplification ratios of both results are in exact agreement at t =0. The observation that the difference between the two points decreases in one case and is exponentially amplified in the other signifies a symmetric correspondence between the dissipating (structural loss) and growing (structural gain) states of the system. Moreover, maintaining the unique features of countless EPs along the two trajectories through stiffness tuning confirms that the system possesses multi-invariance.
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Figure 3. Topological jump and imprint. a. As parameters such as stiffness and damping are finely tuned, the normalized amplitudes increase linearly on a logarithmic scale. The system exhibits an anomalous topological spike at a specific decimal precision, where the state jumps between two branches and instantly returns, thereby demonstrating the nature of the topological jump. Notably, the frequency of these topological jumps exhibits a strong dependence on the magnitude of stiffness. b. Whether obtained by increasing real parameters at a fixed 100-decimal-place root precision with zero imaginary difference ( Γ I ) or by simultaneously increasing both parameters and roots from the integer level ( Γ I I ), the amplification curves smoothly converge into a single locus except for the first few decimal places. This convergence provides decisive evidence of a topological imprint.
Figure 3. Topological jump and imprint. a. As parameters such as stiffness and damping are finely tuned, the normalized amplitudes increase linearly on a logarithmic scale. The system exhibits an anomalous topological spike at a specific decimal precision, where the state jumps between two branches and instantly returns, thereby demonstrating the nature of the topological jump. Notably, the frequency of these topological jumps exhibits a strong dependence on the magnitude of stiffness. b. Whether obtained by increasing real parameters at a fixed 100-decimal-place root precision with zero imaginary difference ( Γ I ) or by simultaneously increasing both parameters and roots from the integer level ( Γ I I ), the amplification curves smoothly converge into a single locus except for the first few decimal places. This convergence provides decisive evidence of a topological imprint.
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Figure 4. To illustrate the effect of stiffness on the occurrence frequency of topological jumps, the results for k 1 = 9 N/m and k 1 = 400 N/m (with m 1 = 1 kg) are shown in panels a and b, respectively. In addition, the frequency of these topological jumps exhibits pronounced sensitivity to variations in the mass and damping ratios.
Figure 4. To illustrate the effect of stiffness on the occurrence frequency of topological jumps, the results for k 1 = 9 N/m and k 1 = 400 N/m (with m 1 = 1 kg) are shown in panels a and b, respectively. In addition, the frequency of these topological jumps exhibits pronounced sensitivity to variations in the mass and damping ratios.
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Figure 5. Topological nucleation at odd and even mass ratios. Maximum amplitudes of x i j at t =0 as a function of decimal precision of the real parameters. All results are presented on a logarithmic scale with ζ 1 = 0.5 , k 1 = 90000   N / m , and m 1 = 1   k g . To verify the consistency of the topological nucleus formation, infinitesimal values, extending from 10 − 9 down to 10 − 36 , were employed through an asymptotic approach. The objective of utilizing such extreme scales is to rigorously demonstrate the topological robustness of the system. a. Results for odd mass ratios ( 10 − 9 , 10 − 23 , and 10 − 35 ) were shifted and aligned with the 10 − 35 data to illustrate scale invariance. b. For even mass ratios ( 10 − 10 , 10 − 24 , and 10 − 36 ), the data were translated to coincide with the 10 − 36 baseline, thereby demonstrating a consistent scale-invariant behavior. The fact that the amplitude profiles, from which the topological nucleus originates, coincide with extraordinary precision at the initial decimal significant figures of panels a and b, proves that they remain invariant across an immense range of scales. While the profiles exhibit variations depending on other parameters, they consistently align according to the mass ratio, confirming that the scale invariance is uniquely governed by the parity of the mass ratio.
Figure 5. Topological nucleation at odd and even mass ratios. Maximum amplitudes of x i j at t =0 as a function of decimal precision of the real parameters. All results are presented on a logarithmic scale with ζ 1 = 0.5 , k 1 = 90000   N / m , and m 1 = 1   k g . To verify the consistency of the topological nucleus formation, infinitesimal values, extending from 10 − 9 down to 10 − 36 , were employed through an asymptotic approach. The objective of utilizing such extreme scales is to rigorously demonstrate the topological robustness of the system. a. Results for odd mass ratios ( 10 − 9 , 10 − 23 , and 10 − 35 ) were shifted and aligned with the 10 − 35 data to illustrate scale invariance. b. For even mass ratios ( 10 − 10 , 10 − 24 , and 10 − 36 ), the data were translated to coincide with the 10 − 36 baseline, thereby demonstrating a consistent scale-invariant behavior. The fact that the amplitude profiles, from which the topological nucleus originates, coincide with extraordinary precision at the initial decimal significant figures of panels a and b, proves that they remain invariant across an immense range of scales. While the profiles exhibit variations depending on other parameters, they consistently align according to the mass ratio, confirming that the scale invariance is uniquely governed by the parity of the mass ratio.
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Figure 6. The process of topological nucleation varies depending on changes in the real parameters. To obtain the data for panels a and b, the damping ratio of the primary system was adjusted to ζ 1 = 0.2 , while all other input parameters remained identical to those used for Figure 5. Although the nucleation processes for both the odd and even mass ratios remained consistent, they exhibited a distinct departure from the original nucleation pathways observed in Figure 5.
Figure 6. The process of topological nucleation varies depending on changes in the real parameters. To obtain the data for panels a and b, the damping ratio of the primary system was adjusted to ζ 1 = 0.2 , while all other input parameters remained identical to those used for Figure 5. Although the nucleation processes for both the odd and even mass ratios remained consistent, they exhibited a distinct departure from the original nucleation pathways observed in Figure 5.
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Figure 7. LCC-surface: Hyperplane in a critically damped state. a. Under the condition of critical damping, characterized by four roots comprising two negative real double roots or two positive real double roots, the locus of the EPs for a constant mass ratio aligns perfectly with the locus on the LCC-surface. The locus between the two points exhibits EP characteristics, whereas the locus beyond these points exhibits critical damping characteristics. The upper two surfaces ( R I ) represent the negative real parts, whereas the lower two surfaces ( R I I ) represent the positive real parts. Alignment of EP-based antisymmetric loci onto a single ellipse demonstrates that these loci are topologically protected against continuous parameter variations, thereby validating the topological invariance of the system. b. This panel represents the LCC-surface, characterized by pairs of negative and positive real double roots. As the complex degenerate roots of the EP invariably possess either purely positive or negative real parts for purely real parameters (Figure 1), the locus of the critical damping points remains decoupled from the locus of the EPs on this plane ( R I I I ).
Figure 7. LCC-surface: Hyperplane in a critically damped state. a. Under the condition of critical damping, characterized by four roots comprising two negative real double roots or two positive real double roots, the locus of the EPs for a constant mass ratio aligns perfectly with the locus on the LCC-surface. The locus between the two points exhibits EP characteristics, whereas the locus beyond these points exhibits critical damping characteristics. The upper two surfaces ( R I ) represent the negative real parts, whereas the lower two surfaces ( R I I ) represent the positive real parts. Alignment of EP-based antisymmetric loci onto a single ellipse demonstrates that these loci are topologically protected against continuous parameter variations, thereby validating the topological invariance of the system. b. This panel represents the LCC-surface, characterized by pairs of negative and positive real double roots. As the complex degenerate roots of the EP invariably possess either purely positive or negative real parts for purely real parameters (Figure 1), the locus of the critical damping points remains decoupled from the locus of the EPs on this plane ( R I I I ).
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