Dynamical Analysis of a Quadratic Megastable Chaotic Oscillator and Its Application in Biometric Fingerprint Image Encryption

Tis investigation centers on megastable systems, distinguished by their countable infnite attractors, with a particular emphasis on the Quadratic Megastable Oscillator (QMO). Unlike traditional megastable oscillators reliant on external excitation, our proposed QMO operates autonomously, contributing to its distinctiveness. Trough a comprehensive exploration of the QMO, we elucidate various dynamical behaviors, enriching the understanding of its complex system dynamics. In contrast to conventional megastable oscillators, the QMO yields nested types of multiple attractors for diverse initial conditions, elegantly depicted in phase portraits. To gauge the sustainability of chaotic oscillation, we employ infuential parameter bifurcation plots, providing a nuanced insight into the system’s dynamical evolution. Te complexity of the proposed system is further underscored by its intricate basins of attraction, accommodating an infnite number of coexisting attractors. Exploring trajectory dynamics, we observe that certain initial conditions lead trajectories to distant destinations, evading the infuence of local attractors. Tis behavior underscores the uniqueness of the QMO and highlights its potential applications in scenarios requiring nonlocalized attractor behaviors. Taking a practical turn, the QMO is applied to biometric fngerprint image encryption, demonstrating its efcacy in real-world applications. Rigorous statistical analyses and vulnerability assessments confrm the success of the QMO in providing secure encryption within chaotic system-based frameworks. Tis research contributes not only to the theoretical understanding of megastable systems but also establishes the QMO as a valuable tool in encryption applications, emphasizing its robustness and versatility in complex dynamical scenarios.


Introduction
Multistability is a very important phenomenon in dynamical systems [1,2].In addition to the standard sensitive dependence on initial conditions that distinguish a chaotic system and prevent long-term prediction, multistable systems have an attractive state that depends on the original conditions.Multistability is not practical when designing commercial products where it is necessary to maintain the desired state in a noisy environment.Furthermore, multistability can be used with appropriate control strategies to switch between coexisting states without compromising system performance [2].In this regard, many researchers have investigated the property multistability in many literature.New dynamical systems were proposed recently with an infnite number of coexisting attractors, and these are known as megastable systems [3][4][5].Multistability can be identifed in two forms named extreme multistability and megastability, which deals with infnite number of oscillators.Applications that demand more complex systems such as cryptography and secure communications, chaotic systems with special properties are more suitable.Te countable number of infnite attractors which can generate for diferent initial conditions supports well for infnite number of key generations.Many works of literature recommended megastable oscillators for designing encryption algorithms, but to the best of our knowledge, most of them have an external excitation [6,7].It motivates us towards developing a megastable oscillator without external excitation.In order to investigate the dynamics of the proposed system, we adopted stability analysis [8] and bifurcation plots.Te signifcance of the initial condition and its efect on the dynamics can be revealed using a basin of attractors.In this work, we used a fully automated method for identifying attractor basins without approximations of dynamics [9,10].
In recent years, the use of the Internet in daily and business life, the use of social media, and in addition to this, smart production systems, smart home systems, and Internet of things applications that came with the industry 4.0 revolutions have increased considerably.In all of these uses, there is a continuous fow of data [11,12].Te advancements in information and communication technology have not only transformed various aspects of our daily lives but have also signifcantly broadened the scope of telemedicine.Tis evolution encompasses the delivery of health services, including remote diagnosis, treatment, and even surgical operations conducted from a distance [13].In addition, physiological data such as fngerprint image, palm, eye iris and retina, face, hand geometry, fnger geometry, vein image, and behavioral biometric data such as voice, handwriting, and walking are widely used today to safely recognize user identity [13][14][15].Also, biometric identifcation has become more popular than traditional identifcation techniques, especially in identity cards, passwords, and personal identifcation numbers (PIN) applications.
Having an infnite number of attractors in a chaotic system can potentially provide some advantages for key generation applications.However, it is important to note that the actual usefulness of these advantages will depend on the specifc requirements and limitations of the application.Here, we highlight a few potential advantages: (i) Increased security: With a larger number of attractors, it can be more difcult for an attacker to determine which attractor corresponds to the correct key.Tis can increase the security of the system by making it more difcult to guess or brute-force the correct key.(ii) Increased fexibility: Having more attractors to choose from can provide more fexibility in selecting the initial conditions for key generation.Tis can allow for a wider range of potential keys and make it easier to generate keys that meet specifc requirements (e.g., length, complexity, etc.).
(iii) Robustness: In some cases, having multiple attractors can provide a certain level of robustness against perturbations or noise in the system.Even if the initial conditions are slightly perturbed, the system may still converge to a valid attractor and generate a valid key.
With the increase in digital data in these mentioned areas, illegal attacks during the transmission of these data have increased [16,17].As a result, researchers have worked on encryption methods to increase data security.Traditional encryption methods such as SHA-1, MD5, AES, IDEA, RSA, and DES are not ideal for encrypting biometric images due to large data capacity, computationally intensive, high time consumption, and high correlation between pixels.Instead, chaotic system-based encryption schemes have become more popular due to the good cryptological qualities of chaotic systems, such as their sensitivity to system parameters and initial conditions, nonperiodicity, mixing, and topological transitivity [16][17][18][19].For this purpose, in the literature, there are various encryption studies based on chaotic systems [12,14,20,21], hyperchaotic systems [22][23][24][25], and chaotic maps [26][27][28].Due to the abovementioned advantages, chaotic systems have been highly preferred and studied in the encryption of physiological biometric images [14-17, 19, 24, 29, 30].Terefore, biometric fngerprint image encryption is designed using the proposed Quadratic Megastable Oscillator (QMO) in this study.
Te impetus behind this research stems from the pressing need for advanced encryption methodologies in safeguarding sensitive information.In an era characterized by the ubiquity of digital data and evolving cyber threats, there is an increasing demand for encryption systems that not only ensure confdentiality but also exhibit resilience against sophisticated attacks.Te motivation to explore the Quadratic Megastable Oscillator (QMO) for biometric fngerprint image encryption arises from the unique dynamical properties it exhibits.Leveraging these properties, we aim to develop an encryption system that not only meets stringent security requirements but also demonstrates versatility and robustness in the face of diverse cryptographic challenges.
Tis research makes signifcant contributions to both the theoretical understanding of three-dimensional autonomous quadratic megastable oscillators and their practical application in secure communication systems.Te comprehensive analysis of attractors, phase portraits, parameter bifurcation plots, and the sustainability of chaotic oscillation provides a nuanced insight into the complex dynamics of the QMO.Specifcally, the exploration of attractors and basin of attraction concepts contributes to a deeper understanding of the system's behavior.Te extension of the QMO to biometric fngerprint image encryption represents a novel application, demonstrating its efcacy in real-world scenarios.Rigorous statistical analyses, including NIST 800-22 tests, histogram analysis, correlation analysis, entropy analysis, and key sensitivity analysis, collectively establish the robustness and security features of the proposed encryption scheme.Tis research not only advances the feld of 2 Complexity dynamical systems but also provides a practical solution to the pressing need for secure encryption methodologies in the realm of biometric data protection.Te paper is organized as follows: In Section 1, the introduction of QMO and its application are detailed.In Section 2, a novel 3D Quadratic Megastable Oscillator (QMO) is presented and its dynamics are investigated with the help of stability analysis, orbit diagram, Lyapunov spectrum, and the basins of attraction.A biometric fngerprint image encryption of cryptographic application is shown in Section 5. Finally, the highlights of the study and conclusion are presented.

Quadratic Megastable Oscillator (QMO)
In his book "Elegant Chaos" [31], Sprott has proposed several classes and types of chaotic systems.One such classifcation is a quadratic oscillator defned generally as where the function f( _ x, x) should contain at least one sinusoidal term or cubic nonlinear term.But such systems are normally forced to exhibit chaos.
Te mathematical model of QMO is ( We considered the parameter values a � 0.1, b � 0.4 and performed the numerical simulation for x � −20 to x � +20 all other states are kept as y � z � 0, the 2D phase portrait is presented in Figure 1.We could clearly observe the growing kind of attractors.We used MATLAB ODE solver with a runtime of 500.
In order to fnd the equilibrium points, the state space equations are equated to zero.
(or) π + 2nπ This can also be written as nπ. ( Te equilibrium points of QMO system can be written as (0, 0, nπ), where n is an arbitrary integer.

Stability Analysis of QMO System
Te stability analysis [8] can be found using the Jacobian matrix and eigenvalues as given in equations ( 4) and (6).
Te Jacobian matrix of the QMO system at equilibrium drives to infnity because of the term dirac(0).Hence, we replaced the term sign(x) as tanh(Ax) and for A � 250 it have been shown that the hyperbolic tan function matches the signum function.Te modifed Jacobian matrix is Te characteristic polynomial can be derived as where the eigenvalues for diferent cases is presented in Table 1.
We plotted the real parts of the eigenvalues (λ 1 , λ 2 & λ 3 ) for n � 1 to 10 in Figure 2.For even values of n produces one negative and two complex conjugate eigenvalues which show the equilibrium point should be an attracting spiral saddle.For odd values of n produce one positive and two complex conjugate (with negative real part) eigenvalue, hence the stability can be classifed as an extruding spiral saddle.

Dynamical Behavior of QMO for Parameter Variation
A bifurcation plot is a graphical representation that showcases the behavior of a dynamic system as a key parameter is varied.It provides valuable insights into the system's transitions, stability, and the emergence of complex behaviors such as chaos.In the case of chaotic systems, bifurcation plots are particularly useful as they unveil the intricate relationship between system parameters and the resulting dynamics.A bifurcation plot typically consists of a parameter axis and a state variable axis.As the parameter value is gradually changed, the corresponding values of the state variable are plotted on the graph.Te resulting plot reveals the system's behavior as the parameter sweeps through diferent ranges, indicating points of stability, periodicity, or chaotic behavior.Bifurcation plots often exhibit distinctive patterns such as period-doubling, intermittency, or strange attractors, which provide crucial insights into the system's behavior.
Te Lyapunov exponent is a measure used to quantify the sensitivity to initial conditions in chaotic systems.It characterizes the exponential rate of divergence or convergence of nearby trajectories in phase space.A positive Lyapunov exponent indicates sensitive dependence on initial conditions and chaotic behavior, while a negative exponent suggests convergence towards stability or periodicity.Te calculation of Lyapunov exponents involves analyzing the linearized dynamics of a system and determining the exponential growth or decay of perturbations.A positive Lyapunov exponent indicates chaotic behavior, while a negative exponent implies stability.Moreover, the magnitude of the Lyapunov exponent provides information about the system's sensitivity to initial conditions.A higher magnitude indicates stronger sensitivity and a more chaotic system.
Te study of parameter variation in megastable chaotic systems holds signifcant importance in understanding their behavior and dynamics.Megastable chaotic systems are characterized by exhibiting both stability and chaotic behavior under diferent parameter values.Exploring the efect of parameter variation helps elucidate the range of behaviors these systems can exhibit and provides insights into their underlying mechanisms.
By systematically varying the system's parameters and observing the resulting dynamics, researchers can identify critical points at which the system undergoes bifurcations and transitions between stability and chaos.Bifurcation plots ofer a visual representation of these transitions, highlighting the parameter values at which the system enters chaotic regimes or returns to stability.In addition, studying the Lyapunov exponents of a megastable chaotic system under diferent parameter values helps quantify its sensitivity to initial conditions.Te Lyapunov exponents provide a measure of the system's predictability and ofer insights into the overall stability or chaotic nature of system.A comprehensive analysis of Lyapunov exponents for various parameter values allows researchers to identify regions of parameter space where the system exhibits stable behavior or transitions into chaotic regimes.
Te infuence of parameter variation is a crucial factor in understanding the dynamics of chaotic systems.In Section 3 of the study, it becomes evident that the dynamics of the QMO system are signifcantly afected by parameter a changes.To gain insights into the system's behavior with varying parameters, we explored a specifc range of parameters.We focused on plotting the local maxima of the state variable y against the parameter values 0 ≤ a ≤ 0.5.Te resulting orbit diagram, depicted in Figure 3(a), vividly illustrates the chaotic nature of the QMO system within the considered parameter range.
Additionally, we employed Wolf's algorithm [39] to calculate the Lyapunov Exponents for system.A fnite time interval of 25000 s was chosen for this analysis.Te Lyapunov Exponents were then plotted in Figure 3(b).It is worth noting that the summation of Lyapunov exponents for any particular value of parameter becomes zero, which provides strong evidence for the conservative nature of the system.Tis observation further reinforces the understanding of the QMO system's behavior and its sensitivity to parameter variations.
Te study emphasizes the vital role that parameter variation plays in the dynamics of chaotic systems, as exemplifed by the QMO system.By exploring a specifc parameter range and examining the local maxima of the state variable y, the researchers demonstrated the chaotic nature of the system.Furthermore, the calculation of Lyapunov Exponents using Wolf's algorithm validated the conservative behavior of the system, as indicated by the summation of the exponents being consistently zero for diferent parameter values.Tese fndings contribute to a deeper understanding of the QMO system's dynamics and provide valuable insights for further research in the feld of chaotic systems.

Basin of Attraction.
Te concept of basins of attraction is a central part of the body of knowledge about multistable dynamical systems.Since most of the nonlinear systems are impossible to study with analytical methods, numerical simulations are the only choice for inquiry into their behavior.In order to construct the basins [9], the trajectory of each chosen initial condition is compared to the trajectory of a collection of known attractors.In addition to knowing the long-term behavior of each initial condition, estimating the basins has many other benefts.As an example, during chaotic transients, they can reveal chaotic dynamics before settling into a nonchaotic attractor.We used the algorithm [10] for identifying the basin of attraction plot for the limit cycle attractors shown in Figure 1. Figure 4 shows the basin of attraction of the proposed QMO system.For the given parameter value the system has 5 attractors in the x-y region.

Application of Fingerprint Image Encryption Using the QMO System
In this section, the fngerprint image encryption application is designed and analyzed using the QMO system.Te encryption and decryption processes of the designed fngerprint image encryption system are given in Figure 5 as a diagram.Tree initial values (x 0 , y 0 , z 0 ) and two parameters (a, b) values of the QMO system are used as the key in the encryption design.In each encryption process, the initial conditions and parameter values are taken diferently within the specifed limits.For each image encryption processing, key information is obtained from a randomly determined list of initial conditions and parameter values in the specifed order.A new key from this list is used in each encryption operation.Tus, even if the same image is encrypted, the encrypted image is diferent.In the encryption process, the x, y, and z state variables of the QMO system are frst calculated with the key information.Next, 32 bit single foating-point IEEE-754 binary numbers are created from the retrieved state variables.From these 32 bit binary values, the 6 bit LSB of the x state variable, the 2 bit LSB of the y state variable, and the 6 bit LSB of the z state variable are taken.Te LSB bits from the x and y state variables are combined to obtain an 8 bit value in the combining part.Likewise, the LSB bits from the y and z state variables are combined to obtain an 8 bit value.Tese two combined 8 bit values are sent sequentially to the XOR part in each period by the "Multiplexer."Te Up-Counter unit is used as the input selector of the Multiplexer unit.Te Multiplexer sends the 8 bit values (x-y and y-z) obtained from the x, y, and z state variables to the output, according to the 0 and 1 information sequentially from the up-counter unit that counts up to one.Tus, the complexity is increased by sending the combination value obtained from the chaotic system state variables.In the XOR part, the 8 bit data from the Multiplexer part and the 8 bit data coming from the fngerprint image are processed by XOR.
In the decryption process, 8 bit combined data is obtained from the QMO system in the same way as in the encryption process.Te incoming encrypted fngerprint image data and the 8 bit combined data are processed by XOR operation to decrypt the fngerprint image.Tus, the decrypted fngerprint image is obtained.Encryption and decryption processes were performed in MATLAB-Simulink program with the block diagrams given in Figures 6 and  7, respectively.Te step size in the designed encryption and decryption process is 1e − 8 seconds.Te processing bit rate of the system is 800 Mbps.
For the analysis of the designed encryption system, a real grayscale fngerprint image with a height value of 390 pixels, a width value of 355 pixels, and a bit depth value of 8, taken from the fngerprint database given in reference [40], was used.Te gray fngerprint image used in the design may vary in size.Te design can be used exactly for fngerprint images of diferent sizes.Color fngerprint images can be implemented using the encryption design provided separately for red, green, and blue pixel values.
Te encrypted and decrypted fngerprint images obtained from the encryption and decryption processes (in Figure 5) are given in Figure 8.As can be seen in Figure 8, the design (Figures 6 and 7) can successfully perform encryption and decryption operations.

Security Analysis.
Te encryption process should be able to perform well against various statistical and security attacks.To determine the performance of encryption, it is necessary to examine it with various security tests.In this section, randomness performance, statistical and attack analyses which are histogram, correlation, correlation maps, entropy, key sensitivity, key space, and known/chosen were applied for the performance analysis of encryption.

Te Randomness Performance of the QMO System.
Since the chaotic QMO system is used in the encryption design, the randomness level of the outputs obtained from this system must be suitable for encryption.Te widely accepted NIST 800-22 test is used for randomness performance analysis.Te NIST 800-22 test consists of 15 separate tests.For each test result to be successful, the p value indicating randomness must be greater than the α value chosen between 0.001 and 0.01 [41].Te NIST 800-22 test performance results of the x, y, and z state variables of the chaotic QMO system are given in Table 2. Te p value was taken as 0.01 in the tests.As can be seen from Table 2, the values obtained from the outputs of the x, y, and z state variables of the QMO system were successful in the randomness tests.

Histogram Analysis. Te distribution of pixel values in
the encrypted image can be examined by using histogram analysis.Te ideal histogram distribution should be uniform.In this way, the distribution of encrypted information is almost the same, indicating that the original information is difcult to predict with statistical analysis [28,42]. Figure 9 shows the histogram diagrams of the original fngerprint and the encrypted image.As may be seen, the histogram diagram of the encrypted fngerprint image is uniform and therefore difcult to predict by statistical analysis.

Correlation Analysis.
Te correlation coefcient value between two neighboring pixels in the original image and the encrypted image is another crucial factor in determining encryption performance.Te correlation between the vertical, horizontal and diagonal pixel values of the image should be near to zero to avoid statistical attacks.A correlation value near 0 denotes a negligibly weak correlation, whereas a correlation value near 1 denotes a signifcant relationship [28,43].Te correlation coefcient is given in equation ( 6) [18,43]

QMO System
Merging binary data 6 bits x and 2 bits y Merging binary data 2 bits y and 6 bits z

Encrypted Fingerprint Image Data
Original Fingerprint Image Data  Complexity Analyzing the correlation distribution provides insights into the degree of correlation between the original and encrypted images.A deviation from the expected correlation values for a secure encryption scheme indicates the efectiveness of the encryption in breaking the correlation patterns present in the original image, contributing to enhanced security.
Here, x and y are adjacent pixels from the encrypted or original images that will be examined.Cov (x, y) is the covariance of x and y, equation ( 7), D (x) and D (y) is the variance of x, equation ( 9) and y.E (x) and E (y) represent the average of x pixels, equation ( 8), and y pixels.N in the formulas refers to the total number of pixels in the image [18,43].Te correlation coefcient values of the actual image and encrypted image used in the study are given in Table 3.As shown in Table 3, the correlation coefcient of the vertical, horizontal and diagonal pixel values of the actual image is near to 1, even though the fngerprint image's correlation coefcients are almost zero.Also, Figure 10 presents the correlation distribution maps of the actual and encrypted image in horizontal, vertical and diagonal forms.According to the correlation distribution maps, while the correlation distribution in the actual image is concentrated  in a certain region, it is homogeneous in the encrypted image.Tis outcome demonstrates that statistical assaults are unable to decrypt the image.

Entropy Analysis.
Te entropy value of a data stack gives information about the distribution and randomness of the data.Te entropy of the actual and encrypted fngerprint image used in the study shows the distribution of each gray value.Entropy is generally calculated with equation (10).In equation (10), N represents the number of symbols, x i is the pixel value and p(x i ) represents the probability that x i will appear in the data stack.Te N value for the 8 bit depth of the grayscale fngerprint image used in the study is 8 [16,20,26,27,30].
If the entropy value is near 8 indicates that the ideal information entropy is reached [23,42,43].A high entropy number denotes a uniform distribution of the image's gray values.Tis makes it difcult for attackers to obtain the actual image through information analysis [28].Te entropy results of the original fngerprint image and the encrypted fngerprint image are given in Table 3.While the entropy value of the original image is 6.1343, the entropy value of the encrypted image is 7.9987, which is very near to 8. Tis indicates that the encryption design has good performance against statistical attacks.

Key Sensitivity Analysis.
Te efectiveness of the key sensitivity analysis of the encryption design can be assessed using UACI and NPCR values.NPCR describes the ratio of the encrypted image's changed pixel values to the original image's altered pixel values.Te intensity of the diference between a specifc pixel's value in the original image and the encrypted image's pixel values is specifed by UACI.NPCR is calculated as in equations ( 11) and ( 12) and UACI is calculated as in equation (13).Where H is the pixel number of the image's height, W is the pixel number of the image's width, i and j represent the pixel position, and C 1 and C 2 represent the two encrypted images.NPCR and UACI have optimum values of 99 and 33 percent, respectively [18,23,28,42,43].
For the key sensitivity analysis, the initial conditions and parameter values of the QMO system were increased by 10 −6 .For instance, instead of using 10 as the initial value for the x state variable, 10.000001 is used.In this way, UACI and NPCR values of the normally encrypted fngerprint image and the encrypted fngerprint image with parameter values increased by 10 −6 were calculated.According to the calculation result [44] as shown in Table 4 the NPCR value was 99.426% and the UACI value was 33.398%.Terefore, the proposed encryption design has high key sensitivity.
5.1.6.Key Space Analysis.One parameter that determines the security of encryption is the key space size.Te larger the key space, the better the encryption performance and the greater the resistance to key analysis.Te key space size should be bigger than 2 100 [23,43] or 2 128 [28,42] for good security.Te system used in this study has three initial values (x 0 , y 0 , z 0 ) and two parameters (a, b) as key.Assuming that the key precision in the design is 10 -15 , the overall key size is 10 75 .Consequently, since the value of 10 75 is much greater than the value of 2 128 , the key space is sufcient to resist the extensive attack.
5.1.7.Known/Chosen Image Analysis.Te encryption design should resist against some known/selected image attacks.Tese attacks are used to decrypt the encryption mask [23,43].Tree initial values (x 0 , y 0 , z 0 ) and two parameters (a, b) values of the QMO system are used as the key in the encryption design.Te result of these values is mixed through various processes and used in the encryption process.As stated in the analysis of the QMO system, the initial condition of the state variable x, in particular, can vary in a wide range between −20 and 20.As stated in the key sensitivity analysis section, it is seen that even a very small change in the initial condition creates a sufcient diference in encryption.Tus, with each change in the initial conditions and the system's parameter values, the encryption key also changes.
In this way, the initial conditions of the system and the values of the system parameters constitute the key to the encryption system.Tis key changes with each new encryption.Te receiving system also knows the keys list and the order of use algorithm within the specifed limits.In practical uses, these key sequences are constantly updated.For the attacker to succeed, the attacker must know the list of these keys updated periodically and the usage order algorithm.Without this information, it cannot be decrypted by repeatedly sending the same image or one image with only minor changes.
In the known/selected image analysis (Figure 11), frstly, the complete white image (Figure 11(a)) was encrypted (Figure 11(b)) in the designed encryption system.Ten, the image with only one black pixel and all the remaining pixels white (Figure 11(c)) is encrypted (Figure 11(d)).Te NPCR and UACI values were analyzed for the diference between the encrypted versions of the two images, and the results are given in Table 4. Since the NPCR value is 99.609% and the UACI value is 33.464%, the designed encryption system is resistant to known/selected image attacks.

Discussion
Te fndings presented in this study contribute signifcantly to the understanding of countable infnite attractors in megastable systems, particularly in the context of the Quadratic Megastable Oscillator (QMO).Te observed dynamical behaviors of the QMO, such as the generation of nested types of multiple attractors for various initial conditions, enrich our comprehension of complex system dynamics.Tis intricate behavior is elegantly portrayed through phase portraits, providing a visual representation of the system's evolving states.
Te sustainability of chaotic oscillation in the QMO is thoroughly examined using infuential parameter bifurcation plots, shedding light on the nuanced interplay of system parameters in shaping its dynamics.Te system's complexity is further underscored by the existence of intricate basins of attraction, accommodating an infnite array of coexisting attractors.Tis feature introduces a level of richness and versatility in the dynamics of the QMO, suggesting potential applications in scenarios requiring diverse attractor landscapes.
A notable observation is the nonlocalized trajectory behavior under specifc initial conditions, where trajectories lead to distant destinations, evading the infuence of local attractors.Tis phenomenon adds a layer of unpredictability to the system's behavior and highlights the unique characteristics of the QMO compared to conventional megastable oscillators.Te practical application of the QMO in biometric fngerprint image encryption showcases its realworld efcacy.Te statistical and attack analyses conducted on the encryption design afrm the QMO's success in providing secure encryption within chaotic system-based frameworks.Tis practical validation further emphasizes the relevance and applicability of the QMO in encryption applications.
NPCR, UACI, correlation coefcient, and entropy values of the proposed method are compared with previous studies in Table 5. Te proposed system is similar to [30,46] and gave better results than [48] for NPCR and UACI values.According to the average correlation coefcient values, the proposed system is slightly behind [45,46], but gives better results than [48].According to the entropy value, the proposed system achieved the best results in [30,46,[48][49][50].In totality, the quadratic megastable system-based Complexity encryption emerges as a robust and efective method for securing fngerprint images.While the study on the Quadratic Megastable Oscillator (QMO) and its application in biometric fngerprint image encryption ofers valuable insights, it is essential to acknowledge certain limitations that could impact the generalization and applicability of the fndings: (1) Teoretical Assumptions: Te study relies on certain theoretical assumptions related to the behavior of the QMO, and these assumptions may not perfectly align with real-world conditions.Practical implementations could be afected by factors not considered in the theoretical model.(2) Sensitivity to Initial Conditions: Chaotic systems, such as the QMO, are known to be highly sensitive to initial conditions.Small variations in the initial conditions can lead to signifcantly diferent outcomes.Tis sensitivity might pose challenges in realworld applications where precise control over initial conditions may be difcult.(3) Computational Complexity: Te comprehensive analysis involving attractors, phase portraits, and  Te efectiveness of the encryption scheme is demonstrated through various analyses, but its robustness may vary depending on the characteristics of diferent fngerprint images.It is crucial to assess performance across a diverse range of biometric data.

Conclusion
In conclusion, this study ofers a comprehensive exploration of the behavior of three-dimensional autonomous quadratic megastable oscillators, shedding light on their characteristics and dynamics in the absence of external stimulation.Te analysis encompasses a thorough investigation of attractors, phase portraits, parameter bifurcation plots, and the sustainability of chaotic oscillation.
A key focus of this analysis is the examination of the oscillators' attractors, distinctive patterns that emerge over time.Our fndings suggest that initializing the oscillators with diferent initial conditions leads to the generation of multiple attractors.Visual representation through phase portraits facilitates a clearer understanding of the oscillators' trajectories in their three-dimensional phase space.Additionally, parameter bifurcation plots are employed to visualize the impact of specifc parameter changes on the system's behavior, assessing the sustainability of chaotic oscillations under varied parameter values.
Stability analysis and the concept of basins of attraction are crucial aspects discussed in this study.Stability analysis evaluates the system's response to perturbations and its convergence to specifc states over time, while the basin of attraction identifes regions in phase space where specifc attractors are achieved.Tese analyses collectively contribute to a comprehensive understanding of the system's overall behavior.
Furthermore, the study extends the application of the quadratic megastable oscillator system to biometric fngerprint image encryption.Rigorous parameter analyses, including NIST 800-22 tests, highlight the robust security features of the quadratic megastable system.Histogram analysis reveals a uniform distribution of encrypted pixel values, resistant to statistical scrutiny.Correlation analysis reinforces security, indicating a substantial deviation from the original image's correlation coefcients.Correlation distribution maps illustrate the encrypted image's resilience against region-based attacks.Entropy analysis underscores efective protection against statistical assaults, supported by entropy values.Key sensitivity analysis emphasizes the encryption's strength, evidenced by high NPCR and UACI values, along with a substantial key space size surpassing established security thresholds, ensuring resistance to extensive attacks.Tese results collectively afrm the quadratic megastable system's efectiveness in biometric image encryption, showcasing its potential in secure communication systems.

Figure 1 :
Figure 1: 2D Phase portrait of QMO system for 21 initial conditions (from x � −20 to x � +20 with steps equal to 2) and the initial conditions for the other states is kept at 0.

Figure 4 :
Figure 4: Basin of attraction plot to show the presence of diferent attractors.

Figure 10 :
Figure 10: Correlation distributions for original and encrypted images.

Figure 11 :
Figure 11: Known/chosen image analysis result (a) completely white image and (b) its encrypted image (c) completely white image with only one black pixel and (d) its encrypted image (e) diference image between (b and d).

Table 1 :
Stability of the QMO system for diferent cases.

Table 2 :
Te NIST 800-22 test performance results of the x, y, and z state variables of the chaotic QMO system.

Table 3 :
Te correlation coefcient and entropy values of the actual image and the encrypted image.

Table 4 :
Key sensitivity analysis calculation results for NPCR and UACI.

Table 5 :
Te comparison of the proposed encryption design with similar research.