1 Introduction

The development of calculus, which emerged from Newton’s idea, has expanded our knowledge of everything from the dynamics of extremely small-scale systems to massive galaxies in space. In the past, a variety of analytical and numerical mathematical tools were developed to solve the Newton, Lagrange, and Hamilton governing equations. In the 1700s, there was an increase in the analysis of integer order calculus. At the same time, the idea of fractional order derivative (FOD) was initiated by Leibniz and L’Hopital [1] for order \((q=\frac{1}{2})\) as well. Initially, mathematicians like Laplace gave an integral representation for fractional order derivatives. Consequently, Riemann–Liouville, Grunwald–Letnikov, and Caputo developed the standard method for solving fractional order differential equations (FDEs) using the integrodifferential operator and with stable numerical techniques [1]. Many systems that exist in nature are fractional in dimension. Therefore, the study and analysis of FDE are mandatory and unavoidable [2,3,4,5,6,7,8]. However, the topology of fractional order and its physical intuition have not been well realized well till now. Naturally, some of the fractional order characteristics are observed in the voltage-current relation of a semi-infinite lossy transmission line [9], diffusion of heat through a semi-infinite solid [1], fractional capacitance in chaotic electronic circuit [10] and the significance of the fractional order are realized and implemented carefully in areas of engineering [11, 12], signal processing [13], Bio-medical engineering [14] processing, analog electronic oscillator [10].

Chaotic attractors in dynamic systems are highly sensitive to initial conditions, with a positive Lyapunov exponent, while strange non-chaotic attractors (SNAs) exhibit aperiodic behavior but are insensitive to initial conditions. Generically, the formation of SNAs is robust in quasi-periodically driven systems. This was first observed and reported by Grebogi et al. [15] in 1984. SNAs were found to exhibit intermediate dynamical behavior between quasi-periodicity and chaos. Theoretically, behaviors of SNAs are observed in quasiperiodically forced pendulums[16], quantum particles in quasiperiodic potentials [17], the quasiperiodically driven Duffing-type oscillators [18], and in memresitive circuits [19], biological oscillators [20]. Also, it is observed in many experiments, such as quasiperiodically forced buckled magnetoelastic ribbon [21], neon glow discharge [22], electrochemical cells, electronic circuits [23], and it was also shown that SNAs can also be utilized in logical computation [24]. Pulsating Lyrae variables star [25] was the first non-chaotic dynamical system observed in nature; Grebogi et al. demonstrated the occurrence of SNA in piecewise linear oscillator [26]. Also, non-chaotic extreme events were explored in the quasi-periodically forced Morse oscillator [27], the co-existence of chaotic and non-chaotic attractors in the ecological slow-fast system was explored [28].

The routes to the birth of SNAs are studied extensively and notably, there are four routes. Heagy Hammel route [29], Fractalization route [30], Type-III Intermittency [31] and Bubble route to formation SNAs [32]. The Heagy–Hammel route involves a doubled torus colliding with a truncated torus, leading to the formation of SNA. Gradual fractalization involves a sudden wrinkling without interaction with the parent torus, and crisis-induced intermittency results from the wrinkled torus colliding with its boundary. Type-III intermittency results in torus doubling bifurcation dominated by subharmonic bifurcation, leading to the formation of SNA. In the bubble route, torus strands arise when parameters change, causing instability and wrinkles. Recent research reveals SNAs emerge in periodically forced smooth and non-smooth systems with noise [33, 34]. New route to SNAs was explored by Zhao et al. [35], where smooth quasiperiodic torus gets non-smooth due to border-collision and got fractalised leading to the SNAs.

This study explains how fractional orders affect the behavior of a SNA in a quasi-periodically forced Chua system. It is the first study to look at SNAs in fractional-order systems, which is an area that isn’t well covered in the existing literature. By systematically varying fractional orders \(q_1\) and \(q_2\), the research provides new insights into system behavior, differing from previous works focused on integer-order systems. Using numerical integration with the Grunwald–Letnikov algorithm, the study characterizes dynamical behaviors through bifurcation studies, Poincarémaps, and FFT analyses. Advanced methods, including the 0–1 test for chaos and recurrence analysis, are employed. The 0–1 test distinguishes between chaotic and non-chaotic dynamics by assessing the growth rate of a test statistic, \(K\), while recurrence analysis measures system determinism, revealing underlying temporal structures. Singular continuous spectrum analysis and random walk in the complex plane demonstrate the fractal nature of SNAs. The novelty lies in systematically exploring SNAs in fractional-order systems and developing methodologies to characterize their complex behaviors. This comprehensive approach, including the 0–1 test and recurrence analysis, offers a fresh perspective on fractional dynamics in nonlinear systems, enhancing the understanding of SNAs and their transition routes.

The manuscript is organized as follows: Sect. 2 constructs the quasi-periodically forced fractional-order Chua system and integrates the state equations using the Grunwald-Letnikov algorithm. Bifurcation studies for variations in \(q_1\) and \(q_2\), and their combinations, are presented. Section 3 characterizes the dynamical behaviors using Poincarémaps and FFT analysis, tracing the route to SNA by varying \(q_1\). Instead of Lyapunov exponents, we analyze the separation of nearby trajectories to emphasize the divergence-less property of SNAs. Singular continuous spectrum analysis and random walk in the complex plane demonstrate the fractal nature of SNAs. Section 4 varies \(q_2\) to study dynamical behaviors. Section 5 explores the quasiperiodic route to SNA by adjusting \(q_1\) and \(q_2\) while fixing the forcing parameter in the chaotic region.

1.1 Fractional Calculus

Generally, Derivatives in integer order calculus are well formulated and established by Newton and Leibniz. Similarly, the derivatives and anti-derivatives for fractional order are represented by

$$\begin{aligned} _{a}D_{t}^{\alpha } = \left\{ \begin{array}{ll} \frac{d}{dt^{\alpha }},& \alpha > 0\\ 1 ,& \alpha = 0 \\ \int _{a}^{t} (d\tau )^{\alpha } ,& \alpha <0 \end{array} \right\} \end{aligned}$$
(1)

The derivative of a function in fractional order is represented by an integrodifferential operator. Riemann–Liouville’s definition of the fractional derivative is equated as

$$\begin{aligned} _{a}D_{t}^{\alpha }f(t) = \frac{1}{\Gamma (n-\alpha )}\frac{d^{n}}{dt^{n}}\int _{0}^{t}\frac{f(\tau )}{(t-\tau )^{\alpha -n+1}}d\tau \end{aligned}$$
(2)

The range of \(\alpha\) must agree with (\(n-1<\) \(\alpha\) \(< n\)). where a and t are the bounds for operation \(_{a}D_{t}^{\alpha }\). Apart from Riemann–Liouville definitions, Grunwald–Letnikov (GL) and Caputo algorithms are implemented for computing fractional order derivatives and solving differential equations [1, 38]. In our work, we have used the GL algorithm for integrating fractional ODE with the memory of 2000 throughout our analysis.

2 State Equation

In the literature, studies on Chua’s system have been critically analyzed and well-established. It is known that to generate chaos, a system must possess two energy-pumping elements, a dissipative element, and a nonlinear element. To achieve quasi-periodicity, two incommensurate forcing frequencies are required. Constructing fractional-order elements in electrical circuits is experimentally challenging. Since SNA is an intermediary behavior between quasi-periodicity and chaos, our work focuses on the normalized equations of a quasiperiodically forced Chua system with fractional order [36].

$$\begin{aligned} \frac{d^{q_1}x}{dt}= & \alpha (y-x-h(x)), \nonumber \\ \frac{d^{q_2}y}{dt}= & x-y+z, \nonumber \\ \frac{d^{q_3}z}{dt}= & -\beta y + f_{1} sin(\theta ) + f_{2} sin(\phi ), \nonumber \\ \frac{d^{q_4}\theta }{dt}= & \Omega _{1}, \nonumber \\ \frac{d^{q_5}\phi }{dt}= & \Omega _{2}. \end{aligned}$$
(3)

In the Chua system, the nonlinear element incorporated is piece-wise continuous and is given by the equation

$$\begin{aligned} h(x) = bx + 0.5(a-b)(|x+1|-|x-1|). \end{aligned}$$
(4)

Here, the variables x and y are normalized variables that correspond to the voltages across capacitors, while the variable z is the normalized variable of the current through the inductor. The external forcing is represented by the normalized variables \(\theta\) and \(\phi\). The system have a discontinuity at two points \(x = -1\) and \(x =1\). The normalized parameter are a = – 1.1862, b = – 0.6400, \(\alpha\) = 5.8333, \(\beta\)=12.8012, \(\Omega _{1}\)=2.7454, \(\Omega _{2}\)=3.4306, \(f_{2}\) =0.1280 and we take \(f_{1}\) = 0.7835 for behavioural study and \(f_{1}\) = 0.9728 for demonstrating effectiveness of the fractional order \(q_{1}\) and \(q_{2}\). We keep other orders \(q_3=q_4=q_5=1\) for the study. The state equations are integrated using the Grunwald-Letnikov algorithm for a fractional order system with a time step (h = 0.0045).

$$\begin{aligned} x_{k+1}&= \left( \alpha (y_{k}-x_{k}-h(x_{k}))\right) hq_{1} - \sum \limits _{j=v}^k C_{j}^{q_{1}}x_{k-j}, \nonumber \\ y_{k+1}&= \left( x_{k+1}-y_{k}+z_{k}\right) hq_{2} - \sum \limits _{j=v}^k C_{j}^{q_{2}}y_{k-j},\nonumber \\ z_{k+1}&= \left( -\beta y_{k+1} + f_{1} sin(\theta _{k}) + f_{2} sin(\phi _{k})\right) hq_{3} - \sum \limits _{j=v}^k C_{j}^{q_{3}}z_{k-j},\nonumber \\ \theta _{k+1}& = (\Omega _{1})hq_{4} - \sum \limits _{j=v}^k C_{j}^{q_{4}}\theta _{k-j},\nonumber \\ \phi _{k+1}&= (\Omega _{2})hq_{5} - \sum \limits _{j=v}^k C_{j}^{q_{5}}\phi _{k-j}. \end{aligned}$$
(5)

Here the control parameter is taken as the fractional order itself \(q_{1}\) and \(q_{2}\) and all other orders are fixed to (\(q_{3} = q_{4} =q_{5} = 1.0\).)

3 Study of Fractional Order Chua System with Quasiperiodic Forcing

In order to understand the dynamics of the system under different fractional orders, we employed various analyses, like determinism from the recurrence analysis and 0–1 test. Recurrence Analysis is a technique that is employed to investigate dynamical systems by analyzing the moments at which a system reverts to a previous state or approaches it within a specified threshold. It quantifies and visualizes the recurrence of states, thereby lending insight into the system’s chaotic and deterministic characteristics. The predictability of the system is quantified by determinism, a measure that is derived from Recurrence Plot (RP). A high determinism value indicates predictable dynamics, while a low determinism value suggests chaotic behavior. To quantify the deterministic behaviour of SNA from the chaotic behaviour, we have employed the recurrence analysis technique by Ngamga et al. [43]. Determinism (DET) in Recurrence Analysis is quantified using the formula:

$$\begin{aligned} DET = \frac{\sum _{l=l_{\min }}^{N} l \cdot P(l)}{\sum _{l=1}^{N} l \cdot P(l)} \end{aligned}$$
(6)

The histogram of diagonal line lengths, denoted as P(l) , is used in this formula to represent the length of a diagonal line in the RP. This histogram indicates the number of diagonal lines that are precisely l. \(l_{\min }\) is the minimum length of diagonal lines that are taken into account in order to prevent the enumeration of very short, potentially spurious, lines. In the numerator, \(\sum _{l=l_{\min }}^{N} l \cdot P(l)\), the sum of the lengths of all diagonal lines that are at least \(l_{\min }\) long is taken. The system’s dynamics are deterministic, as evidenced by the longer lines, which implies that they can be predicted. The denominator, \(\sum _{l=1}^{N} l \cdot P(l)\), is the sum of the lengths of all diagonal lines, including those that are extremely brief and may represent noise or transient structures. Higher DET values indicate a greater proportion of longer diagonal lines in the RP, reflecting more predictable, deterministic dynamics, whereas lower DET values suggest a higher proportion of brief, isolated diagonal lines, reflecting more chaotic or stochastic dynamics. The peculiar nonchaotic dynamics exhibits a fully deterministic process, and as a result, it is deterministic. The DET value can be used to assess the degree of determinism in the system’s behavior.

The 0–1 test is a powerful yet relatively simple method employed to ascertain whether a system demonstrates unpredictable behavior. It includes the examination of the system’s response to a particular driving force and the determination of whether the motion is regular or chaotic [45]. Define a parameter c that is within the range of \((0, 2\pi )\). For each value of c, the following two new series are computed: p(n) and q(n) .

$$\begin{aligned} p(n)= & \sum _{j=1}^{n} x_j \cos (jc)\end{aligned}$$
(7)
$$\begin{aligned} q(n)= & \sum _{j=1}^{n} x_j \sin (jc) \end{aligned}$$
(8)

where \(x_j\) represents the time series data. One can find the Mean Square Displacement (MSD) using the following formula.

$$\begin{aligned} MSD(n) = \langle \Delta p(n)^2 + \Delta q(n)^2 \end{aligned}$$
(9)

where \(\Delta p(n) = p(n+k) - p(k)\) and \(\Delta q(n)\) are equal. The angle brackets represent the average over k. To analyze the data set MSD(n) vs n, a linear model should be applied. The slope of this line represents the test statistic K. The system is non-chaotic if K is close to 0. The system is chaotic if K is close to 1. For unusual nonchaotic dynamics, K is probably near to 0, but not exactly 0. The system has complicated, non-periodic, non-chaotic behavior.

Our primary objective in this study is to comprehend the influence of fractional order on strange nonchaotic behavior. Previous work by Suresh et al. [36] on a quasiperiodically forced Chua system, using a forcing parameter \(f_{1} = 0.7835\), reported the occurrence of SNA behavior when the system order was an integer, specifically \(q_{1} = q_{2} = 1.0\). To investigate the impact of system order on SNA behavior, we systematically varied the system order \(q_1\) from 0.999 to 0.98 while keeping \(q_2\) fixed at 1.0 (Fig. 1a). Similarly, we varied \(q_2\) fractionally within the same range, with \(q_1\) fixed at 1.0 (Fig. 1b). In the third case, we set \((q_1, q_2) = (1.0, 0.96999)\) to observe a torus attractor. We then systematically decreased \(q_1\) from 0.999 to 0.970 for bifurcation studies, revealing the localized doubling of the torus route to chaos via SNA, as depicted in Fig. 1c. The results of the bifurcation studies are presented in Fig. 1a(i)–c(i).The bifurcation studies demonstrated that the onset of chaos and the destruction of Strange Nonchaotic Attractor (SNA) behavior were the results of a decrease in the fractional order \(q_1\). The subsequent sections will explore the stages of transition from SNA to disorder. On the other hand, bifurcation studies on \(q_2\) variation demonstrated that the SNA underwent reverse torus doubling, resulting in a quasiperiodic attractor. Figure 1a(ii)–c(ii) illustrate the determinism that is derived from Eq. (6). The determinism of the system demonstrates that it has a value of one in the SNA regime and a value of less than one in the chaotic regime. This is evident in all three plots of determinism. Similarly, we have plotted the 0–1 test corresponding to the three bifurcation diagrams in Fig. 1a(iii)–c(iii). These plots show that in the SNA regime, the K value is neither 0 nor 1, thereby proving the existence of SNA in the selected parameter regime. Throughout the subsequent sections, we visualize the transitions in dynamical behavior through Poincarésections for four specific bifurcation system orders. The characterization of the dynamical behavior is analyzed using appropriate numerical and statistical measures. The implementation of fractional order in electronic circuits can be feasible only with capacitors rather than inductors.

Fig. 1
figure 1

Effects of fractional orders \(q_1\) and \(q_2\) on the behavior of SNAs are shown. a Bifurcation diagrams varying \(q_1\) from 0.999 to 0.98 with \(q_2\) fixed at 1.0, illustrating SNA to chaos transition. b Bifurcation diagrams varying \(q_2\) fractionally within the same range, \(q_1\) fixed at 1.0, showing system’s bifurcation response. c Bifurcation study decreasing \(q_1\) from 0.999 to 0.970, revealing torus to chaos via SNA doubling. Subfigures (i) depict specific bifurcation scenarios, (ii) show determinism plots indicating dynamics (DET = 1 for SNA, DET < 1 for chaos), and (iii) display 0–1 test results confirming SNA presence (K values neither 0 nor 1)

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3.1 Fractional Order \(q_{1}\)

By fixing \(q_{2}=1.0\) and on varying \(q_{1}\) down from the system order \(q_{1}=0.999\), we observe that chaotic nature is dominant when lowering the dimensionality. To demonstrate the behavior, we fix the parameter \(f_{1}\) = 0.7835 in the SNA region as reported by Suresh et al. [36]. Figure 2a(i) shows that when \(q_{1}\) =0.999 the SNA persists with its stability, on lowering to \(q_{1}\) = 0.997 the SNA bands start to merge with one another as shown in Fig. 2b(i) and at \(q_{1}\) = 0.991 in Fig. 2c(i) the band got merged and on close observation, we can also notice the onset of fractalization in the attractor. On varying \(q_{1}\) = 0.98 the fractalized one band SNA becomes unstable, leading to chaotic behavior as shown in Fig. 2d(i). Along with the Poincaré section, as supporting evidence we included the FFT spectra. Many works of literature and study show that for periodic and quasi-periodic orbits the FFT has Dirac delta peaks, while for irregular (chaotic) motions the spectrum is continuous [39]. Since SNAs exhibit the intermediate dynamical behavior between quasiperiodicity and chaos the spectrum has both discrete and continuous peaks as shown in Fig. 2a(ii)–c(ii). From Fig. 2d(ii) of the FFT spectrum the continuous spectrum dominates the Dirac delta peaks when moving to chaotic regimes (i.e) at \(q_{1}\) = 0.98.

Fig. 2
figure 2

(i) Maximum of the variable x in mod(\(x_{max},2\pi\)) plane and (ii) FFT spectra. a for \(q_{1}\)= 0.999, b for \(q_{1}\)= 0.997, c for \(q_{1}\)= 0.991, and d for \(q_{1}\)= 0.980

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3.2 Separation of Nearby Trajectories

To demonstrate the divergenceless property of SNAs and sensitivity with the diverging property of the chaotic orbits, the separation of nearby points gives supportive evidence. We initially define two arbitrary sets of initial conditions(\([x_{1},y_{1},z_{1}]\) and \([x_{2},y_{2},z_{2}]\)), then allowing the system to evolve to achieve asymptotic behavior. Euclidean distance(\(\Delta\)) is calculated between the trajectories using the algorithm

$$\begin{aligned} \Delta (t) = \sqrt{\left( x_{1}(t) -x_{2}(t)\right) ^2 + \left( y_{1}(t) -y_{2}(t)\right) ^2 + \left( z_{1}(t) -z_{2}(t)\right) ^2 }. \end{aligned}$$
(10)

For SNAs, the nearby trajectory starts converging to the flow of reference trajectory and the \(\Delta\) between them reduces to zero when \(t\rightarrow { \infty }\) this clearly exhibits the quasiperiodic character [40]. The divergence property of the chaotic attractor makes the \(\Delta\) between the trajectories positive. To demonstrate the sensitivity to the initial condition, we compare this test with SNA and chaotic attractor. For the fractional order parameter \(q_{1}\) = 0.999 Fig. 3a the euclidean norm goes to zero after transient time and this confirms the attractor is strange and non-chaotic. In Fig. 3b for chaotic attractor, \(q_{1}\) =0.980 the nearby trajectories do not settle down for a long time run and thus confirm the sensitivity to the initial condition.

Fig. 3
figure 3

Separation of nearby trajectories, a SNA for \(q_{1}=0.999\), and b chaos for \(q_{1}=0.980\)

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3.3 Singular Continuous Spectrum

In the earlier literature, singular continuous spectrum analysis was incorporated for the study of quasiperiodic lattice and quasiperiodically forced quantum systems [41]. Later it was introduced as the statistical tool for the characterization of SNA and its fractal properties by Feudal et al. [42]. Generally, when the magnitudes of partial Fourier coefficients of time series for variable x is plotted against time index(N), it obeys power law and can be written as \(\Psi (\beta , N)\) \(\propto\) \(N^{\alpha }\). The spectrum shows \(\Psi (\beta , N)\) \(\propto\) N for regular motions and for chaos the magnitude of partial Fourier sum \(\Psi (\beta , N)\) \(\propto\) \(N^{2}\). For SNA \(\alpha\), the value falls between (\(1<\alpha <2\)), indicating that the dynamic behavior of the SNAs is in between quasiperiodic and chaotic. As a result, it exhibits both discrete peaks and a continuous spectrum. In this case, \(\beta\) and \(\Omega _{1}\) are proportionate. The partial Fourier sum [44] is given by

$$\begin{aligned} \Psi (\beta , N) = \Sigma _{n=1}^{N} \left\{ x_{n}\right\} e^{(i2\pi \beta )}. \end{aligned}$$
(11)

The singular continuous spectrum is effective for confirmation of SNA. Figure 4a is plotted for \(log|\Psi (\beta ,N)|^{2}\) vs log(N) for \(q_{1}=0.999\) and the slope of the linear fit (\(\alpha\)) is found to be 1.4679 and it is in between 1 and 2 this confirms the attractor is strange and non-chaotic. Since SNA is fractional in dimension, as an additive test, it produces a nontrivial self-similar random path in the complex plane (Re(\(\Psi\)), Im(\(\Psi\))). For \(q_{1}=0.999\) the real and imaginary Fourier coefficients are mapped for every time iteration and it exhibits the fractal nature of the attractor as shown in Fig. 4b.

Fig. 4
figure 4

a Singular continuous spectrum and b fractals in complex plane for \(q_{1}=0.991\)

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4 Fractional Order \(q_{2}\)

In this section, we vary the system order \(q_2\) by fixing \(q_{1}=1.0\) and study the system’s dynamical transition. As in the previous section, we fix the forcing amplitude \(f_{1}\) = 0.7835. In Fig. 5a, for \(q_{2}=0.999\), the SNA does not lose its stability and persists. The occurrence of SNA is confirmed by the singular-continuous spectrum analysis, i.e., the magnitude of the partial Fourier sum obtained is directly proportional to \(N^{1.204}\), and the power of N, i.e., \(\alpha\), lies between 1 and 2 as in Fig. 6a. The fractal nature of SNA is confirmed by a random walk in the complex plane (Re(\(\Psi\)), Im(\(\Psi\))) as given in Fig. 6b. Additionally, in Fig. 7a, there is the separation of nearby trajectories, where the Euclidean distance between the trajectories becomes zero after a long time run, confirming insensitivity to initial conditions. Further, for \(q_{2}=0.9988\) in Fig. 5b, SNA loses its stability and reaches the critical fractalized doubled torus. If \(q_{2}=0.9985\) in Fig. 5c, the system goes through torus doubling bifurcation in the localized regime of the torus strands. If \(q_{2}=0.997\), the system goes through the torus Fig. 5d. In Fig. 5c(ii), we can see a higher number of peaks compared to Fig. 5d(ii) because of the localized torus doubling. In each localized torus, doubling leads to a greater number of peaks in the FFT spectrum. This clearly reveals that on reducing \(q_{2}\) dimensionality, the quasiperiodic nature is increasing, as observed by an increasing number of delta peaks and decreasing continuous peaks in the FFT spectrum.

Fig. 5
figure 5

(i) Local peaks of the variable x in mod(\(x_{max},2\pi\)) plane and (ii) FFT spectra. a for \(q_{2}\)= 0.999, b for \(q_{2}\)= 0.9988, c for \(q_{2}\)= 0.9985, and d for \(q_{2}\)= 0.997

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Fig. 6
figure 6

a Singular continuous spectrum and b fractal path in the complex plane for \(q_{2}=0.999\)

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Fig. 7
figure 7

Separation of nearby trajectories SNA for \(q_{1}=0.999\)

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5 Bubble Doubling in Fractional Order

The bubbling route is previously studied in a quasiperiodically forced Chua oscillator by Suresh et al. [36]. The forcing parameter \(f_{1}\) is used as the control parameter in the Poincaré section to capture different stages of evolution of quasiperiodically obtained torus to SNA and to Chaos.

Fig. 8
figure 8

(i) Local peaks of the variable x in mod(\(x_{max},2\pi\)) plane and (ii) FFT spectra with \(q_{2}=0.96999\) fixed. a for \(q_{1}\)= 0.997, b for \(q_{1}\)= 0.99439, c for \(q_{1}\)= 0.991, and d for \(q_{1}\)= 0.970

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Keeping the system in the chaotic region at \(f_{1} =0.9728\) as previously reported, we first show the efficacy of the fractional order parameter (\(q_{1}\) and \(q_{2}\)). Based to the behavioral analysis of the fractional order in the previous section, the system transits to a chaotic state as the \(q_{1}\) dimensionality decreases, and quasiperiodic behavior emerges as the \(q_{2}\) dimension decreases. Since SNA is known to display behavior that lies in between quasiperiodicity and chaos, we construct the bubble doubling route to SNA by balancing the fractional orders (\(q_{1}\), \(q_{2}\)).

To identify the emergence of SNA, we tune the order \(q_{1}\) as a control parameter by keeping \(q_{2}\) fixed to 0.96999. In Fig. 8a(i), a torus is observed when \(q_{1}\) is set to 0.997, and its corresponding FFT spectrum is shown in Fig. 8a(ii), indicating peaks corresponding to quasiperiodic oscillations and system frequencies. Upon reducing \(q_{1}\) to 0.994, Fig. 8b(i) illustrates torus doubling bifurcation. In the FFT corresponding to the doubled torus, it can be observed that additional small-amplitude peaks appear as a result of the new torus doubling in the localized regime. An unstable torus got fractalized and at \(q_{1}\) = 0.991 the SNA occurs. SNA is confirmed from various characterization techniques such as the FFT spectrum which has both continuous and Dirac delta peaks as shown in Fig. 8c. Singular continuous spectrum analysis shows the growth rate (\(\alpha\)) of the magnitude of partial Fourier sum(\(\Psi (\beta ,N)\)) is found to be 1.181 from linear fit and it is between 1 and 2 in Fig. 9a, as well as the fractal random walk in complex plane suggest us to confirm that the attractor is indeed strange and non-chaotic in Fig. 9. Additionally, the insensitiveness to initial conditions is obeyed in the separation of nearby trajectories Fig. 10. This confirms SNA at the order \(q_{1}\) = 0.991 and \(q_{2}\) = 0.969 as the attractor topologically appeared to be chaotic but exhibits strange nonchaotic behavior as shown in Fig. 8c. For \(q_{1}\) = 0.970 in Fig. 8d(i) the SNA bands got merged and fractalized and chaos occurs, also the FFT spectrum has a large number of continuous bands in Fig. 8d(ii) and high sensitivity to the initial condition in Fig. 10a admits the attractor is chaotic. Observing the emergence of SNA by varying the fractional orders \(q_{1}\) and \(q_{2}\) gives uniqueness to the present work.

Fig. 9
figure 9

a Singular continuous spectrum and b fractals in complex plane for \(q_{1}=0.991\) & \(q_{2}=0.96999\)

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Fig. 10
figure 10

Separation of nearby trajectories, a(i) SNA for \(q_{1}=0.991\) & \(q_{2}=0.96999\), a(ii) chaos for \(q_{1}=0.970\) & \(q_{2}=0.96999\)

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6 Conclusion

In this paper, we analyzed the dynamical behavior of the quasiperiodically driven fractional Chua’s system by varying the fractional order \(q_{1}\) and \(q_{2}\) individually. When varying \(q_{1}\) parameter the SNA bands merged and got fractalized leading to chaos while varying \(q_{2}\) as the parameter the SNA takes a reverse route such as torus doubling and torus to quasiperiodicity. We found the bubble doubling route by a suitable combination of \(q_{1}\) and \(q_{2}\) through Poincaré section and corresponding results agree with characterization tools such as fast Fourier transform, singular continuous spectrum, separation of nearby trajectories and fractal formation in the complex plane. To differentiate between the torus, SNA, and chaos, we used the 0–1 test. By evaluating the growth rate of a test statistic, K, the 0-1 test differentiates between chaotic and non-chaotic dynamics, while recurrence analysis quantifies system determinism and uncovers underlying temporal structures. On the whole, this study contributes by investigating the impact of fractional order on SNAs and the bubble doubling route.