Bifurcation and Hysteresis Analysis in a Memristor Circuit under Periodic Excitations

Nonlinear Phenomena in Complex Systems, vol. 28, no. 3 (2025), pp. 298 - 307 Bifurcation and Hysteresis Analysis in a Memristor Circuit under Periodic Excitations M. Anisha Nashrin,1, ∗S. Siva Sakthi Pitchammal,1S. M. Abdul Kader,1A. Zeenath Bazeera,1and V. Chinnathambi1 1Department of Physics, Sadakathullah Appa College, Tirunelveli 627 011, Tamilnadu, INDIA (Received 9 June, 2025) In this paper, we analyze bifurcation and hysteresis behaviours in a simple memristive circuit model subjected to various periodic excitations. The external periodic inputs considered include sine wave, square wave, symmetric saw-tooth (SST) wave, and asymmetric saw-tooth (AST ) wave. To analyze the circuit’s dynamical behavior, we employ a range of nonlinear simulation tools, including phase portraits, Poincar´e maps, bifurcation diagrams, and maximal Lyapunov exponent diagrams. Several intriguing phenomena related to chaos theory are observed. Notably, the transition to chaotic behavior via antimonotonicity is identified. Additionally, the hysteresis phenomenon and the presence of coexisting attractors, influenced by initial conditions and system parameters, are explored. Furthermore, the period-doubling (PD) route to chaos, reverse period-doubling (RPD) and crisis phenomena are also observed. AMS Subject Classification: 37J20, 65P20, 94c60, 93d23 Keywords: memristive circuit, periodic inputs, chaos, hysteresis, antimonotonicity DOI: https://doi.org/10.5281/zenodo.17236202 1. Introduction A memristor is a type of nonlinear resistor with a memory function, first proposed by Leon Chua [1]. Unlike a conventional linear resistor, a memristor exhibits a dynamic relationship between current and voltage, retaining a memory of past voltages or currents [2]. Since its discovery, the memristor has generated widespread interest due to its significant potential in various engineering fields, including intelligent computing, non-volatile memory, image processing, and chaotic circuits. Its unique nonlinear characteristics facilitate the generation of chaos and other complex dynamical behaviors, making it a valuable component in chaotic circuits and systems, where it enhances chaotic properties and broadens their application scope [3-5]. The memristor’s potential extends to various other domains, such as artificial neural networks [6-8], flash memory [9,10], electronic circuits [11-13] and secure communication ∗E-mail: [email protected] systems [14-16]. In recent years, research on memristors has gained significant momentum, leading to numerous important findings. For instance, Laskaridis et al. [17] reported the occurrence of antimonotonicity, hysteresis, and coexisting attractors in a Shinriki circuit using a physical memristor as a nonlinear resistor. Wang et al. [18] presented a rigorous computer-assisted verification of horseshoe chaos using topological horseshoe theory. Peng et al. [19] explored chaotic and hysteresis behaviors in a discrete memristor-based system with fractional-order differences. Elwakil et al. [20] demonstrated pinched hysteresis loops in simple nonlinear resonant circuits containing a single diode acting as a voltage-controlled switch. Additionally, Li et al. [21] developed a memristor-based hyperchaotic Bao-like system and confirmed the presence of chaotic behaviors within the system. Furthermore, several researchers have designed multistable chaotic systems by introducing memristors into existing chaotic attractors. For instance, Ramakrishnan et al. [22, 23] proposed a three-dimensional memristive 298 Bifurcation and Hysteresis Analysis in a Memristor Circuit under Periodic Excitations 299 oscillator based on a simple damped Duffing oscillator and further developed a higherorder memristive oscillator circuit derived from the Van der Pol-Duffing oscillator. They identified symmetric coexisting chaotic or periodic attractors using bifurcation diagrams and attraction basins. Kengene et al. [24] introduced a novel memristive jerk system exhibiting symmetric and asymmetric coexisting attractors. Similarly, Tchiedjo et al. [25] proposed a second-order memristive emulator comprising two diodes, two resistors, and two capacitors, which was applied to a jerk circuit to generate new memristive chaotic circuits. In addition, some researchers have focused on generating memristive chaotic systems with distinct equilibria and dynamics derived from simple autonomous polynomial systems. For example, Liu and Tu [26] developed a memristive chaotic system with no equilibria, featuring hidden attractors, coexisting attractors, and offset boosting properties. This was achieved by incorporating an ideal flux-controlled memristor into a five-term system, revealing diverse behaviors depending on initial values and parameters. Jia and Lai [27] introduced a non-ideal memristor into the Sprott A system, creating a new memristive system characterized by three equilibria, coexisting chaotic attractors, and an amplitude control feature. Singh et al. [28] investigated a memristive-based hyperchaotic system with line equilibria, exploring its hyperchaos, coexisting attractors, hardware circuit implementation, and active adaptive projective synchronization control, supported by both theoretical and numerical results. The remainder of this paper is organized as follows. Section 2 introduces the models of both autonomous and non-autonomous memristor circuits. This section also includes an analysis of the equilibrium points and the stability of the autonomous memristive system. In Section 3, various types of periodic excitations and their corresponding mathematical representations are presented. Section 4 focuses on the dynamical analysis of the non-autonomous memristive system under these periodic excitations. Finally, the conclusions are summarized, and prospects for future work are discussed. 2. Autonomous and nonautonomous memristor circuits Memristive chaotic circuits and systems exhibit a wide range of intriguing nonlinear dynamics and hold significant value for various engineering applications, which has led to growing research interest. Muthuswamy et al. [29] demonstrated that chaotic attractors can exist in an autonomous memristive circuit. The dynamical model of the simple autonomous memristive circuit proposed by Muthuswamy and Chua [29] is defined by the following equations: ˙x=y, ˙y=−x−β(z2−1)y, (1) ˙z=−y−αz +yz. In this model, the parameters satisfy α > 0 and β > 0. The state variables x(t)and y(t) represent the internal state variables of the memristor, while their rates of change (dx/dt) and (dy/dt)depend on the applied voltage (v), the resulting current (i), and the current values of the state variables. This dependency indicates that the internal state evolves based on the circuit’s electrical history.The third equation ( ˙z)establishes the relationship between the instantaneous voltage and current passing through the memristor, which is modulated by the current values of the internal state variables x(t)and y(t). The system (Eq.1) has a single equilibrium point at (x∗, y∗, z∗) = (0,0,0). From the linear stability analysis, the eigenvalues of the equilibrium point (0,0,0) are determined as:λ1= −αand λ2,3= (β±β2+ 4)/2. For λ1=−α, if α > 0, then λ1=−α < 0(indicating stability), and if α < 0, then λ1=−α > 0 (indicating instability). For λ2,3, the discriminant β2+ 4 >0ensures that the roots are real and distinct. One root is positive (β+β2+ 4)/2, and the other is negative (β−β2+ 4)/2. Since Nonlinear Phenomena in Complex Systems Vol. 28, no. 3, 2025 300 M. Anisha Nashrin, S. Siva Sakthi Pitchammal, S. M. Abdul Kader, A. Zeenath Bazeera, and V. Chinnathambi there is always at least one positive eigenvalue, the equilibrium point (0,0,0) is classified as an unstable saddle point.The dynamical model of the non-autonomous version of the simple memristive circuit is defined as follows ˙x=y+F(t), ˙y=−x−β(z2−1)y, (2) ˙z=−y−αz +yz. where F(t)represents an external periodic driving signal that is time-dependent 3. Types of periodic excitations In this study, we consider four types of external periodic signals: sine waves, square waves, symmetric sawtooth waves, and asymmetric sawtooth waves. The mathematical representations of these periodic signals are given by: Fsin(t) = Fsin (t+ 2π/ω) = fsin ωt, (3) Fsq(t) = Fsq(t+2π/ω) ={f, 0< t < π/ω −f, π/ω < t < 2π/ω, (4) Fsst(t) =      4ft T0,0< t < π/2ω −4ft T0+ 2f, π/2ω < t < 3π/2ω 4ft T0 −4f, 3π/2ω < t < 2π/ω, (5) Fast(t) = {2ft T0,0< t < π/ω 2ft T0 −2f, π/ω < t < 2π/ω, (6) where T0= 2π/ω and tis taken as mod (T0). 4. Dynamical behaviors of the circuit 4.1. Bifurcations and chaos In this section, we numerically investigate the nonlinear dynamics of a simple memristive -1.0 -0.5 0.0 0.5 1.0 0 3 6 1 0 -1 (d) F(t) t 036 (c) -1 1 0 F(t) t 0 3 6 (b) -1 0 1 F(t) t 36 (a) 0 F(t) t FIG. 1: Waveforms of various periodic signals: (a) sine wave, (b) square wave, (c) symmetric sawtooth wave, and (d) asymmetric sawtooth wave. For all signals, the period is T= 2π/ω, with ω= 1 and amplitude f= 1. circuit subjected to various periodic signals, including sine waves, square waves, and symmetric and asymmetric sawtooth waves. For our numerical study, we fix the parameter values as α= 0.6, β = 0.5and ω= 1. Eq.(2) is solved using the fourth-order Runge-Kutta method with a time step of 2π/ω. The numerical solution corresponding to the first 500 drive cycles is treated as transient and discarded. We analyze the nonlinear dynamics of the system (Eq.2) by varying the signal amplitude for each periodic input. Figure 2 presents the bifurcation diagrams and the corresponding maximal Lyapunov exponent (λm)diagrams for the different periodic signals. The maximal Lyapunov exponent (λm)is calculated using the algorithm described in Ref.[30]. Our analysis reveals notable similarities and differences in the bifurcation patterns as the forcing amplitude fis varied. We first examine the effect of the sine wave signal fsin(ωt). Figure 2(a) shows the bifurcation diagram of the system (Eq. 2) driven by the periodic sine wave signal, while Figure 2(b) presents the corresponding maximal Lyapunov exponent diagram. As the amplitude fincreases Нелинейные явления в сложных системах Т. 28, № 3, 2025 Bifurcation and Hysteresis Analysis in a Memristor Circuit under Periodic Excitations 301 0123456 -10 -5 0 0 1 2 3 4 5 6 -0.3 -0.2 -0.1 0.0 0.1 (d)square-wave λ f 0 1 2 3 4 5 6 -14 -7 0square-wave (c) x f 0123456 -0.3 -0.2 -0.1 0.0 0.1 (b) sine wave λ f sine wave (a) x f 0123456 -8 -6 -4 -2 0 2 0 1 2 3 4 5 6 -6 -3 0 3(g) Asy. saw-tooth wave x f 0 1 2 3 4 5 6 -0.3 -0.2 -0.1 0.0 0.1 (h)Asy. saw-tooth wave λ f 0 1 2 3 4 5 6 -0.3 -0.2 -0.1 0.0 0.1 (f) sym. saw-tooth wave x f (e) sym. saw-tooth wave x f FIG. 2: Bifurcation diagrams and the corresponding maximal Lyapunov exponent diagrams of the simple memristive circuit driven by different periodic signals. The parameter values are set as α= 0.6, β = 0.5, and ω= 1.0. -4 -2 0 2 -1 0 1 2 -4 -2 0 2 -3 -2 -1 0 1(d) z x -1 0 1 2 -3 -2 -1 0 1(c) z y -4 -3 -2 -1 0 1 -1.5 0.0 1.5 f = 0.4217688 (b) y x (a) f = 0.4217688 y x FIG. 3: Phase portraits of the system (Eq. 2) driven by a sine wave signal, showing chaotic orbits in (a) the x−yplane, (c) the y−zplane, and (d) the x−zplane. (b) Poincar´e map corresponding to Fig. 3(a). The parameter values are set as α= 0.6, β = 0.5, and ω= 1.0. from zero, chaotic orbits emerge and persist up to f= 0.421769. Beyond this critical value fc, chaotic orbits continue to exist over a range of fvalues, followed by a reverse period-doubling bifurcation. At f= 4.008163, this reverse perioddoubling bifurcation vanishes, and the system eventually settles into periodic behavior. This indicates that the application of a sine wave signal in the system (Eq. 2) can be used to suppress chaotic motion by selecting values of fwithin the interval 0.421769 < f < 6.0. Figure 2 further reveals that chaotic regions are interspersed with Nonlinear Phenomena in Complex Systems Vol. 28, no. 3, 2025 302 M. Anisha Nashrin, S. Siva Sakthi Pitchammal, S. M. Abdul Kader, A. Zeenath Bazeera, and V. Chinnathambi -4 -2 0 2 -2 0 2 -3 -2 -1 0 1 -1 0 1(f) f = 0.677556 y x -3.0 -1.5 0.0 1.5 -1 0 1 2(e) f = 0.677556 y x -2 -1 0 1 -1 0 1(d) f = 0.3688029 y x -3.0 -1.5 0.0 1.5 -1 0 1 2(c) f = 0.3688029 y x -4 -3 -2 -1 0 1 -1.5 0.0 1.5 (b) f = 0.3192262 y x (a) f = 0.3192262 y f FIG. 4: Phase portraits and corresponding Poincar´e maps of the system (Eq. 2) driven by different periodic signals: (a, b) square wave, (c, d) symmetric sawtooth wave, and (e, f) asymmetric sawtooth wave, illustrating chaotic orbits in the x−yplane. The parameter values are set as α= 0.6, β = 0.5, and ω= 1.0. five main periodic windows. When the sine wave signal fsin(ωt)is replaced by other periodic signals, such as square waves, symmetric sawtooth waves, or asymmetric sawtooth waves, a similar dynamic behavior is observed. The bifurcation structures and the corresponding maximal Lyapunov exponent diagrams for these signals are shown in Figures 2(c-h). Despite the general similarity in bifurcation patterns, the ranges of chaotic and periodic behavior differ depending on the type of periodic signal. Specifically, the values of f at which various bifurcations occur vary for each signal type. These ranges are summarized in Table 1. Notably, as fincreases from a small value, a periodic orbit is observed much earlier for the square wave signal, whereas it appears at a relatively higher value of ffor the asymmetric Нелинейные явления в сложных системах Т. 28, № 3, 2025 Bifurcation and Hysteresis Analysis in a Memristor Circuit under Periodic Excitations 303 -4 -2 0 2 -2 0 2 -12 -8 -4 0 4 -2 0 2 4 (d) f = 5.0 y x -8 -4 0 4 -2 0 2 4 (c) f=3.0 y x -5 -4 -3 -2 -1 0.0 0.8 (b) y x (a) f = 1.0 y x FIG. 5: Phase portraits of the system (Eq. 2) driven by a sine wave signal, illustrating periodic orbits: (a) period-4T, (c) period-2T, and (d) period-T. (b) Poincar´e map corresponding to Fig. 5(a). The parameter values are set as α= 0.6, β = 0.5, and ω= 1.0. 0 100 200 300 -4 -2 0 2 0 1 2 3 -10 -5 0 5 10 (h) f=4.0 x t 0 1 2 3 -4 -2 0 2(g) f=0.677556 x t 0 1 2 3 -5 0 5(f) f = 6 x t 0 3 6 -2.0 -1.5 -1.0 -0.5 0.0 0.5 (e) f=0.3688029 x t 0 1 2 3 -4 -2 0 2(c) f=0.3192262 x t 0 1 2 3 -7 0 7(d) f = 2.0 x t 0 100 200 300 -5 -4 -3 -2 -1 (b) f = 1.0 x t (a) f = 0.3 x t FIG. 6: Time evolution of the memristor circuit (Eq. 2) driven by various periodic signals: (a, b) sine wave, (c, d) square wave, (e, f) symmetric sawtooth wave, and (g, h) asymmetric sawtooth wave. The left column illustrates chaotic behavior, while the right column shows periodic behavior. The parameter values are set as α= 0.6, β = 0.5, and ω= 1.0. Nonlinear Phenomena in Complex Systems Vol. 28, no. 3, 2025 304 M. Anisha Nashrin, S. Siva Sakthi Pitchammal, S. M. Abdul Kader, A. Zeenath Bazeera, and V. Chinnathambi -18 -12 -6 0 6 -3 0 3 6 -18 -12 -6 0 6 -3 0 3 6(d) Asy.saw-tooth wave y x -18 -12 -6 0 6 -3 0 3 6(c) sym. saw-tooth wave y x -18 -12 -6 0 6 -3 0 3 6(b) square-wave y x (a)sine wave y x FIG. 7: Phase portraits of the system (Eq. 2) driven by different periodic signals at f= 6.0. The parameter values are set as α= 0.6, β = 0.5, and ω= 1.0. 0.7 0.8 0.9 1.0 1.1 1.2 1.3 -5 -4 -3 0.9 1.2 1.5 1.8 2.1 -5 -4 -3 -2 (d) Asy.saw-tooth wave x f 0.6 1.2 1.8 -5 -4 -3 -2 (c) Sym.saw-tooth wave x f 0.4 0.8 1.2 -5.4 -4.8 -4.2 -3.6 -3.0 -2.4 (b) square-wave x f (a) sine wave x f FIG. 8: Bifurcation diagrams of the system (Eq. 2) driven by various periodic signals, illustrating antimonotonicity behavior. The parameter values are set as α= 0.6, β = 0.5, and ω= 1.0. sawtooth wave signal. We further analyzed the system’s behavior using time evolution plots. In the case of periodic orbits, the time series exhibits a regular, repeating waveform with consistent peaks and troughs occurring at fixed intervals. In contrast, the time series of chaotic orbits appears irregular and nonrepetitive. Figure 6 provides examples of the time evolution of the system (Eq. 2) driven by the different periodic signals at various values of f. Specifically, chaotic oscillations are shown in Figures 6(a), 6(c), 6(e), and 6(g), while periodic oscillations are displayed in Figures 6(b), 6(d), -0.8 0.0 0.8 -0.03 0.00 0.03 -0.3 0.0 0.3 -0.03 0.00 0.03 (d) ω =1.0 ω =1.2 ω =1.5 Asy.saw-tooth wave y x -0.3 0.0 0.3 -0.02 0.00 0.02 ω =1.2 ω =1.0 ω =0.8 (c) Sym.saw-tooth wave y x -0.8 0.0 0.8 -0.06 0.00 0.06 ω =1.0 ω =1.2 ω =1.5 (b) square wave y x sine wave (a) ω=0.8 ω=1.0 ω=1.2 y x FIG. 9: Pinched hysteresis curves of the simple memristor circuit driven by various periodic signals at three different frequencies ω. The parameter values are set as α= 0.6, β = 0.5, and ω= 1.0. 6(f), and 6(h).As the amplitude fis further increased, the long-term behavior of the system eventually settles into periodic motion for all the signals. At f= 6.0, the system consistently exhibits periodic behavior with a period of T. Figure 7 presents the phase portraits at f= 6.0 for the different periodic signals, where cross-well period-Tbehavior is observed for all signals. 4.2. Antimonotonicity phenomenon In many nonlinear dynamical systems, periodic orbits can be both created and destroyed through reverse period-doubling bifurcation sequences as a system parameter is varied monotonically. This phenomenon is known as antimonotonicity, a term introduced by Dawson et al. [31]. Unlike other behaviors, antimonotonicity cannot be eliminated by simply changing variables or reparameterizing the system, making it a fundamental characteristic of certain chaotic systems. It has significant implications for Нелинейные явления в сложных системах Т. 28, № 3, 2025 Bifurcation and Hysteresis Analysis in a Memristor Circuit under Periodic Excitations 305 Table 1: Summary of bifurcation phenomena of the simple memristor circuit in the presence of different shape of periodic signals with α= 0.6, β =.0.5and ω= 1.0. Force Value of fNature of solution sine wave 0< f < 0.421769 Chaos 0.421769 < f < 4.008163 RPD 4.008163 < f < 6.0Period-T orbit square-wave 0< f < 0.319226 Chaos 0.319226 < f < 2.63404 RPD 2.63404 < f < 6.0Period-T orbit SST wave 0< f < 0.3688037 Chaos 0.368803 < f < 5.129383 RPD 5.129383 < f < 6.0Period-T orbit AST wave 0< f < 0.677556 Chaos 0.677556 < f < 4.623124 RPD 4.623124 < f < 6.0Period-T orbit experimental studies that focus on the fine structure of chaotic systems. Dawson et al. [31] proposed a geometric mechanism involving dimple formations as a possible explanation for antimonotonicity in one-dimensional maps containing two critical points within the chaotic attractor. This phenomenon has been observed both experimentally and through numerical simulations in various nonlinear physical systems [32-35]. To demonstrate the existence of antimonotonicity in the memristor circuit driven by various periodic signals, we computed bifurcation diagrams depicting the maxima of x(t)versus the parameter ffor specific values of αand β, as shown in Figure 8. The characteristic primary bubble structure associated with antimonotonicity is clearly visible in Figure 8, providing strong evidence of this phenomenon in the memristive circuit. 4.3. Pinched hysteresis loop behavior under periodic excitations The pinched hysteresis loop is a hallmark of memristor circuits, representing the nonlinear relationship between voltage and current that arises from the system’s memory effect. This behavior is fundamentally linked to the evolution of the memristor’s internal state variable, which changes over time in response to the applied voltage or current. The resulting modulation of resistance (or conductance) leads to a non-instantaneous response, producing the characteristic pinched hysteresis loop in the voltage-current (v-i) plane. Recently, similar hysteretic responses have been reported in diverse systems. For instance, Elwaki et al. [20] observed a pinched hysteresis loop in nonlinear resonators, while Peng et al. [19] demonstrated such behavior in a discrete memristor-based system with fractional-order differences. These studies underscore the broader relevance of pinched hysteresis beyond conventional memristive elements. To explore this phenomenon in our proposed system described by Eq. (2), we examined its response to four different periodic excitations: sine, square, symmetric sawtooth, and asymmetric sawtooth waveforms. The resulting voltage-current hysteresis loops are presented in Fig. 9, plotted in the x-y plane for three different values of the excitation frequency. The simulation results reveal that the system exhibits waveform-dependent memristive behavior. Each excitation type leads to a distinct loop shape, highlighting the sensitivity of the dynamics to the nature of the input signal. Furthermore, an increase in input frequency leads to a corresponding increase in both the area and the geometric complexity of the hysteresis loops. This frequency dependence is a well-known characteristic of memristive systems and further validates the memristive nature of the proposed model. 5. Conclusion In this paper, we numerically analyzed the dynamical behavior of a simple memristive circuit driven by various periodic signals, including sine, square, symmetric sawtooth, and asymmetric sawtooth waveforms. A range of numerical toolsNonlinear Phenomena in Complex Systems Vol. 28, no. 3, 2025 306 M. Anisha Nashrin, S. Siva Sakthi Pitchammal, S. M. Abdul Kader, A. Zeenath Bazeera, and V. Chinnathambi such as bifurcation diagrams, phase portraits, Poincar´e maps, time series analysis, and maximal Lyapunov exponent (MLE) diagrams-was employed to investigate the circuit’s dynamic responses. The non-autonomous memristor circuit exhibited rich dynamical behavior under these different excitations, revealing both chaotic and regular regimes. Bifurcation diagrams showed that chaotic regions, interspersed with periodic windows, appear within a limited parameter range due to the nature of the input signals. Periodic orbits emerged at lower frequency values for the square-wave input, while for the asymmetric sawtooth waveform, they appeared at relatively higher frequency values. Furthermore, both the MLE and bifurcation diagrams indicated the presence of coexisting attractors and the phenomenon of antimonotonicity. The characteristic pinched hysteresis loop, a hallmark of memristive systems, was observed for all the applied signals. Notably, with increasing input frequency, both the area and complexity of the hysteresis loops increased. This enhancement suggests that the internal state variable dynamics-especially the nonlinear coupling terms in the governing equations-may induce frequency-amplified memory effects. Overall, the study highlights the potential of the memristor as a nonlinear element in chaotic circuit design. Specifically, the proposed circuit could find applications in chaotic encryption and secure communication systems. 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