Simulation and Analysis of the Second-Order Memristive System in the CUDAynamics Suite

Abstract

Cycle-to-cycle variability of switching parameters inherent to memristive devices introduces significant problems in the design of neuromorphic systems and non-volatile memory. This study investigates the dynamics of a second-order memristive system incorporating capacitive effects that model parasitic charge within individual memristors, addressing both the technical need for accurate analysis of complex regimes and the demand for exploratory environments. Simulations were performed using CUDAynamics, an interactive software suite developed by the authors, which utilizes parallel computing, primarily via NVIDIA Compute Unified Device Architecture (CUDA). It integrates multiple analysis tools for dynamical systems, including bifurcation diagrams, the largest Lyapunov exponent and periodicity mapping, and interactive navigation in multidimensional parameter spaces. The memristive system was discretized applying multiple integration methods with a fixed time step and various waveforms of the input signal. Analysis tools revealed well-defined regions of chaotic dynamics in the memristor resistance parameter space as functions of input signal properties. Sinusoidal and triangular waveforms produced topologically similar distributions of dynamical regimes, whereas the square waveform, mimicking digital inputs, generated distinct dynamical patterns while still preserving chaotic trajectories under specific conditions. Interactive visualization capabilities of CUDAynamics effectively demonstrate attractor evolution and hysteresis deformation, providing immediate visual feedback that significantly enhances conceptual comprehension of nonlinear feedback mechanisms. Beyond its practical implications for the design of analog and digital memristive devices, CUDAynamics offers a scalable, open-source toolkit to aid researchers and engineers in exploring complex dynamical phenomena.

1. Introduction

The memristor, theoretically predicted by L. Chua in 1971 [1] and experimentally realized in 2008 [2], has since been recognized as a foundational element for neuromorphic computing architectures and next-generation non-volatile memory. Unlike conventional passive components, the memristor exhibits resistance memory dependent on the previous current flow history, effectively serving as a physical analog of biological synaptic plasticity [3,4,5] and neuronal membrane excitability [6]. These intrinsic memory properties have catalyzed extensive research into memristor-based chaotic oscillators, adaptive filters, and bio-inspired neural networks.

Despite their promise, the practical deployment of memristor technologies is hindered by significant challenges in parameter reproducibility and cycle-to-cycle switching variability [7,8,9,10]. Recent advances in resistive switching device modeling have emphasized the critical role of device-to-device and cycle-to-cycle variability in shaping nonlinear dynamical responses [11,12,13]. Concurrently, variability-aware compact models have been developed to bridge experimental characterization and circuit-level simulation [14,15]. These effects manifest both at the single-device level and within crossbar arrays, substantially complicating the design of robust neuromorphic architectures. The primary source of such dynamical instability arises from parasitic capacitive effects caused by charge accumulation in the active memristive layer and interfacial structures [16,17,18]. Incorporating these capacitive components into circuit models increases system dimensionality and frequently induces complex nonlinear phenomena, including multistability, superperiodic oscillations, and deterministic chaos. Recent studies have demonstrated that even under simple harmonic excitation capacitive-augmented memristive models can exhibit robust chaotic dynamics [19,20,21]. Concurrently, memristive models employing built-in nonlinear generators (e.g., Duffing-type oscillators) to achieve controllable chaotization are being actively investigated [22,23,24,25,26,27]. Such regimes are highly valuable for applications in secure communications, cryptographic pseudorandom bit generation, high-sensitivity sensing, and neural networks [28,29,30,31].

Investigating these high-dimensional nonlinear systems demands extensive numerical exploration across multidimensional parameter spaces. Contemporary research increasingly relies on parallel computing architectures, particularly graphics processing units (GPUs), to accelerate the generation of bifurcation diagrams, Lyapunov exponent maps, and synchronization thresholds [32,33,34,35,36]. These workflows are increasingly integrated for the rapid and secure handling of large data sets [37]. Complementary to high-throughput simulation, recent advances in intelligent monitoring and fractional-order multi-rate data fusion provide robust frameworks for real-time state estimation and uncertainty correction in complex nonlinear systems [38]. Integrating such methodological concepts into the analysis of memristive dynamics represents a promising direction for bridging numerical exploration with experimental validation. Complementary to high-throughput simulation, recent advances in intelligent monitoring and fractional-order multi-rate data fusion provide robust frameworks for real-time state estimation and uncertainty correction in complex nonlinear systems [38,39,40,41]. Integrating such methodological concepts into the analysis of memristive dynamics represents a promising direction for bridging numerical exploration with experimental validation. However, existing computational tools are predominantly designed for static, batch-processing workflows and rarely support real-time, interactive parameter exploration [32,42]. This paper introduces CUDAynamics, a computational framework developed for interactive analysis of chaotic and multistable dynamical systems. The platform employs a hybrid parallelization strategy, integrating NVIDIA’s Compute Unified Device Architecture (CUDA) with Open Multi-Processing (OpenMP) to achieve high-throughput numerical simulations. This computational foundation enables rapid exploration of parameter spaces, thereby accelerating hypothesis testing and dynamical regime identification in nonlinear models. User interaction is facilitated through a responsive, immediate-mode graphical interface built upon the Dear ImGui and ImPlot libraries. By decoupling interface rendering from background numerical computations, the system allows researchers to dynamically modify system parameters, initial conditions, and model topologies. In addition to a diverse built-in library of nonlinear systems, CUDAynamics incorporates several analysis tools employing the program’s parallel computational framework, such as the calculation of the largest Lyapunov exponent, periodicity and lifetime of chaotic transients. Visual outputs, including time series, phase portraits, bifurcation diagrams, and heatmaps, are updated continuously, providing researchers with an intuitive platform for exploring nonlinear phenomena. Visual overview of the suite’s graphical user interface can be found in Appendix A. The software supports both rapid exploratory modeling and high-resolution, publication-quality diagram generation, all within a unified, user-friendly framework.

The primary objective of this study is to demonstrate the research utility of CUDAynamics through a systematic investigation of nonlinear dynamics in a second-order memristive system, incorporating capacitive effects that account for parasitic charge accumulation in physical devices [19]. The computational framework and associated numerical experiments were developed as part of an applied research project focused on the nonlinear circuit dynamics. The CUDAynamics program is publicly available at: https://github.com/solawk/cudaynamics, (accessed on 1 May 2026). Microsoft Visual Studio 2022 and CUDA Toolkit 12.6.2 constitute the prerequisite dependencies for compiling the source code.

The main contributions of the study are as follows:

• An open-source, GPU-accelerated computational environment (CUDAynamics) is introduced, featuring a responsive immediate-mode graphical interface that enables interactive, real-time navigation through high-dimensional parameter spaces of nonlinear systems.
• Leveraging this platform, a comprehensive numerical analysis of a second-order memristive system incorporating parasitic capacitance is conducted under sinusoidal, triangular, and square-wave excitations. This reveals the precise parameter configurations and waveform-dependent sensitivities under which deterministic chaos and superperiodic regimes emerge.
• A new finite difference scheme is implemented for the memristor using a composition diagonal method with variable symmetry (VSCD). Compared to conventional explicit schemes, it demonstrates enhanced numerical stability and introduces a tunable symmetry parameter that acts as an additional bifurcation control.

To clarify the narrative focus, the central contribution of this work is the development and validation of CUDAynamics as a scalable research tool. The memristor simulation, numerical scheme comparison, and chaos analysis constitute an applied case study that demonstrates the software’s capacity to resolve complex dynamical phenomena while providing new physical insights into waveform-driven chaotization.

The remainder of this paper is organized as follows. In Section 2, the mathematical formulation of the second-order memristive system is presented, alongside the numerical integration strategies in CUDAynamics. In Section 3, the computational results are detailed, including phase-space trajectory visualizations, and comprehensive bifurcation and stability analyses under sinusoidal, triangular, and square-wave excitations. In Section 4, the physical implications of the observed dynamical regimes are discussed and current methodological limitations are examined. Finally, Section 5 concludes the paper by summarizing the key contributions and outlining future directions for advanced nonlinear system identification.

2. Materials and Methods

We consider a second-order memristive system that includes an internal state variable x, ranging from 0 to 1 and describing the internal channel formation, and a state variable n that incorporates capacitance into the simple memristor model. The original set of equations of the studied second-order mixed system is as follows [19]:

$$\begin{gathered} V(t)=R(x)I(t)+\Delta V(x,n) \\ {\displaystyle \frac{dx}{dt}}=k_2{f}_w(x)I(t) \\ {\displaystyle \frac{dn}{dt}}=k_3{f}_v(n)I(t) \end{gathered}$$

(1)

where V(t) is the input voltage, I(t) is the current passing through the memristor, R(x) = R_ON x + R_OFF(1 − x) is the resistance of the memristor, with R_ON and R_OFF being resistances in states 1 and 0, respectively, and ΔV(x, N) is the function of lumped capacitive effects [19]:

$$\Delta V(x,n)=k_1{\displaystyle \frac{n}{x+a}},$$

(2)

f_ν(N) is the capture function [19]:

$$f_v(n)=\left\{\begin{array}{l}1-\left|{\displaystyle \frac{n}{n_{\max }}}\right|,n\in \left[-n_{\max },n_{\max }\right]\\ 0,n\notin \left[-n_{\max },n_{\max }\right]\end{array}\right.,$$

(3)

k₁, k₂, k₃, a and n_max are constants.

To implement and simulate the mixed second-order memristive system in CUDAynamics, we rewrite the model’s equations into the following:

$$\begin{gathered} I={\displaystyle \frac{V-\left(k_1\frac{n}{x+a}\right)}{R_{ON}\ast x+R_{OFF}\ast \left(1-x\right)}} \\ {\displaystyle \frac{dx}{dt}}=k_{2}I \\ x=\left\{\begin{array}{l}0,x<0\\ 1,x>1\end{array}\right. \\ {\displaystyle \frac{dn}{dt}}=k_3\ast \left\{\begin{array}{l}1-\left|{\displaystyle \frac{n}{n_{\max }}}\right|,n\in \left[-n_{\max },n_{\max }\right]\\ 0,n\notin \left[-n_{\max },n_{\max }\right]\end{array}\right. \end{gathered}$$

(4)

It is important to note that in the original equations [19] a window function f_w is present. It equals 1 when x is within the range [0, 1] and equals 0 otherwise. However, since the value of x is limited to this interval by definition, the function degenerates to a constant 1 and hence can be removed from the equation. If the domain of the state variable x is not limited by the range [0, 1] and the window function is present, then dx/dt becomes zero as soon as variable x leaves the interval. This completely stops the dynamics of the variable x due to lack of additive terms in the right-hand side of its differential equation.

The numerical implementation of the mixed second-order memristive model within the CUDAynamics framework, employing a finite difference scheme to solve Equation (4), is provided in Appendix A.

The stability properties and bifurcation behavior of composition methods with variable symmetry, including VSCD, have been rigorously characterized in our previous works [43,44]. These studies establish that the symmetry parameter acts as a controlled, structure-preserving perturbation: it can induce or suppress multistability without compromising the qualitative dynamics of the underlying continuous system.

Table 1. Comparison of numerical integration methods.

Method	EE	IE	EMP	IMP	VSCD	RK4
Order	1	1	2	2	2 (s = 0), 1 (general s)	4
Stability	Conditionally stable	A-stable	Conditionally stable	A-stable	A(α)-stable (s = 0), tunable via s	Conditionally stable
Preference region	Very small	Small	Moderate	Small	Reasonable size, tunable via s	Large
Symmetry	No	No	No	Yes (symplectic)	Yes (s = 0), tunable via s	No
Applicability to chaos	Prone to dynamical degradation	Suppress chaos due to numerical damping	Limited stability	Good for conservative systems, costly for large-scale	Robust, s serves as a bifurcation parameter	Robust, widely used as reference
Computational cost per step	1 RHS	Newton iterations, Jacobian	2 RHS	Newton iterations, Jacobian	2 RHS, analytical implicitness or simple iterations	4 RHS
Memory footprint	Minimal	High (Jacobian, iteration buffers)	Low (1 stage buffer)	High (Jacobian, iteration buffers)	Low (1 stage buffer)	Moderate (3 stage buffers)
References	[45]	[45]	[45]	[43,45]	[43,44]	[45]

summarizes the key qualities of the VSCD method in comparison with established numerical integration schemes: explicit and implicit Euler methods (EE and IE), explicit and implicit midpoint methods (EMP and IMP), as well as the Runge–Kutta method of fourth order (RK4). The computational cost per integration step is quantified by the number of evaluations of the right-hand side (RHS) function of the underlying system of ordinary differential equations required by a numerical method.

To achieve higher efficiency and robustness of the simulations, we used several numerical integration methods of higher order than the explicit Euler’s method employed in the original paper [19], including the RK4 method, the EMP method, and the VSCD method [44], which underlies the further presented simulation results. The distinctive feature of the latter method is that it consists of two adjoint parts: the first is a semi-explicit adjoint part, and the second is a diagonally implicit adjoint part (solved by two simple iterations). When the symmetry parameter is set to zero, which corresponds to midpoint evaluation over the integration step, the VSCD method falls within the class of symmetric integrators [46]. In addition to the sinusoidal input signal, which was the only one the memristive system has been previously tested with [19], we introduce a triangle wave and a square wave as new possible input signals, see Figure 1.

The considered system is available in the suite’s library under the name “Mixed 2nd-order memristor”. The finite difference schemes in CUDAynaemics are implemented as C++ compute kernels conforming to the CUDA C++ standard. Variables V()/Vnext(), parameters P(), and step-size H are batched in contiguous memory, with dedicated macros provided for efficient data access. The numb macro substitution enables arithmetic operations in either single- or double-precision floating-point formats. These kernels enable seamless execution via both CUDA and OpenMP, eliminating redundant code development without adding computational overhead.

The computational architecture of the CUDAynamics framework is built upon batched data processing via memory buffers. In interactive mode, the system employs a double-buffering strategy, wherein two buffers are computed alternately to enable continuous visual output without interruption. The size of the buffers is usually a compromise between data transfer overhead costs and memory capacity of the computing device. The suite possesses an immediate mode graphical user interface, capable of displaying multiple diagram types interconnected with each other. For example, initial value or parameter configurations can be swept through in a heatmap, updating the phase-space plots, time series and bifurcation diagrams accordingly. Trajectories in a phase-space plot can be assigned colors from a heatmap, both in the single trajectory “orbit” mode and in the trajectory ensemble “particle” mode. The whole suite works and updates together in real time, providing the tools for efficient iterative research.

For extended temporal analyses, CUDAynamics offers a high-resolution (Hi-Res) computational mode that optimizes resource utilization by disabling trajectory data transfer, thereby substantially reducing memory usage and execution time. The computational workload is partitioned into manageable chunks to facilitate scalable processing. As implied by its designation, this mode is specifically designed for the generation of high-resolution phase- and parameter-space heatmap diagrams. Furthermore, CUDAynamics supports command-line interface (CLI) execution for individual Hi-Res tasks, enabling efficient batch processing of multiple long-term simulations without manual intervention.

The largest Lyapunov exponent, serving as an indicator of chaotic behavior in the trajectory, is calculated in CUDAynamics via Wolf’s method [47,48]. The neighboring trajectory is created by displacing the original along the selected axis and is simulated alongside it. We have modified the algorithm by measuring the divergence/convergence rate for fixed time periods and resetting the distance between the trajectory points each time. This approach increased the robustness of the algorithm, implemented within the architecture of the simulation suite.

3. Results

The numerical experiments reported in this section were conducted using the CUDAynamics software suite on a workstation operating under Windows 11 Pro. The experimental platform comprised an Intel Core i7-13700K processor (16 cores, 24 threads, 5.4 GHz maximum turbo frequency), 32 GB of DDR5-6000 system memory, and an NVIDIA GeForce RTX 4070 GPU equipped with 12 GB of GDDR6X VRAM and 5888 CUDA cores.

This section is organized as follows: first, we validate the replication of [19] using the VSCD method; second, we demonstrate the interactive visualization capabilities of CUDAynamics; third, we present original results on waveform-dependent dynamics and symmetry-parameter bifurcations.

The second-order capacitive memristor model and its chaotic behavior under sinusoidal excitation were initially reported in [19], our work replicates these results using higher-order integration schemes to confirm their numerical robustness. Initial conditions for all variables are values of 0. Most of the parameter values have been provided in the paper [19]: a = 0.1, n_max = 10, k₂ = 46.5. The exact resistance values R_OFF and R_ON have not been listed in the source, so the values have been iteratively estimated to match the resulting figures with identical parameter sweep settings. We therefore assume the missing values R_OFF = 1 and R_ON = 20 by default, as we have managed to compute a qualitatively similar period diagram with these values. Parameter k₁ was swept from 0 to 0.05 and parameter k₃ was swept from 0 to 2500. VSCD method was used with a step size h = 0.001 in double precision by default. CUDAynamics is currently limited to using a fixed time step. While this is a noticeable limitation, it did not impede the conducted research and insight gain. While adaptive-step methods are superior for minimizing local truncation error in single-trajectory simulations, they introduce variable temporal resolution that complicates the consistent computation of global dynamical indicators (e.g., Lyapunov exponents, periodicity classification) across massive parameter grids. For the systematic mapping of bifurcation structures, uniform discretization ensures that the distribution of the computed points in parameter space corresponds to a consistent computational effort and temporal sampling, which is critical for reproducible heatmap generation. The variable for feature extraction is the current I, the neighborhood size is chosen as ε = 0.1. Simulation time consisted of 800,000 transient steps and 500,000 steps subjected to analysis. In Figure 2a comparison is presented between the simulation results with different numerical schemes. As can be seen, at an integration step-size h = 0.01, second-order methods lose stability. However, unlike EMP, the VSCD method preserves the proportions of the dynamical mode regions much better. At an integration step-size h = 0.001, the VSCD result is almost indistinguishable from the reference RK4 method.

Comparative accuracy analysis, results of which are provided in Figure 3, demonstrate an advantage of the VSCD method over EMP with the same timestep. The experiment was carried out by constructing diagrams with a resolution of 100 × 100. In Figure 3a, the error was derived as the sum of the difference in the periodicity values of the solution under consideration (VSCD and EMP) compared to the reference RK4 with the step-size h = 0.0001, divided by the diagram resolution. A stability loss is noticeable for the EMP method, unlike VSCD, which maintains accuracy over larger step-sizes. It is especially notable in the case of the structural similarity index measure [49] (SSIM) comparison in Figure 3b, where values closer to 1 are better. With the same accuracy (error e = 0.5165) the solution is achieved by the VSCD method with the step-size h = 0.001 and by the EMP method with the step size h = 0.005825. In CUDAynamics on a computer with the specified configuration, the runtime for constructing the corresponding periodicity diagram using the VSCD method is 16.07 s, and using the EMP method it is 17.51 s. Thus, the finite-difference scheme of the VSCD method is more efficient and also has additional symmetry parameterization.

By enabling instantaneous rendering of phase-space trajectories, CUDAynamics provides visual means to track the emergence and development of the hysteresis loop in the memristor’s I-V curve, as demonstrated in Figure 4.

Numerical simulations featuring triangular and square excitation waveforms were conducted using equivalent parameter sets. The resulting diagrams are presented in Figure 5 and Figure 6.

The location and extent of the memristor’s chaotic behavior regions in the resistances R_ON and R_OFF parameter space depend on the input signal’s amplitude Vamp, as demonstrated in the largest Lyapunov exponent diagram in Figure 7. To calculate the exponent, the value of variable x is displaced by 10⁻⁵ and the neighboring trajectory is simulated for 300 steps before every divergence/convergence rate measurement and the displacement distance reset. To measure the distance between the trajectories in the phase-space the variable values n and x are used. Consistent with theoretical expectations, increased resistance levels require a correspondingly higher excitation amplitude to trigger resistive switching. In the case of amplitude value variation, the location of the chaotic region depends linearly on the resistance value R_OFF, but depends nonlinearly on the value R_ON. As amplitude is increased, the shape of this region is compressed along the R_OFF parameter axis. At low Vamp values, the chaotic region exhibits a comparable morphology across all signal types; however, as Vamp increases, this region progressively narrows under square-wave excitation.

Within the investigated parameter space, high-periodicity regimes, termed “superperiodicity” [19], coexist with chaotic dynamics. Additionally, the relative R_OFF modulates the prominence of the S-shaped negative differential resistance (NDR) region observed during the memristor’s transition from the high-resistance to the low-resistance state, as illustrated in Figure 8.

Decreasing the input signal frequency Vfreq expands the chaotic regimes in the memristor’s parameter space, as shown in Figure 9. This expansion follows a linear trend along both the R_OFF and R_ON axes. Moreover, the morphology of these chaotic regions remains qualitatively similar under sine and triangular excitation, whereas square-wave forcing yields distinctly different structures.

The results presented in Figure 10 confirm that the introduced symmetry parameter inherent to the VSCD framework exhibits bifurcation-dependent behavior. The two-parameter diagrams show that varying the symmetry parameter nonlinearly alters dynamical regions across the entire k₁ plane, but only in the positive k₃ region, where it induces a transition from an initially stable periodic state to chaos. The one-dimensional diagram of the current I inter-peak intervals is intended to illustrate the bifurcation scenario with respect to the symmetry parameter.

4. Discussion

The obtained simulation results confirm the fundamental role of second-order capacitive dynamics in the emergence of complex regimes within memristive systems. Unlike first-order models, which are typically constrained to relaxation or periodic oscillations, the introduction of an additional state variable representing stored charge establishes the necessary conditions for deterministic chaos. The parasitic capacitive component effectively acts as a phase-shift mechanism, disrupting regular periodicity and driving the system into chaotic attractors. This structural transition underscores the necessity of incorporating higher-order parasitic effects into memristor modeling to accurately predict device behavior in neuromorphic and analog circuit applications.

It is important to emphasize that the dynamical regimes reported in this study were obtained through numerical simulation of the model proposed in [19]. While this model incorporates parasitic capacitive effects motivated by physical considerations [16,17,18,50], direct experimental validation against measured I-V curves of specific memristive devices falls outside the scope of the present methodological contribution. However, our research group has previously demonstrated experimental identification procedures for real memristive devices, including Knowm memristors [22] and threshold-switching selectors, using the dynamic route map approach and compact modeling techniques. The models identified in those works are already imported into CUDAynamics for hardware-calibrated simulation, establishing a clear pathway from numerical exploration to experimental verification.

The present findings demonstrate that chaotization is not a numerical artifact but an intrinsic structural property of the extended model. This conclusion is now supported by three convergent lines of evidence: (i) the consistency of dynamical regimes across integration methods of different orders and step-sizes (as shown in Figure 2), confirming numerical robustness; (ii) the preservation of chaotic attractors under variation of the symmetry parameter, in agreement with the stability theory for semi-implicit composition schemes [43]; and (iii) the theoretical framework for variable-symmetry integrators [44], which establishes that the additional capacitive state variable n introduces the minimal phase-space dimensionality required for Shilnikov-type homoclinic bifurcations in memristive systems [19].

Furthermore, this study reveals that the geometry of chaotic parameter regions is highly sensitive to the excitation waveform. Continuous signals, such as sinusoidal and triangular inputs, yield topologically similar bifurcation structures due to their smooth derivatives and absence of discontinuities. Conversely, square-wave excitation introduces abrupt voltage transitions that amplify nonlinear responses and compress the spatial extent of chaotic attractors. This waveform-dependent behavior has direct implications for both analog signal processing and digital switching architectures.

Aside from replicating and clarifying the simulation results from the work [19] with a numerical integration method of higher order, the results hold substantial practical value for hardware design. The identification of stable chaotic regimes facilitates the development of hardware-based pseudorandom number generators, where memristive oscillators offer superior integration density and lower power consumption compared to conventional complementary metal-oxide-semiconductor (CMOS) implementations. Additionally, the persistence of chaotic dynamics under square-wave inputs highlights a critical design consideration for digital memristive and neuromorphic circuits: unintended chaotization must be explicitly accounted for to ensure predictable switching and reliable memory retention. These findings provide design insights for memristive neuromorphic circuits and hardware-based cryptographic systems. While the present study establishes the computational framework and identifies waveform-dependent dynamical phenomena through high-resolution numerical analysis, subsequent work will focus on calibrating the simulated regimes against experimental data from physical memristive devices using identification methodologies developed in our prior studies [22].

The computational framework employed in this study, CUDAynamics, applies GPU-accelerated parallelization to overcome the limitations of traditional sequential simulators (e.g., simulation program with integrated circuit emphasis or SPICE), which are often ill-suited for high-dimensional parameter sweeps and real-time bifurcation analysis. By providing immediate-mode graphical feedback and enabling on-the-fly modification of system parameters, CUDAynamics transforms abstract research on nonlinear dynamics into an observable, exploratory process.

Despite these contributions, the current implementation relies on fixed-step numerical integration and lacks adaptive local error control, which may limit precision in stiff dynamical regimes. Future work will follow a three-phase implementation roadmap. First, to overcome the limitations of fixed-step integration, we will integrate an adaptive time-stepping scheme into CUDAynamics based on the variable-step modified VSCD method. A new framework will be adopted to couple a Hairer-type step-size controller with an adjustable symmetry parameter, enabling dynamic step adaptation while preserving the discrete system’s geometric and chaotic properties. Implementation will include tolerance-driven step control, local error estimation, and GPU-optimized extrapolation tables to reduce computational overhead in stiff regimes. Second, experimental validation will be conducted using our established parameter-identification workflow [22]. We will acquire I-V hysteresis and transient response data from physical Knowm memristive devices under identical sinusoidal, triangular, and square-wave excitations, followed by inverse optimization to quantify deviations between simulated and measured regimes. It should be noted that the memristor model employed here assumes deterministic threshold dynamics, whereas physical devices exhibit cycle-to-cycle variability due to filamentary stochasticity [10]. While this limitation does not invalidate the structural chaos observed in the idealized model, it implies that under parameter sweeps experimental results would likely display noisy boundaries rather than sharp bifurcation curves. Third, to facilitate engineering applications, the validated discrete models will be exported as SPICE-compatible macro-models and benchmarked in hardware-in-the-loop prototypes for neuromorphic applications. Each phase will be accompanied by open-source code updates, benchmark datasets, and reproducibility documentation to ensure practical utility for the nonlinear dynamics and memristive engineering communities.

5. Conclusions

The present study demonstrates that incorporating parasitic capacitive effects into a second-order memristor model serves as a sufficient condition for the emergence of deterministic chaos and superperiodic regimes, thereby validating and extending prior numerical investigations. Through high-order numerical integration in the CUDAynamics environment, we established that the geometry and spatial extent of chaotic parameter regions are highly sensitive to the excitation waveform. While continuous sinusoidal and triangular inputs yield topologically analogous bifurcation structures, square-wave excitation compresses chaotic attractors. The GPU-accelerated parallel architecture of the suite enabled rapid, high-resolution mapping of these multidimensional parameter spaces, overcoming the computational bottlenecks inherent in traditional sequential simulators and confirming the structural, rather than numerical, nature of the observed chaotization. These results yield essential design guidelines for memristive neuromorphic circuits and hardware cryptographic systems, wherein both intentional and emergent chaotic dynamics must be rigorously characterized to guarantee operational reliability.