Memory capacitors that never forget: a six-dimensional circuit settles into chaos

Chaos has a new home in an unusual corner of electronics. In a study published in the journal Discover Electronics, researcher I. O. Ajayi reports a detailed numerical and analytical investigation of a six-dimensional series-connected memcapacitor system, a circuit built around two nonlinear memcapacitive elements coupled through an active nonlinear feedback network. The analysis, published on 12 October 2026 as an open-access article, confirms that at the investigated parameter set the system behaves as a bounded deterministic chaotic oscillator with exactly one robust expanding direction in its dynamics, rather than the two or more that would mark it as hyperchaotic.

Memcapacitors are the capacitive cousins of memristors. Whereas an ordinary capacitor has a fixed capacitance, a memcapacitor’s capacitance depends on its internal state, which carries the imprint of everything the device has experienced. Its instantaneous electrical response depends not only on the present excitation but on the entire history of previous inputs. That memory introduces extra state variables into any circuit that contains it, and those extra degrees of freedom substantially enrich the nonlinear behaviour that the circuit can produce, from multistability to high-dimensional chaos.

The specific system examined here was proposed by Adelakun and colleagues, who connected two nonlinear memcapacitor emulators in series to create one of the first experimentally validated high-dimensional circuits relying purely on memcapacitive memory. The state vector of the model has six components: four electrical state variables of the active nonlinear circuit, labelled x, y, z and w, plus two internal memory-state variables, v1 and v2, associated with the two memcapacitive elements. The governing equations couple these variables through linear terms and nonlinear products such as y multiplied by powers of the memory variables, so the memory coordinates feed information from the system’s past directly back into its instantaneous electrical dynamics.

Before running any simulations, the new study established several structural facts analytically. Setting all state derivatives to zero reveals that the system does not possess a single isolated equilibrium point. Instead, it has a three-dimensional equilibrium manifold, a continuum of equilibrium states parameterised by the free coordinates w, v1 and v2, with x, y and z pinned at zero. The study also identified a conserved quantity: because both memory variables evolve according to the same equation, their difference v1 minus v2 never changes. This invariant constrains the accessible phase space and, as the numerical results later show, explains part of the structure of the Lyapunov spectrum.

Dissipation provides the third structural pillar. The divergence of the vector field, which measures how infinitesimal volumes of phase space contract or expand on average, is not constant in this system; it depends explicitly on the memory variables. The study therefore evaluated the time-averaged divergence along the actual trajectory. At the baseline operating point, with the parameter set adopted from the original study and the reference initial condition (0.6, 0.5, 0.45, 0.1, 0.5, 0.5), the mean divergence came out as approximately minus 0.5254, indicating net contraction of phase-space volume and supporting the formation of a bounded dissipative attractor.

The numerical protocol was deliberately conservative. All simulations used a classical fourth-order Runge–Kutta integrator with a step size of 0.001, a total integration time of 10,000, and removal of the first 5,000 time units as transient dynamics, leaving 5,000,001 retained points. The Lyapunov spectrum, the set of six numbers that quantify how rapidly infinitesimal perturbations grow or shrink along independent directions, was computed from the variational dynamics using periodic QR re-orthonormalisation to keep the tangent vectors numerically separable over long integrations. The validated spectrum at the baseline parameter set reads, in descending order: 0.08244, 0.00009897, 0.0000000258, minus 0.000125, minus 0.000347 and minus 0.6075.

Only one exponent exceeds the adopted positivity threshold of ten to the minus four, so the system is classified as chaotic rather than hyperchaotic. The separation between the leading exponent and the rest is striking: the largest exponent is roughly 833 times bigger than the second, which means the conclusion does not hinge on interpreting any borderline value as positive. The near-zero exponents are consistent with the expected neutral directions of the system, one belonging to the autonomous flow itself and a second arising from the conserved v1 minus v2 relation. The sum of all six exponents agrees with the independently computed time-averaged divergence to within an absolute discrepancy of only 5.878 times ten to the minus nine, an internal consistency check that strongly supports the reliability of the calculation.

Complementary diagnostics painted a coherent picture of the attractor. Time series after transient removal show all six state variables fluctuating indefinitely without settling to equilibrium or repeating periodically, yet remaining bounded: the largest excursion, in the variable x, reaches about 19.21, while the memory variables stay within about 2.59 in magnitude. The study notes these are empirical trajectory bounds rather than rigorous global bounds for every possible initial condition. Two- and three-dimensional phase portraits reveal elongated, folded structures in the electrical variables, the geometric signature of repeated stretching and folding, while the projection of the two memory variables collapses onto a diagonal line, a direct visual confirmation of the invariant. A Poincaré section at the plane z equals zero produced a structured, non-discrete set of intersections rather than the finite set expected of a periodic orbit, and the power spectral density of the signal showed broadband content rather than isolated spectral lines. Each of these observations was treated as supporting evidence rather than a stand-alone proof of chaos.

The convergence analysis reveals an honest hierarchy of confidence. The leading exponent remained positive across integration times of 500 to 10,000 time units, with screening estimates ranging from about 0.0725 to 0.0977, and across integration step sizes from 0.010 down to 0.001, where the calculated values were approximately 0.0744, 0.0725, 0.0883 and 0.0717. The sign of the dominant exponent is therefore robust, even though its precise finite-time value varies with the numerical choices. The smaller exponents, by contrast, fluctuate near zero and show much greater sensitivity, so the study explicitly refrains from assigning them precise asymptotic values or using them to infer a second expanding direction.

Physically, the picture that emerges is of chaos arising from a memory-mediated balance between stretching and contraction. Nonlinear feedback through the memcapacitive memory variables creates local exponential expansion along one direction, the negative divergence contracts phase-space volume in the complementary directions, and the invariant memory relation restricts the geometry of the accessible manifold. With a largest exponent of about 0.0824, small uncertainties in initial conditions amplify roughly as e raised to the power of 0.0824 times time before the bounded nonlinear dynamics limits their growth, producing deterministic unpredictability in which the future is fully specified by the equations yet practically sensitive to tiny initial errors. The study frames this as a foundation for future work on parameter-dependent transitions, attractor organisation, basin structure, multistability, hidden attractors, synchronisation and control, and notes potential relevance to secure signal generation and chaos-based information processing, while emphasising that application-specific performance would require separate engineering analysis. For now, the defensible conclusion is precise: at the validated baseline, this six-dimensional memcapacitive circuit is a bounded, dissipative, genuinely chaotic flow with one robust expanding direction, constrained memory geometry and strong transverse contraction.

Subject of Research: Numerical and analytical dynamical analysis of a six-dimensional series-connected memcapacitor system exhibiting bounded chaotic behaviour

Article Title: Chaotic dynamics, basin structure and global dynamical analysis of a six-dimensional series memcapacitor system

Article References: Ajayi, I. O. (2026). Chaotic dynamics, basin structure and global dynamical analysis of a six-dimensional series memcapacitor system. Discover Electronics, 3(1), Article 147. https://doi.org/10.1007/s44291-026-00297-6

Image Credits: AI Generated

DOI: 10.1007/s44291-026-00297-6

Keywords: memcapacitor, chaos, Lyapunov exponents, nonlinear dynamics, dissipative systems, attractor, Poincaré section, Runge–Kutta integration, memory circuit elements, multistability, phase-space contraction, electronic circuits