Complex Dynamics and Structure-Preserving Characteristics of a Three-Dimensional Discrete Cosine Memristive Map

A three-dimensional discrete map with state-dependent memristive feedback is constructed to examine complex dynamics persistence under finite parameter variations and perturbations. Fixed-point analysis, bifurcation diagrams, Lyapunov exponents (LEs), the 0-1 test, spectral entropy, and C0 complexity characterize the dynamics. At a=0.8, b=0.6, and c=0.4, positive finite-time maximum Lyapunov exponents support chaotic behavior while simulated trajectories remain finite. The original 1000-point scan yields 233 samples with λmax>10-3. A finer scan with step 0.0002 resolves four non-positive-LE samples within the previously reported positive-exponent region, and long-run validation retains their signs; thus, the finite grid does not establish a continuous chaotic interval. Feedback ablation distinguishes the combined model from isolated cosine and exponential components under the tested conditions. Initial-state diagnostics separate periodic initial-state equivalence and central inversion symmetry from translated attractor self-reproduction. Under selected parameter perturbations and Gaussian noise up to σ=10-3, the principal trajectory structure remains identifiable. Comparisons with the logistic and Hénon maps provide context. In National Institute of Standards and Technology (NIST) SP 800-22 testing, 146 of 148 non-overlapping template matching subtests meet the nominal 96/100 threshold, while two yield 95/100. These findings do not prove global boundedness, strict robust chaos, or cryptographic security.