Chaos Seeding in Nonlinear Systems

• Chaos seeding is the practice of specifying initial conditions, control parameters, or local perturbations that direct deterministic nonlinear systems into chaotic regimes.
• It is applied in chaos-based pseudorandom number generators by initializing both state and update rules to achieve sensitive dependence and desirable statistical properties.
• Chaos seeding is engineered across disciplines—from digital PRNGs to quantum and physical systems—to induce complex dynamics and control chaotic behavior for practical applications.

Chaos seeding denotes the specification of initial conditions, control parameters, or local perturbations that place a deterministic nonlinear system into a trajectory or regime whose later evolution exhibits sensitive dependence, mixing, or many-body spectral complexity. In the chaos-based PRNG literature, it usually refers to the variables that initialize a map, an asynchronous Boolean dynamical system, or a family of coupled maps; in exact-arithmetic constructions it also includes rules for selecting many non-merging initial points; in physical and quantum settings it often means tuning a device or applying a local defect so that chaos is generated autonomously rather than injected by noise (Wang et al., 2010, Saito et al., 2017, Xu et al., 22 Jun 2026, Santos et al., 2020).

1. Conceptual scope

The term is not used uniformly. In discrete chaotic-iteration generators, the seed is the initial Boolean state together with whatever hidden variables determine the strategy of component updates and the schedule of output extraction. In low-dimensional map-based generators, the seed is usually one or more initial real values and, in some designs, control parameters or coefficient increments. In physically realized chaos sources, “seeding” can shift from state initialization to parameter seeding: the system is tuned into a positive-Lyapunov or period-doubling regime, after which microscopic perturbations are amplified by deterministic dynamics. In many-body quantum systems, a single-site defect can act as a “speck of chaos,” locally breaking integrability and globally inducing chaotic spectral correlations (Wang et al., 2010, Orue et al., 2010, Irfan et al., 2024, Santos et al., 2020).

This heterogeneity matters because the mathematical object being seeded is different across subfields. Some papers seed an orbit on a fixed map; some seed both the orbit and the map law itself; some seed a family of exact irrational initial points for parallel streams; and some do not inject a random seed at all, but instead move a device into a regime where infinitesimal mismatches become dynamically relevant (Saito et al., 2017, Bahi et al., 2011, Xu et al., 22 Jun 2026).

2. Seed structures in chaos-based pseudorandom generators

In PRNG design, the seed is often larger than a single scalar. Several representative constructions make this explicit.

System	Independent seed components	Seed role
Discrete chaotic iterations PRNG (Wang et al., 2010)	$x^0$, $y^0$	$x^0$ initializes Boolean state; $y^0$ drives the logistic map that generates strategy $S^n$ and spacing $m^n$
Trident (Orue et al., 2010)	$x_0,y_0,z_0$, initial coefficients, coefficient increments, optionally $s$	Initialization fixes both starting states and the evolving law of three coupled maps
Cotangent-square PRNG (Mondal et al., 2012)	$x_0$	The initial real state is expanded into bytes through iteration and IEEE-754 extraction
Robust chaotic tent map (Irfan et al., 2024)	$x_0$, $y^0$0	$y^0$1 are derived from $y^0$2; thresholding at $y^0$3 yields bits
CryptoChaos (Song et al., 11 Apr 2025)	passphrase	The passphrase seeds four maps before SHA3-256, X25519, Blake3, and HKDF fusion

A recurrent pattern is that one seed component often controls more than one downstream object. In the discrete chaotic-iteration generator based on vectorial negation, the effective seed is the pair $y^0$4, because $y^0$5 determines both the strategy sequence $y^0$6 and the return-spacing sequence $y^0$7, while $y^0$8 initializes the Boolean state (Wang et al., 2010). In the robust chaotic tent map, the independent seeded quantities are $y^0$9 and $x^0$0, whereas $x^0$1 and $x^0$2 are derived quantities, not extra independent secret inputs (Irfan et al., 2024). In Trident, initialization is still broader: security-relevant state includes not only $x^0$3, but also the initial coefficients and their per-step increments, because those increments continually alter the affine maps and therefore the future orbit structure (Orue et al., 2010).

This suggests that chaos seeding in PRNGs is often state-plus-law initialization rather than mere placement of a point in state space. That reading is explicit in the coupled-map and chaotic-iteration papers, where the seed fixes not only where evolution starts but also which update mechanism is repeatedly applied (Orue et al., 2010, Bahi et al., 2011).

3. Exact admissibility, non-merging seeds, and engineered seed spaces

A distinct line of work treats chaos seeding as a mathematically controlled selection problem. The Bernoulli-map generator on cubic algebraic integers does not merely ask for a chaotic initial point; it constructs seed families represented exactly by integer triples $x^0$4 so that the corresponding irrational roots are nearly uniformly distributed in $x^0$5 and the resulting orbits do not merge (Saito et al., 2017). The admissible set is defined by

$x^0$6

and the practical family

$x^0$7

is used because, for large $x^0$8, the corresponding roots become almost equidistant. Non-merging is guaranteed by choosing source points, characterized by explicit parity and modular conditions; the simplest sufficient recipe is to choose $x^0$9 and $y^0$0 with opposite parity (Saito et al., 2017). Here, chaos seeding is not just initialization but controlled multi-stream placement in phase space.

The chaotic-iteration literature introduces a different engineering layer. In these generators the effective initialization consists of an internal Boolean state $y^0$1, hidden auxiliary PRNG states that generate strategy and iteration-count sequences, and an iterate function $y^0$2. The decisive theorem is that $y^0$3 is chaotic according to Devaney iff the iteration graph $y^0$4 is strongly connected (Bahi et al., 2011, Bahi et al., 2011). This shifts part of “seeding” from the runtime seed to the design-time choice of $y^0$5: selecting $y^0$6 seeds the topology of reachable asynchronous updates. The later balanced-map constructions preserve this by modifying the image vector of vectorial Boolean negation under pairing constraints such as

$y^0$7

so that new maps remain both strongly connected and statistically balanced (Bahi et al., 2011).

Trident pushes the same idea into arithmetic parameter classes. For the full perturbed map, the reported best parameter class is

$y^0$8

$y^0$9

and the three coupled maps are required to use different coefficients and increments (Orue et al., 2010). Here, admissible seeding is constrained by modular arithmetic because the aim is not only sensitivity to initial conditions, but also long digital periods and avoidance of short cycles.

4. Sensitivity, statistical validation, and cryptographic limits

Most chaos-seeding papers justify their designs through sensitivity and statistical tests. The discrete chaotic-iteration PRNG states that chaos “implies a deterministic but unpredictable system very sensitive to its initial conditions,” and argues that different initial values should have zero cross-correlation; under its six extraction schemes, Scheme 6 passes all 15 NIST SP800-22 tests using 100 sequences of 1,000,000 bits each (Wang et al., 2010). The cotangent-square PRNG treats $S^n$0 as the seed, gives the example that $S^n$1 and $S^n$2 produce byte streams differing from the third byte onward, and reports passing Diehard and NIST batteries after extracting the lower six bytes of each IEEE-754 double state (Mondal et al., 2012). The robust chaotic tent map studies perturbations of size $S^n$3 in either $S^n$4 or $S^n$5, reports nearly zero inter-sequence correlations under tiny seed changes, and claims NIST, ENT, and TestU01 success for thresholded bitstreams (Irfan et al., 2024). The CIPRNG optimization work is more explicit that chaotic iterations act as a post-treatment layer: Old and New CIPRNG improve many defective generators to NIST $S^n$6 and DieHARD $S^n$7, whereas the Xor variant improves quality progressively as the functional power $S^n$8 increases (Bahi et al., 2017).

At the same time, the literature repeatedly stops short of a modern cryptographic seeding treatment. The discrete chaotic-iteration PRNG based on $S^n$9 does not quantify practical seed entropy, collision probability, backtracking resistance, or resistance to adversarial seed guessing, and its worked example derives both components from the decimal part of the current time (Wang et al., 2010). The cotangent-square generator gives no min-entropy estimate for clock-based seeding and notes that reproducibility depends on floating-point implementation and transcendental libraries (Mondal et al., 2012). The robust chaotic tent map claims an effective key space near $m^n$0, but its count includes $m^n$1 and $m^n$2, which are derived from $m^n$3, not independent seed variables (Irfan et al., 2024). CryptoChaos makes the architectural move of seeding four maps from a passphrase and then fusing the result with an ephemeral X25519 shared secret through SHA3-256, Blake3, and HKDF, but it does not specify how the passphrase is converted into each map’s initial condition or parameter set (Song et al., 11 Apr 2025).

A plausible implication is that many chaos-based schemes use seeding as an entropy-expansion front end or state-diversification mechanism, while leaving seed ingestion, nonce handling, and weak-seed analysis to implicit assumptions or to conventional cryptographic primitives layered on top.

5. Parameter-seeded chaos in physical and quantum systems

Outside PRNGs, chaos seeding often means creating a control regime in which deterministic dynamics amplify infinitesimal perturbations. In the hybrid molecular optomechanical system formed by a plasmonic nanocavity coupled to a whispering-gallery-mode resonator, the dominant chaos-enabling knob is the plasmon–vibration coupling $m^n$4, while the WGM detuning $m^n$5 and inter-cavity coupling $m^n$6 open, shift, or suppress chaotic windows (Xu et al., 22 Jun 2026). The semiclassical equations

$m^n$7

$m^n$8

lead to a period-doubling route to chaos as $m^n$9 increases. No special random seed is injected; instead, the system is parameter-seeded into a positive-$x_0,y_0,z_0$0 regime.

A closely related hardware perspective appears in weakly coupled GaAs/(Al,Ga)As superlattices. There, deterministic chaos is generated under dc bias by the nonlinear feedback among resonant tunneling, local charge accumulation, and the global voltage constraint. The paper shows that 10-period structures with higher barriers exhibit stronger and wider deterministic chaos than earlier 50-period devices, and reports that in the 10-period GaAs/Al$x_0,y_0,z_0$1Ga$x_0,y_0,z_0$2As superlattice the lowest harmonic can reach frequencies up to $x_0,y_0,z_0$3 GHz (Ruiz-Garcia et al., 2016). In this setting, chaos seeding is bias-triggered rather than value-initialized.

Physical white-chaos generation by optical heterodyning of two delayed-feedback lasers provides yet another variant. The beat signal

$x_0,y_0,z_0$4

becomes spectrally white when the two external-cavity delays are unequal and effectively non-resonant, suppressing both relaxation-oscillation signatures and cavity-delay fingerprints (Wang et al., 2014). Experimentally, the resulting white chaos shows $x_0,y_0,z_0$5-dB fluctuation in a band of $x_0,y_0,z_0$6 GHz, a $x_0,y_0,z_0$7 dB bandwidth of $x_0,y_0,z_0$8 GHz, delay-lag correlations below the background-noise standard deviation, and an experimental correlation dimension reported as $x_0,y_0,z_0$9 (Wang et al., 2014). Here, the “seed” is a broadband, low-signature physical chaotic carrier.

In many-body quantum systems, the same language becomes perturbative. A single local impurity $s$0 is enough to seed chaos in the interacting integrable spin-$s$1 XXZ chain, the transverse-field Ising chain, and the spin-1 Lai–Sutherland chain (Santos et al., 2020). The onset of chaos is diagnosed by GOE-like level repulsion, a correlation hole in the mean survival probability, and ETH-compatible off-diagonal matrix elements of local observables. For the XXZ chain, the defect strength needed to reach fixed Brody thresholds scales approximately as

$s$2

for $s$3, respectively (Santos et al., 2020). In this literature, chaos seeding is the local breaking of integrability whose global effect strengthens with system size.

6. Diagnostics and terminological boundaries

Because seeded chaos is often claimed through sensitivity rhetoric, diagnostics matter. The sequential test for chaos proposes an orbit-based criterion tied to unpredictability and Poincaré chaos rather than directly to Devaney or Li–Yorke notions (Akhmet et al., 2019). For continuous systems, it looks for sequences $s$4 such that

$s$5

while

$s$6

and uses the discrete analog

$s$7

for maps (Akhmet et al., 2019). This is not a proof from finite data, but it formalizes the idea that seeded chaos should exhibit close recurrence together with later separation.

A separate misconception is terminological. In tournament design and social-network diffusion, “seeding” names allocation policies rather than the initialization of chaotic dynamics. The tournament-seeding literature studies paradoxical sensitivity of downstream pots or coefficients and proposes the rule

$s$8

to restore incentive compatibility (Csató, 2020). The network-diffusion literature treats a stochastic seeding strategy as a probability distribution over seed sets and compares one-hop targeting with uniform random seeding via importance-weighted estimators (Chin et al., 2018). These are technically rigorous “seeding” problems, but they are not chaos seeding in the Devaney, Lyapunov, PRNG, or physical-chaos sense.

Across the dynamical-systems literature, the most consistent synthesis is therefore narrow: chaos seeding is the act of choosing an initial condition, parameter tuple, admissible seed family, or local perturbation class that determines whether a deterministic nonlinear system will remain regular, enter a chaotic regime, or generate statistically useful pseudo-random or physical-random output. The strongest papers make that choice explicit and structurally constrained; weaker ones rely mainly on sensitivity claims and post hoc statistical testing.