An interactive companion to Zelinka & Daher (2021) · Modeling the Sustainable Development Nexus
Every system — a swinging pendulum, a national economy, Earth’s climate, the seventeen Sustainable Development Goals — can be described by three numbers: how it’s built, how it moves through time, and how unpredictable it is. This page turns that idea into a space you can move through and test for yourself. Every system — a pendulum, an economy, the climate, the seventeen Sustainable Development Goals — occupies a region of a space defined by three dimensions: structure (interconnectivity and hierarchy), temporality (feedback, cycles, delays), and chaos (aperiodic randomness). This page turns Figure 1 of the chapter into a navigable, testable state space. The dimensionality of systems (Zelinka & Daher, 2021) posits structure, temporality, and chaos as three continuous, interacting dimensions sufficient to classify any system. This page operationalizes Figure 1 as an explorable state space in which systems occupy credible regions rather than deterministic points.
01 · Explore the space
Up means more parts. Right means time matters more. Toward you means more surprise. No system sits at an exact spot — each one is a little cloud of “probably around here,” and the cloud grows when a system is more chaotic or harder to measure. Drag to spin it; click anything that glows. Structure rises, temporality runs right, chaos comes toward you — three co-equal coordinates. But no real system occupies an exact point. Following the chapter — structure and dynamics describe a system's average state, chaos its variance and deviation from that mean — every system is drawn as a : a cloud of plausible positions around a mean-state marker. Clouds widen with chaos and with reliance on expert judgment; even the family boundaries are bands, not lines. Axes are normalized to [0,1]. Each system is rendered as a mean-state vector μ = (S, T, C) with a Gaussian credible region of spread σ = ε + λC, where ε is epistemic placement uncertainty (expert judgment; cf. the chapter's cross-impact discussion) and λC the chaos-driven deviation about the mean state. Family boundaries follow the superellipse separatrices of Fig. 1, rendered as bands.
02 · The dimensions, in your hands
Reading about the axes is one thing; driving them is another. Build a network and feel structure grow. Push a system through time and feel temporality. Then watch many simple things become one complex thing. Each dimension gets a laboratory. Structure: wire a network and watch interconnectivity raise the score. Temporality: combine the chapter's building blocks of dynamics — a reinforcing loop, a balancing loop, and a delay — and watch growth, goal-seeking, and oscillation appear. Then : when enough parts interact, the whole exceeds the sum of its parts (1 + 1 = 3). Operational micro-models per dimension: (i) structure as edge density over user-placed nodes in a unit-disk graph; (ii) temporality as the first-order delayed-feedback system xt+1 = xt + r·xt + b·(G − xt−d), exhibiting growth, convergence, and delay-induced oscillation (cf. Sterman, 2000); (iii) emergence via local-rule agents (Reynolds, 1987) with order parameter φ = |⟨v̂⟩| (Vicsek et al., 1995).
Click the dark canvas to add parts. Slide “reach” to let them link up. A pile becomes a web. Structure is interconnectivity: parts and the links among them. Add nodes and widen the linking reach — density drives the structure score, the same quantity the vertical axis of the space measures. The score grows with node count and edge density E/C(N,2) over a user-radius disk graph — an operational proxy for the interconnectivity-plus-hierarchy construct of Ch. 2.
Fig. D-2 (interactive). Structure as interconnectivity: density over a growing network.
Three dials: a loop that feeds growth, a loop that chases a goal, and a delay. The delay is the troublemaker — push it up and the line starts to swing. The chapter builds temporality from and . R compounds the state; B steers it toward a goal; the delay makes B act on old information — and delayed balancing feedback is where oscillation comes from. xt+1 = xt + r·xt + b·(G − xt−d): with d = 0 the balancing loop converges monotonically; a sufficient delay–gain product destabilizes it into oscillation — the canonical system-dynamics result (Forrester, 1961; Sterman, 2000).
Fig. D-3 (interactive). Temporality from first principles: reinforcing growth, balancing goal-seeking, and the oscillations born of delay.
Every bird follows tiny rules about its nearest neighbors. Nobody is in charge. Turn up the interaction — and a flock appears out of nowhere. The chapter's starlings: each bird senses only its nearby flockmates, yet a murmuration organizes itself — behavior of the whole that no single part contains. Watch φ climb as local interaction creates global order: 1 + 1 = 3. Local alignment, cohesion, and separation (Reynolds, 1987), with interaction weight k scaling all coupling terms. Global order is measured by φ = |⟨v̂⟩| ∈ [0,1] (Vicsek et al., 1995): φ near 0 for independent agents, φ → 1 for a coherent flock. Order without central control — (Ch. 2).
Fig. D-4 (interactive). Emergence: independent agents become a flock as local interaction strengthens — the whole exceeding the sum of its parts (Ch. 2).
03 · Locate your own system
Score any system you care about — your commute, your workplace, a forest, a city — on the three dials. The page tells you what kind of system it is, and because nobody scores these things perfectly, it answers in percentages, not absolutes. Score any system you work with — a supply chain, a watershed, a classroom, a policy portfolio — along the three dimensions. The space classifies it and, following the chapter's central argument, recommends which modeling tradition fits: a snapshot, a simulation, or a distribution. Because placements rest on judgment, the verdict is fuzzy: 400 positions are sampled from your scores' credible region and membership is reported by degree (after Zadeh, 1965). Slider scores parameterize μ; placement confidence sets ε. Classification is Monte-Carlo: N = 400 draws from N(μ,σ²) clipped to [0,1]³, each assigned to a family by the Fig. 1 partition, yielding fuzzy membership degrees (Zadeh, 1965). Method recommendations follow the chapter's soft-to-hard sequence: → → code-level simulation, with distributional treatment when chaos dominates.
Fuzzy verdict · 400 sampled positions
Five systems. Read each one, set the three dials where you think it lives, then reveal the chapter’s placement. Land inside the cloud to score. Commit to a placement before seeing the canonical one. You score per axis by landing within the system’s credible region — full points inside ±1σ, partial within ±2σ. Committing first is the point: it surfaces your model of the system so the reveal can correct it. A predict–observe–explain exercise over the Fig. 1 space. Scoring per axis against the system’s spread σ = ε + λC: |guess − μ| ≤ σ → 2 pts; ≤ 2σ → 1 pt. Five rounds, 30 points.
○ your guess · ┃ canonical mean · band = ±1σ credible region
Activity A-1 (interactive). Predict-then-reveal: place a system before the chapter does, scored against its credible region.
04 · The chaos lab
Here's the wild part: chaos doesn't need complexity. One tiny formula, repeated over and over, can become completely unpredictable. The two lines below start almost identical — slide r past 3.57 and watch them go their separate ways. The chapter's key counterintuition: chaos doesn't require complexity. The logistic map is about as simple as a dynamical system gets — one variable, one parameter — yet past a threshold it produces behavior that never repeats. Two runs below start 0.0001 apart. Push r past 3.5699… and watch them part ways: the , live. The logistic map xn+1 = r·xn(1−xn) follows the period-doubling route to chaos, accumulating at r ≈ 3.5699; beyond it trajectories are aperiodic with sensitive dependence on initial conditions (Lorenz, 1963; Feldman, 2019). Twin runs differ by Δx₀ = 10⁻⁴; the terminal gap operationalizes the Butterfly Effect.
Fig. D-5a (interactive). Twin trajectories of the logistic map, Δx₀ = 10⁻⁴.
λ is the mean log-derivative along the orbit; λ > 0 ⇒ chaos (Feldman, 2019). Sent to the classifier as C ≈ 0.5 + λ/(2 ln 2) — the lab as a measurement instrument for one axis.
The bifurcation map · click to set r
Fig. D-5b (interactive). Bifurcation diagram, r ∈ [2.6, 4.0].
Each vertical stripe shows where the system ends up for that r. One line splits into two, then four, then a blur — order sliding into chaos. Each vertical slice shows the long-run values the system visits at that r. One branch becomes two, then four, then a dust of points: the period-doubling road to deterministic chaos — fully determined, never repeating, hypersensitive to where it started. Long-run attractor values versus r (160 transient iterations discarded, 130 plotted per column). Period-doubling cascade with periodic windows (e.g., period-3 near r ≈ 3.83): deterministic yet aperiodic — chaos as a dimension independent of structural complexity.
Both charts look wild. The left one is pure formula — run it again and it repeats exactly. The right one is pure chance — it never repeats. The chapter splits chaos in two. Deterministic: an exact rule — rerun with the same start and the path is identical, but nudge x₀ by 0.0001 and it diverges. Stochastic: genuine randomness — no two runs ever match. Deterministic: the logistic map at r = 3.9 — trajectories are reproducible functions of x₀ with sensitive dependence on initial conditions (Feldman, 2019), graded by the (Ch. 2). Stochastic: a Bernoulli walk — realizations are non-reproducible; only distributions describe it.
Fig. D-5c (interactive). Deterministic versus stochastic chaos: identical reruns versus unrepeatable ones.
05 · Lineage
Classifying problems by their kind of difficulty is an old ambition. Each generation added a dimension; the Dimensionality of Systems turns the categories into coordinates.
Splits science's problems into simplicity, disorganized complexity (tamed by statistics), and the unconquered middle: organized complexity — many factors interrelated into an organic whole.
The first axis: how many parts, how organized.
At MIT, stocks, flows, and feedback loops make time itself modelable. Behavior becomes a consequence of structure — and simulation becomes a policy instrument.
The second axis: temporality, operationalized.
Three small weather equations prove that fully determined systems can be aperiodic and hypersensitive to initial conditions — slightly different starts, considerably different futures.
The third axis: chaos, discovered hiding in determinism.
Membership in a class becomes a matter of degree rather than yes-or-no, giving categories with soft edges a formal mathematics.
Why every system here is a cloud, and every verdict a percentage.
Emergence, self-organization, and nonlinearity become a transdisciplinary research program — the flock that no ornithologist can predict from the bird.
The vocabulary this space is drawn in.
Clear, complicated, complex, chaotic: a sense-making compass telling leaders which decision style fits their context. Powerful — but categorical, qualitative, and flat.
The nearest neighbor: domains, not dimensions.
Structure, temporality, and chaos become three continuous, interacting axes — a state space in which every system occupies a credible region: located, compared, and tracked as it moves. Categories become coordinates with uncertainty attached; the SDG nexus becomes a region you can model.
This page is that figure, made navigable.
Appendix · Methods & provenance
Everything labeled “Ch. 2” paraphrases Zelinka & Daher (2021). The constants below are this page’s operationalization — chosen for pedagogy, exposed for tuning, and kept deliberately separate from the chapter’s claims.
Coordinates. Each system’s mean state μ = (S, T, C) is an editorial reading of the chapter’s qualitative descriptions onto axes normalized to [0, 1]; every placement carries a “placed here” rationale in the inspector, citing the passage it rests on.
Spread. σS = σT = ε + 0.085·C and σC = ε + 0.05·C. The chaos-proportional term implements the chapter’s claim that structure and dynamics describe the average state while chaos describes variance and deviation from it. ε is per-system epistemic placement uncertainty (0.015–0.10), motivated by the chapter’s discussion of expert judgment in cross-impact estimation: equations carry the smallest ε, social systems the largest.
Families. The Fig. 1 separatrices are drawn as superellipses (p = 2.7; k = 4.6 and 8.2 on the structure–temporality face) and rendered as bands rather than lines — the families shade into one another.
Classifier. Monte-Carlo over N = 400 draws from N(μ, σ²) clipped to [0, 1]³, each draw assigned by the partition above; memberships are reported as degrees following Zadeh (1965).
Playgrounds. Structure: score over node count and edge density in a unit-disk graph. Temporality: xt+1 = xt + r·xt + b·(G − xt−d). Chaos lab: the logistic map with Δx₀ = 10⁻⁴ and 160 transients discarded. Emergence: Reynolds (1987) rules with order parameter φ = |⟨v̂⟩| (Vicsek et al., 1995). The chaos lab’s Lyapunov readout uses λ = ⟨ln|r(1−2x)|⟩ over the orbit and maps to the chaos coordinate as C ≈ 0.5 + λ/(2 ln 2) — an operational bridge, not a chapter claim.
Status. The spread model and the ε table are the author’s to tune — they are implementation, not chapter. If they are later formalized in a follow-up paper, this page becomes its demonstrator. All constants live in one commented block at the top of the page’s script.
Sources
Zelinka, D. (2026). The Dimensionality of Systems — An Interactive Explorer. Civitas Systems: Systems Atlas, Paper 01. Adapted from Zelinka, D., & Daher, B. (2021), Modeling the Sustainable Development Nexus as a Complex-Coupled System, IGI Global. doi:10.4018/978-1-7998-5788-4.ch002