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Positive feedback closes a causal loop whose net effect reinforces an initiating change. A loop with two inhibitory links can therefore have positive feedback. In signal transduction, a product that increases the activity of the enzyme making it illustrates reinforcement; the consequences depend on the full mathematical model, not just the loop sign. See signaling feedback.

A biological example is cell-cycle activation feedback. Cyclin B binds Cdc2, whose activity promotes activation of Cdc25; Cdc25 removes inhibitory phosphorylation from Cdc2, permitting further activation. Cdc2 also inhibits Wee1, an enzyme that inhibits Cdc2 through phosphorylation. That second return contains two inhibitions and therefore also has positive feedback. These are regulatory paths, not a claim that every connection is a single direct molecular encounter. ^[ref]

Reinforcement can preserve a decision after its initiating stimulus weakens: returning from the high state may require a lower input than entering it. The cell-cycle hysteresis experiment below tests this history dependence at matched cyclin B concentrations. A steep response alone is insufficient evidence of bistability, and finite observation times can make slow relaxation look like memory. A proposed switch must specify its input, readout and observation interval.

Pressure calibration assigns pressure values to a sensor's output by comparison with a reference under specified conditions. It is not the same as making the output more finely resolved. A smooth electrical resistance curve can be repeatable yet have an incorrect pressure scale.

Bridgman used pressure-dependent electrical resistance of manganin and established fixed points, including a bismuth transition. He reported poorer accuracy above the better calibrated range.^[ref]

A fixed-point specimen and a property-measurement sample have distinct roles: one helps assign the scale, while the other is studied on that scale. The reference transition must itself have an assigned pressure at the relevant temperature. Agreement at a single point does not determine a nonlinear sensor response elsewhere. Comparing increasing and decreasing pressure checks hysteresis; returning to a reference checks drift. These are checks one can design from the measurement model, not additional historical controls asserted for the lecture's experiments.

As an original stipulated low-field model, take the axial magnetic field intensity $H$ in $A/m$ and prescribeAll strain components are dimensionless, and their trace vanishes. This is an even, quasistatic, unsaturated toy law with a fixed measuring axis. It is distinct from the fixed-magnitude directional model in saturated isotropic magnetostrictive strain. In particular, one must not replace local unit magnetization by normalized net magnetization in a many-domain specimen and assume this law follows.

For $H(t)=H_0\sin (2\pi ft)$, $H_0=20kA/m$, and $f=50\mathrm{Hz}$, a trigonometric identity givesThe axial strain ranges from zero to $20\times 10^{-6}$, with mean $10\times 10^{-6}$ and a $100\mathrm{Hz}$ component of amplitude $10\times 10^{-6}$. The cosine coefficient is negative with the chosen time origin. For $L=100\mathrm{mm}$ the length ranges from zero to $2\mu \mathrm{m}$ relative to $H=0$. There is no $50\mathrm{Hz}$ term in this unbiased model.

A static bias $H_b$ changes the conclusion. Expanding $\kappa [H_b+H_0\sin (2\pi ft){]}^2$ produces a fundamental-frequency term $2\kappa H_b{H}_0\sin (2\pi ft)$ as well as the doubled-frequency term. This is an incremental linear response about a state with a chosen magnetic axis; it does not contradict isotropic linear magnetostriction vanishes at an unbiased isotropic reference. Real hysteresis, thermal expansion from Joule heating, and mechanical resonance can add further components or change their phases. Frequency alone cannot identify the cause of motion: Joule heating also contains an even-in-electric current contribution.

A resistance pressure gauge is a pressure sensor whose electrical resistance is calibrated against pressure. Even a simple local response needs both pressure and temperature. For an original illustrative model definewhere $R_0$ is the reference electrical resistance at $p=0$, $T=T_0$, and zero history offset. The derivatives $\alpha$ and $\beta$ are local sensitivities with units $\mathrm{P}{\mathrm{a}}^{-1}$ and ${\mathrm{K}}^{-1}$, respectively; $h$ is a dimensionless history offset. Constant sensitivities are an approximation over a chosen range, not material constants valid at every pressure.

Choose synthetic coefficients $\alpha =0.020\mathrm{GP}{\mathrm{a}}^{-1}$ and $\beta =2.0\times 10^{-4}{\mathrm{K}}^{-1}$, with $T-T_0=10\mathrm{K}$ and $h=0$. A measured ratio $R/R_0=1.022$ then gives $p=(0.022-0.002)/0.020=1.00\mathrm{GPa}$. Ignoring the temperature change gives $1.10\mathrm{GPa}$, a $0.10\mathrm{GPa}$ error. This model is deliberately stipulated; these numbers are not measured manganin coefficients or Bridgman's data.

If two histories differ by $\Delta h=0.0004$, their outputs at the same pressure differ by $0.0004R_0$; interpreting both with a single history-free curve gives an apparent pressure separation $\Delta h/\alpha =0.020\mathrm{GPa}$. The half separation is $0.010\mathrm{GPa}$. Reversing the drive can additionally cause mechanical friction hysteresis, independently of this sensor model. Record both the electrical calibration history and the loading direction.

For independent small standard measurement uncertainties $u_y,u_T,u_h$ and exact stipulated $\alpha ,\beta$, first-order propagation givesFor $u_y=10^{-4}$, $u_T=0.5\mathrm{K}$, and $u_h=10^{-4}$, this is $u_p=0.00866\mathrm{GPa}$. Real calibration also includes coefficient measurement uncertainties, correlations, reference-point measurement uncertainty, and any unresolved nonhydrostatic loading. A sensor calibrates its local environment; it cannot by itself certify the absence of deviatoric stress throughout the sample.

Saturating positive feedback limits the reinforcing rate as activity increases. Here is an explicitly synthetic ordinary differential equation for normalized nonnegative input $u$, response $x$, and return strength $\beta$:The return is an increasing, nonproportional function of $x$, while removal is proportional to $x$. Time and activities are dimensionless variables. At $x=0$ the vector field points into the nonnegative domain, and $dx/dt\le u+\beta -x$ prevents unbounded growth there. This makes the toy useful for asking about a bounded switch; it supplies no molecular mechanism for the chosen rate law.

At an equilibrium point, $f=0$. For a simple equilibrium point, the sign of the derivativedetermines local asymptotic stability. If $\beta =1$, the maximum reinforcing slope is $3\sqrt{3}/8<1$, so $f$ decreases strictly: every $u\ge 0$ has one asymptotically stable equilibrium point despite positive feedback.

Three equilibrium points become possible only if the reinforcing slope exceeds removal's slope somewhere. Its maximum is $3\sqrt{3}\beta /8$, at $x=1/\sqrt{3}$. Consequently this toy requiresfor two distinct folds. This is a necessary threshold for a bistability interval in this toy, not a universal biochemical number. At a fold, both $f=0$ and $f_x=0$, givingFor $\beta =1.8$, the folds are at $(x,u)\simeq (0.350003,0.153563)$ and $(0.887240,0.094405)$. Between these input values there are two asymptotically stable equilibrium points separated by an unstable one. At $u=0.12$, the three activities are approximately $0.171331$, $0.621230$ and $1.127439$; the low and high ones are stable.

A sufficiently slow increasing sweep follows the low branch to $u_{\mathrm{on}}\simeq 0.153563$ and then jumps upward. A decreasing sweep follows the high branch to $u_{\mathrm{off}}\simeq 0.094405$ and then jumps downward: hysteresis. These are limiting quasistatic thresholds, with relaxation between input changes and negligible perturbations that could cause earlier switching. Finite sweep rates and fluctuations can alter observed thresholds. A switchlike response with only one stable equilibrium point does not have this equilibrium hysteresis mechanism.

A soft magnetic material changes its magnetization relatively easily and has a low coercive field. High useful permeability and small hysteresis losses favour such materials in electromagnets and alternating-field cores. “Soft” describes magnetic response rather than mechanical softness; electrically resistive ferrimagnets can also be magnetically soft.

A motivating measurement used the McMath solar optical telescope at Kitt Peak. The Sun's image drifted across a $10\mathrm{mm}$-high, $0.2\mathrm{mm}$-wide optical slit; a double-pass spectrograph selected wavelength intervals from $0.0015\mathrm{nm}$ wide at $303\mathrm{nm}$ to $0.007\mathrm{nm}$ at $1099\mathrm{nm}$. An EMI 9750 QB photodetector or silicon photodiode supplied a digitized signal: $1\mathrm{kHz}$ samples were summed into 4096 positions per crossing, including dark and off-limb sky readings. ^[ref]

Controls included daily realignment of optical telescope mirrors, wavelength settings between spectral lines insensitive to the rotational Doppler shift, selection of detectors without detectable hysteresis, and measured electronic nonlinearity corrections below $0.1\%$ of maximum signal. Limb-position calibration combined centres of gravity of the edge derivative profiles with almanac semidiameters; the preliminary edge shifts were consistent with astronomical seeing of approximately $5$ arcseconds full width at half maximum. ^[ref]

The decreasing centre-to-limb readout gives a specific intensity profile, constraining stratification of the photosphere under a radiative transfer model. Astronomical seeing, instrumental scattering of light, and solar granulation limit this inference; brightness alone does not uniquely determine temperature versus geometric height. These observations need not follow the stipulated linear limb-darkening law.

In a subcritical Hopf bifurcation, the small periodic orbit repels within the critical centre manifold and lies on the side where the equilibrium point attracts within that manifold. With the critical eigenvalue real part increasing through zero, the positive cubic sign gives the exact radial Hopf normal formFor $\mu <0$, the nonzero radius is $r_{\ast }=\sqrt{-\mu }$. Radii below $r_{\ast }$ shrink to zero, whereas radii above it grow. The radial derivative at the cycle is $\mu +3r_{\ast }^2=-2\mu >0$, proving that the periodic orbit repels. For example, $\mu =-0.04$ gives an unstable circle of radius $0.2$: a radius of $0.1$ decays while one of $0.3$ grows. The unstable circle shrinks into zero as $\mu \uparrow 0$; at $\mu =0$, $\dot{r}=r^3$ already drives every small positive radius outward.

For $\mu >0$ there is no nonzero constant radius in this cubic model, and zero repels. In a general dynamical system, this says only that the nearby orbits leave the local neighbourhood. It does not guarantee a distant attracting limit cycle, a jump to a particular amplitude, or hysteresis. Additional nonlinear or global information is needed; the quintic radial Hopf equation illustrates one possible stabilizing extension. Reversing the parameter orientation reverses which numerical side contains the branch, so the defining distinction between supercritical Hopf bifurcation and subcritical Hopf bifurcation is their reduced stability and their relation to the equilibrium point, rather than the bare sign of an arbitrarily named parameter.

A magnetic energy preference for an axis, with the two opposite directions equivalent in a purely uniaxial model. An easy-axis energy can be written $K{\sin }^2\theta$ for $K>0$. A unidirectional term distinguishes the two directions and can instead shift a hysteresis.