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Hysteresis refers to systems that may exhibit path dependence, or "rate-independent memory". In a deterministic system with no dynamics or hysteresis, it is possible to predict the system's output at an instant in time given only its input at that instant in time. In a system with hysteresis, this is not possible; the output depends in part on the internal state of system and not only on its input. There is no way to predict the system's output without looking at the history of the input (to determine the path that the input followed before it reached its current value) or inspecting the internal state of the system.

A system may be explicitly designed to exhibit hysteresis, especially in control theory. An example is the switching element of a thermostat that controls a furnace; this element has hysteresis. Schmitt triggers are examples of electronic circuits that exhibit hysteresis, in this case by introducing a positive feedback. Many physical systems naturally exhibit hysteresis. Hysteresis phenomena occur in magnetic materials, ferromagnetic materials and ferroelectric materials, as well as in the elastic, electric, and magnetic behavior of materials, in which a lag occurs between the application and the removal of a force or field and its subsequent effect. Electric hysteresis occurs when applying a varying electric field, and elastic hysteresis occurs in response to a varying force.

The word hysteresis is often used specifically to represent rate-independent state. This means that if some set of inputs X(t) produce an output Y(t), then the inputs X(αt) produce output Y(αt) for any α > 0. The magnetized iron or the thermostat have this property. Not all systems with state (or, equivalently, with memory) have this property; for example, a linear low-pass filter has state, but its state is rate-dependent.

The term "hysteresis" is sometimes used in other fields, such as economics or biology, where it describes a memory, or lagging, effect. In the field of structural analysis it is strongly related to the concept of elastic and plastic materials.

History

The term "hysteresis" is derived from Template:Polytonic, an ancient Greek word meaning "deficiency" or "lagging behind". It was coined by Sir James Alfred Ewing.

Some early work on describing hysteresis in mechanical systems was performed by James Clerk Maxwell. Subsequently, hysteretic models have received significant attention in the works of Preisach (Preisach model of hysteresis), Neel and Everett in connection with magnetism and absorption. A simple parametric description of various hysteretic loops may be found in ref. (with the model, substitution of rectangle, triangle or trapezoidal pulses instead of the harmonic functions also allows piecewise-linear hysteresis loops frequently used in discrete automatics to be built, see example in Fig. 4). More formal mathematical theory of systems with hysteresis was developed in 1970s by a group of Russian mathematicians led by Mark Krasnosel'skii, one of the founders of nonlinear analysis. He suggested an investigation of hysteresis phenomena using the theory of nonlinear operators.

Hysteresis was initially seen as problematic, but is now thought to be of great importance in technology. For example, the properties of hysteresis are applied when constructing non-volatile storage for computers; as hysteresis allows most superconductors to operate at the high currents needed to create strong magnetic fields. Hysteresis is also important in living systems. Many critical processes occurring in living (or dying) cells use hysteresis to help stabilize them against the various effects of random chemical fluctuations. Magnetic core losses (iron losses) in transformers are due to hysteresis.

Terminology

Rate-independent hysteresis

The memory, or lagging, effects which constitute hysteresis should not necessarily be interpreted as a simple time lag. Even relatively simple systems such as an electric circuit containing resistors and capacitors exhibit a time lag between the input and the output, but are not normally considered hysteretic. Typically there is a very short time scale over which the dynamic behavior and various related time dependences are observed. In magnetism, for example, the dynamic processes occurring on this very short time scale have been referred to as Barkhausen jumps.

Also, if observations are carried out over very long periods, creep or slow relaxation (typically toward true thermodynamic equilibrium, or other types of equilibria that depend on the nature of the system) can be noticed. This is also not normally considered to be hysteresis.

Hysteresis occurs when observations are carried out without regard for very swift dynamic phenomena or very slow relaxation phenomena and the system appears to display irreversible behavior whose rate is practically independent of the driving force rate. This rate-independent irreversible behavior is the key feature that distinguishes hysteresis from most other dynamic processes in many systems.

Hysteresis loop

If the displacement of a system with hysteresis is plotted on a graph against the applied force, the resulting curve is in the form of a loop. In contrast, the curve for a system without hysteresis is a single, not necessarily straight, line. Although the hysteresis loop depends on the material's physical properties, there is no complete theoretical description that explains the phenomenon. The family of hysteresis loops, from the results of different applied varying voltages or forces, form a closed space in three dimensions, called the hysteroid.

Manifestations

In engineering

Control systems

Hysteresis can be used to filter signals so that the output reacts slowly by taking recent history into account. For example, a thermostat controlling a heater may turn the heater on when the temperature drops below A degrees, but not turn it off until the temperature rises above B degrees (e.g., if one wishes to maintain a temperature of 20 °C, then one might set the thermostat to turn the furnace on when the temperature drops below 18 °C, and turn it off when the temperature exceeds 22 °C). This thermostat has hysteresis. Thus the on/off output of the thermostat to the heater when the temperature is between A and B depends on the history of the temperature. This prevents rapid switching on and off as the temperature drifts around the set point. The furnace is either off or on, with nothing in between. The thermostat is a system; the input is the temperature, and the output is the furnace state. If the temperature is 21 °C, then it is not possible to predict whether the furnace is on or off without knowing the history of the temperature.

Similarly a pressure switch also exhibits hysteresis. Its pressure setpoints are substituted for those of temperature corresponding to a thermostat.

Electronic circuits

A Schmitt trigger is a simple electronic circuit that also exhibits this property. Often, some amount of hysteresis is intentionally added to an electronic circuit to prevent unwanted rapid switching. This and similar techniques are used to compensate for contact bounce in switches, or noise in an electrical signal.

A latching relay uses a solenoid to actuate a ratcheting motion that keeps the relay closed even if power to the relay is terminated.

Hysteresis is essential to the workings of some memristors (circuit components which "remember" changes in the current passing through them by changing their resistance).

User interface design

A hysteresis is sometimes intentionally added to computer algorithms. The field of user interface design has borrowed the term hysteresis to refer to times when the state of the user interface intentionally lags behind the apparent user input. For example, a menu that was drawn in response to a mouse-over event may remain on-screen for a brief moment after the mouse has moved out of the trigger region and the menu region. This allows the user to move the mouse directly to an item on the menu, even if part of that direct mouse path is outside of both the trigger region and the menu region. For instance, right-clicking on the desktop in most Windows interfaces will create a menu that exhibits this behavior.

In mechanics

Elastic hysteresis

Elastic hysteresis was one of the first types of hysteresis to be examined.

A simple way to understand it is in terms of a rubber band with weights attached to it. If the top of a rubber band is hung on a hook and small weights are attached to the bottom of the band one at a time, it will get longer. As more weights are loaded into it, the band will continue to extend because the force the weights are exerting on the band is increasing. When each weight is taken off, or unloaded, it will get shorter as the force is reduced. As the weights are taken off, each weight that produced a specific length as it was loaded onto the band now produces a slightly longer length as it is unloaded. This is because the band does not obey Hooke's law perfectly. The hysteresis loop of an idealized rubber band is shown in Fig. 3.

In one sense the rubber band was harder to stretch when it was being loaded than when it was being unloaded. In another sense, as one unloads the band, the cause (the force of the weights) lags behind the effect (the length) because a smaller value of weight produces the same length. In another sense more energy was required during the loading than the unloading; that energy must have gone somewhere, it was dissipated or "lost" as heat.

Elastic hysteresis is more pronounced when the loading and unloading is done quickly than when it is done slowly. Some materials such as hard metals don't show elastic hysteresis under a moderate load, whereas other hard materials like granite and marble do. Materials such as rubber exhibit a high degree of elastic hysteresis.

A word of caution: rubber behaves like a gas. When the rubber band is stretched it heats up. If it is suddenly released, the rubber cools down, very easy to perceive just by touching. So, there is a large hysteresis from the thermal exchange with the environment and a smaller hysteresis due to internal friction within the rubber. This proper, intrinsic hysteresis could be measured only if adiabatic isolation of the rubber band is imposed.

Contact angle hysteresis

The contact angle formed between a liquid and solid phase will exhibit a range of contact angles that are possible. There are two common methods for measuring this range of contact angles. The first method is referred to as the tilting base method. Once a drop is dispensed on the surface with the surface level, the surface is then tilted from 0° to 90°. As the drop is tilted, the downhill side will be in a state of imminent wetting while the uphill side will be in a state of imminent dewetting. As the tilt increases the downhill contact angle will increase and represents the advancing contact angle while the uphill side will decrease; this is the receding contact angle. The values for these angles just prior to the drop releasing will typically represent the advancing and receding contact angles. The difference between these two angles is the contact angle hysteresis. The second method is often referred to as the add/remove volume method. When the maximum liquid volume is removed from the drop without the interfacial area decreasing the receding contact angle is thus measured. When volume is added to the maximum before the interfacial area increases, this is the advancing contact angle. As with the tilt method, the difference between the advancing and receding contact angles is the contact angle hysteresis. Most researchers prefer the tilt method; the add/remove method requires that a tip or needle stay embedded in the drop which can affect the accuracy of the values, especially the receding contact angle.

Adsorption hysteresis

Hysteresis can also occur during physical adsorption processes. In this type of hysteresis, the quantity adsorbed is different when gas is being added than it is when being removed. The specific causes of adsorption hysteresis are still an active area of research, but it is linked to differences in the nucleation and evaporation mechanisms inside mesopores. These mechanisms are further complicated by effects such as cavitation and pore blocking.

In physical adsorption, hysteresis is evidence of mesoporosity-indeed, the definition of mesopores (2-50 nm) is associated with the appearance (50 nm) and disappearance (2 nm) of mesoporosity in nitrogen adsorption isotherms as a function of Kelvin radius. An adsorption isotherm showing hysteresis is said to be of Type IV (for a wetting adsorbate) or Type V (for a non-wetting adsorbate), and hysteresis loops themselves are classified according to how symmetric the loop is. Adsorption hysteresis loops also have the unusual property that it is possible to scan within a hysteresis loop by reversing the direction of adsorption while on a point on the loop. The resulting scans are called "crossing," "converging," or "returning," depending on the shape of the isotherm at this point.

Matric potential hysteresis

The relationship between matric water potential and water content is the basis of the water retention curve. Matric potential measurements (Ψ_m) are converted to volumetric water content (θ) measurements based on a site or soil specific calibration curve. Hysteresis is a source of water content measurement error. Matric potential hysteresis arises from differences in wetting behaviour causing dry medium to re-wet; that is, it depends on the saturation history of the porous medium. Hysteretic behaviour means that, for example, at a matric potential (Ψ_m) of 5 kPa, the volumetric water content (θ) of a fine sandy soil matrix could be anything between 8% to 25%.

Tensiometers are directly influenced by this type of hysteresis. Two other types of sensors used to measure soil water matric potential are also influenced by hysteresis effects within the sensor itself. Resistance blocks, both nylon and gypsum based, measure matric potential as a function of electrical resistance. The relation between the sensor’s electrical resistance and sensor matric potential is hysteretic. Thermocouples measure matric potential as a function of heat dissipation. Hysteresis occurs because measured heat dissipation depends on sensor water content, and the sensor water content–matric potential relationship is hysteretic. As of 2002, only desorption curves are usually measured during calibration of soil moisture sensors. Despite the fact that it can be a source of significant error, the sensor specific effect of hysteresis is generally ignored.

In materials

Magnetic and electrical hysteresis

The phenomenon of hysteresis in ferromagnetic and ferroelectric materials can conceptually be explained as follows: a system can be divided into subsystems or domains, much larger than an atomic volume, but still microscopic. Such domains occur in ferroelectric and ferromagnetic systems, since individual dipoles tend to group with each other, forming a small isotropic region. Each of the system's domains can be shown to have a metastable state. The metastable domains can in turn have two or more substates. Such a metastable state fluctuates widely from domain to domain, but the average represents the configuration of lowest energy. The hysteresis is simply the sum of all domains, or the sum of all metastable states.

Magnetic hysteresis

Hysteresis is well known in ferromagnetic materials. When an external magnetic field is applied to a ferromagnet, the atomic dipoles align themselves with the external field. Even when the external field is removed, part of the alignment will be retained: the material has become magnetized. For example, a piece of iron that is brought into a magnetic field retains some magnetization, even after the external magnetic field is removed. Once magnetized, the iron will stay magnetized indefinitely. To demagnetize the iron, it would be necessary to apply a magnetic field in the opposite direction. This is the effect that provides the element of memory in a hard disk drive.

The relationship between magnetic field strength (H) and magnetic flux density (B) - Fig. 2, is not linear in such materials. If the relationship between the two is plotted for increasing levels of field strength, it will follow a curve up to a point where further increases in magnetic field strength will result in no further change in flux density. This condition is called magnetic saturation.

If the magnetic field is now reduced linearly, the plotted relationship will follow a different curve back towards zero field strength at which point it will be offset from the original curve by an amount called the remanent flux density or remanence.

If this relationship is plotted for all strengths of applied magnetic field the result is a sort of S- shaped loop. The width of the middle section of the loop describes the amount of hysteresis, related to the coercivity of the material.

Its practical effects might be, for example, to cause a relay to be slow to release due to the remaining magnetic field continuing to attract the armature when the applied electric current to the operating coil is removed.

This curve for a particular material influences the design of a magnetic circuit,

This is also a very important effect in magnetic tape and other magnetic storage media like hard disks. In these materials it would seem obvious to have one polarity represent a bit, say north for 1 and south for 0. However, to change the storage from one to the other, the hysteresis effect requires the knowledge of what was already there, because the needed field will be different in each case. In order to avoid this problem, recording systems first overdrive the entire system into a known state using a process known as bias. Analog magnetic recording also uses this technique. Different materials require different biasing, which is why there is a selector switch for this on the front of most cassette recorders.

In order to minimize this effect and the energy losses associated with it, ferromagnetic substances with low coercivity and low hysteresis loss are used, like permalloy.

In many applications small hysteresis loops are driven around points in the B-H plane. Loops near the origin have a higher µ. The smaller loops the more they have a soft magnetic (lengthy) shape. As a special case, a damped AC field demagnetizes any material provided it is initially sufficiently intense.

Magnetic field hysteresis loss causes heating. This is an energy loss mechanism in power transformers and electric motors and other apparatus using ferromagnetic cores.

Electrical hysteresis

Electrical hysteresis typically occurs in ferroelectric material, where domains of polarization contribute to the total polarization. Polarization is the electrical dipole moment (either C·m or C·m).

Liquid-solid phase transitions

Hysteresis manifests itself in state transitions when melting temperature and freezing temperature do not agree. For example, agar melts at 85 °C and solidifies from 32 to 40 °C. This is to say that once agar is melted at 85 °C, it retains a liquid state until cooled to 40 °C. Therefore, from the temperatures of 40 to 85 °C, agar can be either solid or liquid, depending on which state it was before.

In biology

Cell biology and genetics

Cells undergoing cell division exhibit hysteresis in that it takes a higher concentration of cyclins to switch them from G2 phase into mitosis than to stay in mitosis once begun.

Darlington in his classic works on genetics discussed hysteresis of the chromosomes, by which he meant "failure of the external form of the chromosomes to respond immediately to the internal stresses due to changes in their molecular spiral", as they lie in a somewhat rigid medium in the limited space of the cell nucleus.

In developmental biology, cell type diversity is regulated by long range-acting signaling molecules called morphogens that pattern uniform pools of cells in a concentration- and time-dependent manner. The morphogen Sonic Hedgehog (Shh), for example, acts on limb bud and neural progenitors to induce expression of a set of homeodomain-containing transcription factors to subdivide these tissues into distinct domains. It has been shown that these tissues have a 'memory' of previous exposure to Shh. In neural tissue, this hysteresis is regulated by a homeodomain (HD) feedback circuit that amplifies Shh signaling. In this circuit, expression of Gli transcription factors, the executors of the Shh pathway, is suppressed. Glis are processed to repressor forms (GliR) in the absence of Shh, but in the presence of Shh, a proportion of Glis are maintained as full-length proteins allowed to translocate to the nucleus, where they act as activators (GliA) of transcription. By reducing Gli expression then, the HD transcription factors reduce the total amount of Gli (GliT), so a higher proportion of GliT can be stabilized as GliA for the same concentration of Shh.

Neuroscience

The property by which some neurons do not return to their basal conditions from a stimulated condition immediately after removal of the stimulus is an example of hysteresis.

Respiratory physiology

Lung hysteresis is evident when observing the compliance of a lung on inspiration versus expiration. The difference in compliance (volume/pressure) is due to the additional energy required during inspiration to recruit and inflate additional alveoli.

The transpulmonary pressure vs Volume curve of inhalation is different from the Pressure vs Volume curve of exhalation, the difference being described as hysteresis. Lung volume at any given pressure during inhalation is less than the lung volume at any given pressure during exhalation.

In economics

Economic systems can exhibit hysteresis. For example, export performance is subject to strong hysteresis effects: because of the fixed transportation costs it may take a big push to start a country's exports, but once the transition is made, not much may be required to keep them going.

Permanently higher unemployment

Hysteresis is a hypothesized property of unemployment rates. It's possible that there is a ratchet effect, so a short-term rise in unemployment rates tends to persist.

An example is the notion that inflationary policy leads to a permanently higher 'natural' rate of unemployment (NAIRU), because inflationary expectations are 'sticky' downward due to wage rigidities and imperfections in the labour market.

When some negative shock reduces employment in a company or industry, there are less employed workers left. As usually the employed workers have the power to set wages, their reduced number incentivizes them to bargain for even higher wages when the economy again gets better, instead of letting the wage be at the equilibrium wage level, where the supply and demand of workers would match. This causes hysteresis, i.e., the unemployment becomes permanently higher after negative shocks.

Another channel through which hysteresis can occur is through learning by doing. Workers who lose their jobs due to a temporary shock may become permanently unemployed because they miss out on the job training and skill acquisition that normally takes place.

Hysteresis has been invoked, by Olivier Blanchard among others, as explaining the differences in long run unemployment rates between Europe and the United States.

Game theory and capital controls

Hysteresis occurs in applications of game theory to economics, in models with product quality, agent honesty or corruption of various institutions. Slightly different initial conditions can lead to opposite outcomes and resulting stable "good" and "bad" equilibria.

Another area where hysteresis phenomena are found is capital controls. A developing country can ban a certain kind of capital flow (e.g. engagement with international private equity funds), but when the ban is removed, the system takes a long time to return to the pre-ban state.

Applications

Hysteresis represents states, and the characteristic curve shape is sometimes reminiscent of a two-value state, also called a bistable state. The hysteresis curve really contains infinitely many states, but a simple application is to let the threshold regions (usually to the left and to the right) represent respectively the on and off states. In this way, the system can be regarded as bistable. Note that even if no external field is applied, the position of the hysteresis curve might change with time: it is not necessarily stationary; i.e. the system may not stay in exactly the same state as it had previously. The system might need new energy transfer to be stationary.

The hysteresis effect can be used when connecting complex circuits with the so-called passive matrix addressing. This scheme is praised as a technique that can be used in modern nanoelectronics, electrochrome cells, memory effect, etc. In this scheme, shortcuts are made between adjacent components (see crosstalk) and the hysteresis helps to keep the components in a particular state while the other components change states. That is, one can address all rows at the same time instead of doing each individually.

In economics, hysteresis is used extensively in the area of labor markets. According to theories based on hysteresis, economic downturns (recession) result in an individual becoming unemployed, losing his/her skills (commonly developed 'on the job'), demotivated/disillusioned, and employers may use time spent in unemployment as a screen. In times of an economic upturn or 'boom', the workers affected will not share in the prosperity, remaining long-term unemployed (over 52 weeks). Hysteresis has been put forward as a possible explanation for the poor unemployment performance of many economies in the 1990s. Labor market reform, and/or strong economic growth, may not therefore aid this pool of long-term unemployed, and thus specific targeted training programs are presented as a possible policy solution.

In the field of audio electronics, a noise gate often implements hysteresis intentionally to prevent the gate from "chattering" when signals close to its threshold are applied.

Small vehicle suspensions using rubber (or other elastomers) can achieve the dual function of springing and damping because rubber, unlike metal springs, has pronounced hysteresis and does not return all the absorbed compression energy on the rebound. Mountain bikes have frequently made use of elastomer suspension, as did the original Mini car.

Additional considerations

Models of hysteresis

Each subject that involves hysteresis has models that are specific to the subject. In addition, there are models that capture general features of many systems with hysteresis. An example is the Preisach model of hysteresis, which represents a hysteresis nonlinearity as a superposition of square loops called hysterons.

The Bouc-Wen model of hysteresis is often used to describe non-linear hysteretic systems. It was introduced by Bouc and extended by Wen, who demonstrated its versatibility by producing a variety of hysteretic patterns. This model is able to capture in analytical form, a range of shapes of hysteretic cycles which match the behaviour of a wide class of hysteretical systems; therefore, given its versability and mathematical tractability, the Bouc-Wen model has quickly gained popularity and has been extended and applied to a wide variety of engineering problems, including multi-degree-of-freedom (MDOF) systems, buildings, frames, bidirectional and torsional response of hysteretic systems two- and three-dimensional continua, soil liquefaction, and base isolation systems among others. The Bouc-Wen model and its variants/extensions have been used in applications of structural control, in particular in the modeling of the behaviour of magneto-rheological dampers, base isolation devices for buildings and other kinds of damping devices; it has also been in the modelling and analysis of structures built of reinforced concrete, steel, mansory and timber.

Energy

When hysteresis occurs with extensive and intensive variables, the work done on the system is the area under the hysteresis graph.

See also

• Remanence
• Hysteresivity
• Path dependence
• Backlash (engineering)
• Metastability
• Bean's critical state model

Notes

1. Mielke, A.; Roubicek, T. (2003). "A Rate-Independent Model for Inelastic Behavior of Shape-Memory Alloys". Multiscale Model. Simul. 1 (4): 571–597. doi:10.1137/S1540345903422860.
2. ^ R. V. Lapshin (1995). "Analytical model for the approximation of hysteresis loop and its application to the scanning tunneling microscope" (PDF). Review of Scientific Instruments. 66 (9). USA: AIP: 4718–4730. Bibcode:1995RScI...66.4718L. doi:10.1063/1.1145314. ISSN 0034-6748. (Russian translation is available).
3. ^ Mayergoyz, Isaak D. (2003). "Mathematical Models of Hysteresis and their Applications: Second Edition (Electromagnetism)". Academic Press. {{cite journal}}: Cite journal requires |journal= (help)
4. EETimes April 30, 2008
5. Augustus E. Love (1927). Treatise on the Mathematical Theory of Elasticity (Dover Books on Physics & Chemistry). New York: Dover Publications. ISBN 0-486-60174-9.
6. J.A Ewing, Brit. Assoc. Rep., 1889, p. 502
7. B. Hopkinson and G.T. Williams (1912). "The Elastic Hysteresis of Steel". Proc. Roy. Soc. A. 87: 502. Bibcode:1912RSPSA..87..502H. doi:10.1098/rspa.1912.0104.
8. Gregg and Sing (1982). Adsorption, Surface Area, and Porosity (Second ed.). London: Academic Press.
9. K. S. W. Sing; et al. (1985). "Reporting physisorption data for gas/solid systems with special reference to the determination of surface area and porosity (Recommendations 1984)". Pure and Applied Chemistry. 57 (4): 603–619. doi:10.1351/pac198557040603. {{cite journal}}: Explicit use of et al. in: |author= (help)
10. G. A. Tompsett; et al. (2005). "Hysteresis and Scanning Behavior of Mesoporous Molecular Sieves". Langmuir. 21 (8): 8214–8225. doi:10.1021/la050068y. PMID 16114924. {{cite journal}}: Explicit use of et al. in: |author= (help)
11. "Accuracy of capacitance soil moisture". Retrieved 2007-06-05.
12. Scanlon, B. R., Andraski, B. J., and Bilskie, J. (2002). "Methods of soil analysis: Physical Methods: Miscellaneous methods for measuring matric or water potential" (PDF). Soil Science Society of America. 4: 643–670. ISBN 0-89118-810-X. Retrieved 2006-05-26.{{cite journal}}: CS1 maint: multiple names: authors list (link)
13. Joseph R. Pomerening, Eduardo D. Sontag, James E. Ferrell Jr (2003). "Building a cell cycle oscillator: hysteresis and bistability in the activation of Cdc2". Nature Cell Biology. 5 (4): 346–251. doi:10.1038/ncb954. PMID 12629549.{{cite journal}}: CS1 maint: multiple names: authors list (link)
14. Darlington, C.D. 1937/1988 facsimile. Recent advances in cytology, second edition. Churchill/Garland publishing, New York and London.
15. Rieger, R.; Michaelis, A.; Green, M.M. 1968. A Glossary of Genetics and Cytogenetics : Classical and Molecular.
16. Harfe BD, Scherz PJ, Nissim S, Tian H, McMahon AP, Tabin CJ (2004). "Evidence for an expansion-based temporal Shh gradient in specifying vertebrate digit identities". Cell. 118 (4): 517–28. doi:10.1016/j.cell.2004.07.024. PMID 15315763.{{cite journal}}: CS1 maint: multiple names: authors list (link)
17. Lek M, Dias JM, Marklund U, Uhde CW, Kurdija S, Lei Q, Sussel L, Rubenstein JL, Matise MP, Arnold H-H, Jessell TM, Ericson J (2010). "A homeodomain feedback circuit underlies step-function interpretation of a Shh morphogen gradient during ventral neural patterning". Development. 137 (23): 4051–4060. doi:10.1242/dev.054288. PMID 21062862.{{cite journal}}: CS1 maint: multiple names: authors list (link)
18. Escolar, J.D. (2004). "Lung histeresis: a morphological view" (PDF). Histology and Histopathology Cellular and Molecular Biology. 19 (1): 159–16. PMID 14702184. Retrieved 1 March 2011. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
19. John B. West (2005). Respiratory physiology: the essentials. Hagerstown, MD: Lippincott Williams & Wilkins. ISBN 0-7817-5152-7.
20. Hysteresis and the European Unemployment Problem, Olivier J. Blanchard, Lawrence H. Summers, NBER Working Paper No. 1950 (Also Reprint No. r0808), 1987
21. R. Bouc (1967). Forced vibration of mechanical systems with hysteresis. In Proceedings of the Fourth Conference on Nonlinear Oscillation. Prague, Czechoslovakia, p. 315
22. R. Bouc (1971). Modèle mathématique d'hystérésis. Acustica, 24, p. 16-25 (in French)
23. Y. K. Wen (1976). Method for random vibration of hysteretic systems. Journal of Engineering Mechanics. ASCE. Vol. 102, No. 2. p. 249--263

Further reading

• Mark Krasnosel'skii and Alexei Pokrovskii, Systems with Hysteresis, Springer-Verlag, New York, 1989.
• Isaak D. Mayergoyz, Mathematical Models of Hysteresis and their Applications: Second Edition (Electromagnetism), Academic Press, 2003.
• The Science of Hysteresis (3-volume set), ed. by Isaak D. Mayergoyz, Giorgio Bertotti, Academic, 2005.

External links

• Overview of contact angle Hysteresis
• Preisach model of hysteresis – Matlab codes developed by Zs. Szabó and Gy. Kádár
• Hysteresis
• What's hysteresis?
• Dynamical systems with hysteresis (interactive web page)
• Magnetization reversal applet (coherent rotation)
• Elastic hysteresis and rubber bands

• Systems theory
• Fundamental physics concepts
• Electric and magnetic fields in matter
• Materials science