Buddhism and sexual orientation

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https://en.wikipedia.org/wiki/Buddhism_and_sexual_orientation 

The relationship between Buddhism and sexual orientation varies by tradition and teacher. According to some scholars, early Buddhism appears to have placed no special stigma on homosexual relations, since the subject was not mentioned.

Lay followers and monastic practices

In the early sutras of Buddhism, "accepted or unaccepted human sexual conduct" for laypersons "is not specifically mentioned." "Sexual misconduct" is a broad term, subject to interpretation according to followers' social norms. Early Buddhism appears to have been silent concerning homosexual relations.

According to the Pāli Canon and Āgama (the early Buddhist scriptures), it is not specified that same or opposite gender relations have inherently something to do with sexual misconduct, and some Theravada monks express that same-gender relations do not violate the rule to avoid sexual misconduct, which would mean not having sex with someone underage (thus protected by their parents or guardians), someone betrothed or married and who have taken vows of religious celibacy.

Some later traditions feature restrictions on non-vaginal sex (some Buddhist texts mention non-vaginal sex as sexual misconduct, including men having sex with men or paṇḍaka) though some Buddhist monks interpret such texts as being about forced sex. This non-vaginal sex view is not based on what Buddha said, but from some later scriptures such as the Abhidharma texts.

Regarding Buddhist monks, the Vinaya (code of monastic discipline) bans all sexual activity, but does so in purely physiological terms, making no moral distinctions among the many possible forms of intercourse it lists.

Among Buddhists, there is a wide diversity of opinion about homosexuality. Buddhism teaches that sensual enjoyment and desire in general are hindrances to enlightenment, and inferior to the kinds of pleasure (see, e.g. pīti, a Pāli word often translated as "rapture") that are integral to the practice of jhāna.

Buddhist texts

Early texts

Buddha is often portrayed as a male figure, such as in this painting from a monastery in Laos.

Within the earliest monastic texts such as the Vinaya (c. 4th century BCE), male monks are explicitly forbidden from having sexual relations with any of the four genders: male, female, ubhatovyañjanaka and paṇḍaka; various meanings of these words are given below. Later, the Buddha allowed the ordination of women, forbade ordination to these other types of people, with exceptions to a few particular types of paṇḍaka. The Buddha's proscriptions against certain types of people joining the monastic sangha (ordained community) are often understood to reflect his concern with upholding the public image of the sangha as virtuous; in some cases, this is explicitly stated. Social acceptability was vital for the sangha, as it could not survive without material support from lay society.

Ubhatovyañjanakas

The word ubhatovyañjanaka is usually thought to describe people who have both male and female sexual characteristics: hermaphrodites (intersex). In the Vinaya, it is said that ubhatovyañjanaka should not be ordained, on account of the possibility that they would entice a fellow monk or nun into having sex. Although it has been seen by some that the category of ubhatovyañjanaka is of later addition to the early buddhist texts, since it does not appear in the early suttas, the Pāṭimokkhas, nor in the early parts of the Vinaya.

Paṇḍakas

The paṇḍaka is a complex category that is variously defined in different Buddhist texts. In the earliest texts, the word seems to refer to a socially stigmatized class of trans-feminine and/or cross-dressing people, some of whom may have been sex workers. Paisarn Likhitpreechakul argue that these people are grouped together with groups who are excluded from ordination as well; those with physical disabilities such as deafness or dwarfism, or those who have committed crimes. "The Story of the Prohibition of the Ordination of Pandaka" from the Vinaya claims that the ban is a response to the example of a paṇḍaka monk with a desire to have sex. Being refused by other monks, he had sex with animal handlers, who then told the wider community and brought disgrace upon the sangha. Since the word paṇḍaka does not appear in either of the early Suttas nor in the early parts of the Vinaya, it has been seen by some as a possible indication that pandaka's inclusion in the Vinaya did not happen in the Buddha's lifetime but was added later.

In the Lotus Sutra, it said Bodhisattva should not go near Paṇḍaka, as like what monk rules said in Vinaya. The Theravadin text Milinda Panha, claims that Paṇḍakas let out secrets through their imperfection.

LGBTQ+ people in later traditions

Some modern commentators interpret the word ubhatovyanjañaka as including those who are not physically intersex, but display behavioral and psychological characteristics of both sexes, such as a woman who is attracted to other women. 5th-century Buddhist writer Buddhaghosa describes ubhatobyanjanaka as people with the body of one sex but the "power", or gender of the other. Leonard Zwilling argues that in this account Buddhaghosa does not in fact describe "hermaphroditism" but rather bisexuality or homosexuality. Janet Gyatso pointed out that Zwilling destroys his own argument that pandakas are homosexuals when he writes, "The Vinaya, in fact, goes so far as to distinguish sexual activity between normative males from sexual relations between a socially normative male and a pandaka."

In other texts, the term paṇḍaka can include those born sexually indeterminate or with no sex, eunuchs, those whose impotence changes every half month, males who gain sexual potency by absorbing other men's semen, or spying on other people having sex. It sometimes includes males or females with any sexual dysfunction, such as impotence or irregular menstrual cycles. The common element seems to be those whose sexuality is either limited physiologically, or those who are sexually impotent. Together these impotence types are almost always portrayed negatively as a pariah class, especially in the earliest texts. In modern contexts, paṇḍaka is sometimes alleged to include lesbians, gay men, and transgender and intersex people, although in ancient times, a man who sexually penetrated another man or a paṇḍaka was not himself considered a paṇḍaka.

Some texts of the Abhidharma state that a paṇḍaka cannot achieve enlightenment in their own lifetime, but must wait for reincarnation as a man or woman. Ananda — Buddha's cousin and disciple — was said to be a paṇḍaka in one of his many previous lives, as was the Buddhist nun Isidāsī (from the Therigatha). In both cases, birth as a paṇḍaka was a result of poor karma, and the idea that being a paṇḍaka stems from bad behaviour in a previous life is common in Buddhist literature. Asanga and Vasubandhu discussed if a pandaka was able to be enlightened or not.

In the Samantapasadika, a work of the 5th century CE Theravadin commentator and scholar Buddhaghosa's, paṇḍaka are described as being filled with defiling passions (ussanakilesa), unquenchable lusts (avapasantaparilaha) and are dominated by their libido (parilahavegabhibhuta). The 4th century Mahayana Buddhist writers Vasubandhu and Asanga contend that the paṇḍaka has no discipline for spiritual practice, due to their defiling passions of both male and female sexes. They lack the moral fortitude to counter these passions because they lack modesty and shame. Incapable of showing restraint, such a being is abandoned by their parents and lacking such ties are unable to hold strong views. Asanga, like Vasubandhu, refuses the pandaka recognition as a layman on the grounds that such persons are unfit to associate with or serve the Sangha. Asanga, however, considers them capable to practice the path of a layman individually if they so desire, but without receiving recognition as a layman or being introduced in the sangha. A position similar to Asanga view was also featured in the Lotus Sutra, where sangha members were advised to avoid the paṇḍaka.

Buddha's proscriptions against certain types of people joining the monastic sangha (ordained community) are often understood to reflect his concern with upholding the public image of the sangha as virtuous. Peter Jackson, the scholar of sexual politics and Buddhism in Thailand, speculates that the Buddha was initially reluctant to allow women to join the sangha for this reason. Jackson explains:

Buddhism, the middle path, has always been concerned with the maintenance of social order and since the Buddha's time the sangha has never claimed to provide a universal vehicle for the spiritual liberation of all individuals in society, explicitly excluding those who are considered to reflect badly on the monkhood in terms of prevailing social norms and attitudes.

Social acceptability was vital for the sangha, as it could not survive without material support from lay society.

Several Theravada Buddhist texts state that the members of the paṇḍaka are excluded from a variety of Buddhist practices (in addition to ordination):

• acting as preceptors in ordination ceremonies
• making donations to begging monks
• meditating and
• ability to understand the Dharma.

The Mahakala Ma Ning, a wrathful deity revered in Tibetan Buddhism, especially the Nyingma school, as a defender of the Dharma. The term ma ning has been translated as "genderless" or "eunuch", and equates to paṇḍaka. In this macabre 19th-century image the Ma Ning holds a human heart in their hand, and also a garland of hearts around their waist.

Classical Mahayana scholars like Shantideva and Aśvaghoṣa considered non-vagina sex (including men sex with men) to be sexual misconduct. Shantideva based his views on quotes from the Saddharma-smrtyupasthana Sutra. According to Mizuno Kōgen's study, Saddharma-smrtyupasthana Sutra is related to Ghoṣa's Abhidharmāmṛtarasaśāstra, and is compiled by Sarvastivada sect (possibly from someone related to Ghoṣa after the 2nd century). In the Great Treatise on the Perfection of Wisdom (Sanskrit: Mahā-prajñāpāramitā-śāstra), the Madhyamaka scholar Nagarjuna mentioned the non-vagina sex restriction as based on coerced action toward one's own spouse.

In contrast, later texts, particularly Tibetan Buddhist writings, occasionally value paṇḍaka positively for their "middleness" and balance. The paṇḍaka in these Tibetan works is translated with the term ma ning — "genderless" or "without genitals". The 13th-century Tibetan monk Gyalwa Yang Gönpa, who was one of the significant figures in the early Drukpa Kagyu sect, writes about ma ning as a balanced state between maleness and femaleness. Yang Gönpa describes ma ning as "the abiding breath between male exhalation and female inhalation" and "the balanced yogic channel, as opposed to the too tight male channel, and the too loose female one".

Most Mahayana teachings assert that all beings who correctly practice the dharma may reach enlightenment, since all possess innate Buddha nature. Enlightenment being achievable even in a single life.

Tibetan Buddhism

There are different views among Tibetan Buddhist teachers on acceptable expressions of sexuality.

Historically, Gampopa (12th century), one of the main early masters of the Kagyu school of Tibetan Buddhism, followed the Indian Buddhist tradition, starting with the 3rd-century Hinayana texts of Vasubandhu, and stated that oral and anal sex, whether with a man or a woman, are violations of the third precept regarding inappropriate sexual behavior. Longchenpa, the 13th-century founder of the Nyingma school, citing the 3rd-century Mahayana texts of the Indian master Asanga, elaborated that inappropriate sexual behavior also include the hands among inappropriate parts of the body for sexual activity. In the same way, Gelug predecessor Je Tsongkhapa also adheres to such rules in his studies.

Various contemporary teachers of Tibetan Buddhist lineages, including the 17th Gyalwang Karmapa, have offered understanding towards LGBTQ people while noting that same-sex relationships do not necessarily constitute misconduct for lay people. The Dalai Lama has maintained the views of Je Tsongkhapa

The 14th Dalai Lama has "voiced his support for the full recognition of human rights for all people, regardless of sexual orientation," while noting that from a Buddhist point of view, lesbian and gay sex is "generally considered sexual misconduct." In the most recent interview with the Dalai Lama on this topic (March 10, 2014), the Dalai Lama said gay marriage is OK provided it's not in contradiction with the values of one's chosen religion.

The Dalai Lama has also stated that any sex other than penis-vagina intercourse with one's own monogamous partner, including oral sex, anal sex, and masturbation is improper from the Buddhist perspective. In his 1996 book Beyond Dogma, he states, "A sexual act is deemed proper when the couples use the organs intended for sexual intercourse and nothing else... homosexuality, whether it is between men or between women, is not improper in itself. What is improper is the use of organs already defined as inappropriate for sexual contact."

The Dalai Lama cited the Indian Buddhist texts of Vasubandhu, Asanga, and Ashvaghosha as his sources concerning what constitutes inappropriate sexual behavior. In 1997 he stated: "Sexual organs were created for reproduction between the male element and the female element — and everything that deviates from that is not acceptable from a Buddhist point of view." The Dalai Lama has repeatedly said to LGBT groups that he can't rewrite the texts. He thinks that this is the type of issue that would need to be discussed by a council of Buddhist elders from all Buddhist traditions.

Theravada Buddhism

Peter Jackson, an Australian scholar of sexual politics and Buddhism in Thailand, writes that "Buddhism is a complex tradition and there is no single canonical or scripturally sanctioned position on homosexuality." Thailand is one of several countries with a large population of Theravada Buddhist.

In traditional Thai Buddhist accounts of sexuality, "[sexual] actions and desires have an involuntary cause [and] do not themselves accrue any future karmic consequences. They are the outworking of past karma, not sources for the accumulation of future karma. According to Bunmi, homosexual activity and the desire to engage in homosexual activity fall into this category and are not sinful and do not accrue karmic consequences." Jackson writes that this understanding of homosexuality "prevailed in Thailand until recent decades."

In the 1980s in Thailand, during the AIDS epidemic, there was "a shift in Buddhist attitudes from relative tolerance of homosexuality to condemnation." These views were "unprecedented in recent Thai history." During this time there were two ways Buddhists viewed homosexuality: in the sympathetic view, it was said that homosexuality arose from the karma of previous lifetimes; in the intolerant view, it was seen as arising from immoral conduct in one's present life.

In 1989, the supreme governing body of the Thai sangha affirmed that "gays" (here translated from Thai kathoey) are prohibited from being ordained. Their declaration has apparently gone unheeded in some quarters, as Phra Pisarn Thammapatee (AKA Phra Payom Kalayano), one of the most eminent monks in the country, demanded in 2003 that 1,000 gay monks be ousted from the sangha, and that better screening processes are put in place to keep out any gay postulants.

Recently, Phra Payom Kalayano, an eminent monk and abbot, affirmed the rights of gay monks to join the Sangha: "In the past, katoey had no hope of being ordained because the rules were stricter and society was less open minded. But they have just as much right as anyone else to join the monkhood." This view has been affirmed by other Thai Theravada monks. But it's not known if that statement given by Phra Payom was supported only by some monks of his sangha or by official structures of his Theravada school. There is no information if any other Thai branches of Theravada has changed their attitude towards LGBT ordination because other Theravada branches does not seem to have changed their attitude on this issue.

Japanese Buddhism

See also: Homosexuality in Japan § Monastic same-sex love

Several writers have noted the strong historical tradition of open bisexuality and homosexuality among male Buddhist institutions in Japan. When the Tendai priest Genshin harshly criticised homosexuality as immoral, others mistook his criticism as having been because the acolyte wasn't one's own. Chigo Monogatari (稚児物語), "acolyte stories" of love between monks and their chigo were popular, and such relationships appear to have been commonplace, alongside sex with women.

Western Christian travellers to Japan from the 16th century have noted (with distaste) the prevalence and acceptance of forms of homosexuality among Japanese Buddhists—Jesuit priest Francis Cabral wrote in 1596 that "abominations of the flesh" and "vicious habits" were "regarded in Japan as quite honourable; men of standing entrust their sons to the bonzes to be instructed in such things, and at the same time to serve their lust".

A 17th-century Japanese Buddhist scholar, Kitamura Kigin(北村季吟,きたむら きぎん), wrote that Buddha explained the pursuit of homosexuality over heterosexuality among priests:

It has been the nature of men's hearts to take pleasure in a beautiful woman since the age of male and female gods, but to become intoxicated by the blossom of a handsome youth... would seem to be both wrong and unusual. Nevertheless, the Buddha preached that [Mount] Imose was a place to be avoided and the priests of the law entered this Way as an outlet for their feelings, since their hearts were, after all, made of neither stone nor wood. Like water that plunges from the peak of Tsukubane to form the deep pools of the Minano River, this love has surpassed in depth the love between women and men in these latter days. It plagues the heart not only of courtier and aristocrat but also of brave warriors. Even the mountain dwellers who cut brush for fuel have learned to take pleasure in the shade of young saplings." — Wild Azaleas (1676)

A later Japanese legend attributed the introduction of monastic homosexuality to Japan to Shingon founder Kukai, although scholars now dismiss the veracity of this assertion, pointing out his strict adherence to the Vinaya. Nonetheless, the legend served to "affirm same-sex relation between men and boys in 17th century Japan."

Chinese Buddhism

About Buddhism and homosexuality in China, scholar A. L. De Silva writes, "Generally the attitude has been one of tolerance. Matteo Ricci, the Jesuit missionary who lived in China for 27 years from 1583, expressed horror at the open and tolerant attitude that the Chinese took to homosexuality and naturally enough saw this as proof of the degeneracy of Chinese society."

Venerable Hsing Yun, one of the premier figures in contemporary Chinese Buddhism, has stated that Buddhism should never teach intolerance toward homosexuality, and that people should expand their minds.

Marriage is an institution that reflects the values of the society that supports it. If the people of a society no longer believe that it is important to be married, then there is no reason why they cannot change the institution of marriage. Marriage is a custom. Customs can always be changed. We can find the same core point in this question as we have in others — the ultimate truth of the matter is that individuals can and should decide for themselves what is right. As long as they are not violating others or breaking the laws of the society in which they are living, then they are free to do what they believe is right. It is not for me or anyone else to tell them that they must get married if they want to live together. That is their choice and their choice alone.

The same analysis can be applied to homosexuality. People often ask me what I think about homosexuality. They wonder, is it right, is it wrong? The answer is, it is neither right nor wrong. It is just something that people do. If people are not harming each other, their private lives are their own business; we should be tolerant of them and not reject them.

However, it will still take some time for the world to fully accept homosexuality. All of us must learn to tolerate the behavior of others. Just as we hope to expand our minds to include all of the universe, so we should also seek to expand our minds to include all of the many forms of human behavior.

Tolerance is a form of generosity and it is a form of wisdom. There is nothing anywhere in the Dharma that should ever lead anyone to become intolerant. Our goal as Buddhists is to learn to accept all kinds of people and to help all kinds of people discover the wisdom of the teachings of Shakyamuni Buddha.

— Hsing Yun, Buddhism Pure and Simple, pp. 137–138

Hsing Yun is a best-selling author throughout the Sinosphere, as well as an advocate of Humanistic Buddhism, an approach to reform Chinese Buddhism to match the needs of contemporary laypeople. As such, his views may not be wholly reflective of the older Buddhist views in China. However, at an earlier point (1998), he is quoted as remarking that "according to Buddhism, any emotional involvement, whether homosexual or heterosexual, is a form of attachment ... and, therefore, is a source of suffering," and when the topic of homosexuality, in particular, came up "the master spoke more strongly, calling homosexuality a perverted view (xiejian)."

An example of an older view in opposition to homosexuality is given by the traditional master, Hsuan Hua, an important figure for Buddhism in both China and the United States. Master Hsuan Hua stated that homosexuality "plants the seeds that lead to rebirth in the lower realms of existence". In his commentary on the Sutra of Forty-two Chapters, he described homosexuality as behavior caused by confusion, as the product of sexual desires.

Some Mahayana Buddhist leaders were active in the movement for same-sex marriage rights in Taiwan which legalized same-sex marriages in 2019.

Buddhism in the West

Soeng Hyang, current headmaster of the Kwan Um School of Zen

In contrast to Buddhism in Asia, modern Buddhism in the Western world (European Union, United States of America, Canada, Australia and New Zealand) is typically associated with a concern for social equality—partly as a result of its largely middle-class intellectual membership base, and its philosophical roots in freethought and secular humanism. When applying Buddhist philosophy to the question of homosexuality, western Buddhists often emphasize the importance the Buddha placed on tolerance, compassion, and seeking answers within one's self. They stress these overarching values rather than examining specific passages or texts. As a result, western Buddhism is often relatively gay-friendly, especially since the 1990s. As an interpretation of what is sexual misconduct is an individual decision and not subject to judgment by any central authority, a view of accepting all peoples, but rejecting certain types of sexual acts is more predominant. LGBT people such as Issan Dorsey, Caitriona Reed, Pat Enkyo O'Hara and Soeng Hyang have been ordained as Buddhist monastics and clergy.

An interesting position comes from the western scholar Alexander Berzin,

The texts in Buddhist traditions have been written from the point of view of a heterosexual male. We need to explore the intention of the teachings on sexual misconduct, which ultimately is to eliminate attachment, obsessive desire, and dissatisfaction. If a heterosexual male finds no bounds to these disturbing emotions, he might have sex with someone else's partner, as well as other men. We can apply the same logic and explore what constitutes boundless attachment and dissatisfaction for homosexual and bisexual males or females. For example, having sex with somebody else's partner and so on could be destructive for these types of person as well".

The Dzogchen Ponlop Rinpoche, holder of the Karma Kagyu and Nyingma lineages, in a 2008 talk delivered to LGBT Dharma practitioners at the Shambhala Meditation Center of New York, stressed that for vajrayana lay practitioners, homosexual relationships are no better or worse than heterosexual relationships and that only unhealthy relationships, in general, are to be avoided. Both the Nalandabodhi sangha, which was founded by The Dzogchen Ponlop Rinpoche, and the Shambhala sanghas founded by Chögyam Trungpa Rinpoche have stated that they are welcoming of all sexual orientations. The Shambhala Meditation Center of New York hosts a weekly practice group, Queer Dharma, specifically catering to the needs of the LGBTQ Buddhist community. According to the Danish Karma Kagyu Lama Ole Nydahl, Buddha saw homosexuality as circumstances making life more difficult, but also explained the reason for homosexuality could be aversion against the opposite sex in a former life. Nydahl says however that sexual orientation is not really important in order to practice Buddhism. Well known Zen Buddhist, Thich Nhat Hanh, notes the spirit of Buddhism is inclusiveness and states "when you look at the ocean, you see different kinds of waves, many sizes and shapes, but all the waves have water as their foundation and substance. If you are born gay or lesbian, your ground of being in the same as mine. We are different, but we share the same ground of being."

The U.S. branch of Soka Gakkai International, a Japan-based new religious movement (Japanese new religion) influenced by Nichiren Buddhism, announced in 1995 that they would start holding wedding ceremonies for same-sex couples, and in 2001 established a conference for LGBT members and their supporters. A Buddhist temple in Salt Lake City connected with Jodo Shinshu, another Japanese school of Buddhism, also holds religious rites for same-sex couples.

Another Buddhist organization founded in the West, Juniper Foundation, wrote an article A Buddhist Vote for Same Sex Marriage demonstrating how Buddhist thinking embraces same-sex marriage:

The heart of Buddhist thought is its insight philosophy, which uses critical inquiry to challenge dogma and to reveal how seemingly fixed ideas are more arbitrary than we might think. Applying this philosophy, we see that social customs are not fixed laws but evolving conventions that serve a purpose in a particular culture and time. Marriage is one of these conventions. It is not a rigid law but a social custom that evolves.

In western traditions, there is widescale support for LGBT rights including the European Buddhist Union, the Buddhist Churches of America, many Shin Buddhist groups, and Zen leaders such as Thich Nhat Hanh. In a PEW research poll, 88% of American Buddhists stated that homosexuality should be accepted. This was a higher level of support than any other religious group studied. On 2012, the Australian branch of Buddhism voiced its support for same-sex marriage in a hearing of the Australian Parliament that sought to gather views on whether to legalize same-sex marriage.

Hysteresis

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Hysteresis 

Electric displacement field D of a ferroelectric material as the electric field E is first decreased, then increased. The curves form a hysteresis loop.

Hysteresis is the dependence of the state of a system on its history. For example, a magnet may have more than one possible magnetic moment in a given magnetic field, depending on how the field changed in the past. Plots of a single component of the moment often form a loop or hysteresis curve, where there are different values of one variable depending on the direction of change of another variable. This history dependence is the basis of memory in a hard disk drive and the remanence that retains a record of the Earth's magnetic field magnitude in the past. Hysteresis occurs in ferromagnetic and ferroelectric materials, as well as in the deformation of rubber bands and shape-memory alloys and many other natural phenomena. In natural systems it is often associated with irreversible thermodynamic change such as phase transitions and with internal friction; and dissipation is a common side effect.

Hysteresis can be found in physics, chemistry, engineering, biology, and economics. It is incorporated in many artificial systems: for example, in thermostats and Schmitt triggers, it prevents unwanted frequent switching.

Hysteresis can be a dynamic lag between an input and an output that disappears if the input is varied more slowly; this is known as rate-dependent hysteresis. However, phenomena such as the magnetic hysteresis loops are mainly rate-independent, which makes a durable memory possible.

Systems with hysteresis are nonlinear, and can be mathematically challenging to model. Some hysteretic models, such as the Preisach model (originally applied to ferromagnetism) and the Bouc–Wen model, attempt to capture general features of hysteresis; and there are also phenomenological models for particular phenomena such as the Jiles–Atherton model for ferromagnetism.

Etymology and history

The term "hysteresis" is derived from ὑστέρησις, an Ancient Greek word meaning "deficiency" or "lagging behind". It was coined in 1881 by Sir James Alfred Ewing to describe the behaviour of magnetic materials.

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 Ferenc Preisach (Preisach model of hysteresis), Louis Néel and Douglas Hugh Everett in connection with magnetism and absorption. A more formal mathematical theory of systems with hysteresis was developed in the 1970s by a group of Russian mathematicians led by Mark Krasnosel'skii.

Types

Rate-dependent

One type of hysteresis is a lag between input and output. An example is a sinusoidal input X(t) that results in a sinusoidal output Y(t), but with a phase lag φ:

${\displaystyle {\begin{aligned}X(t)&=X_{0}\sin \omega t\\Y(t)&=Y_{0}\sin \left(\omega t-\varphi \right).\end{aligned}}}$

Such behavior can occur in linear systems, and a more general form of response is

${\displaystyle Y(t)=\chi _{\text{i}}X(t)+\int _{0}^{\infty }\Phi _{\text{d}}(\tau )X(t-\tau )\,\mathrm {d} \tau ,}$

where ${\displaystyle \chi _{\text{i}}}$ is the instantaneous response and ${\displaystyle \Phi _{d}(\tau )}$ is the impulse response to an impulse that occurred ${\displaystyle \tau }$ time units in the past. In the frequency domain, input and output are related by a complex generalized susceptibility that can be computed from ${\displaystyle \Phi _{d}}$; it is mathematically equivalent to a transfer function in linear filter theory and analogue signal processing.

This kind of hysteresis is often referred to as rate-dependent hysteresis. If the input is reduced to zero, the output continues to respond for a finite time. This constitutes a memory of the past, but a limited one because it disappears as the output decays to zero. The phase lag depends on the frequency of the input, and goes to zero as the frequency decreases.

When rate-dependent hysteresis is due to dissipative effects like friction, it is associated with power loss.

Rate-independent

Systems with rate-independent hysteresis have a persistent memory of the past that remains after the transients have died out. The future development of such a system depends on the history of states visited, but does not fade as the events recede into the past. If an input variable X(t) cycles from X₀ to X₁ and back again, the output Y(t) may be Y₀ initially but a different value Y₂ upon return. The values of Y(t) depend on the path of values that X(t) passes through but not on the speed at which it traverses the path. Many authors restrict the term hysteresis to mean only rate-independent hysteresis. Hysteresis effects can be characterized using the Preisach model and the generalized Prandtl−Ishlinskii model.

In engineering

Control systems

In control systems, hysteresis can be used to filter signals so that the output reacts less rapidly than it otherwise would by taking recent system history into account. For example, a thermostat controlling a heater may switch the heater on when the temperature drops below A, but not turn it off until the temperature rises above B. (For instance, if one wishes to maintain a temperature of 20 °C then one might set the thermostat to turn the heater on when the temperature drops to below 18 °C and off when the temperature exceeds 22 °C).

Similarly, a pressure switch can be designed to exhibit hysteresis, with pressure set-points substituted for temperature thresholds.

Electronic circuits

Sharp hysteresis loop of a Schmitt trigger

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 Schmitt trigger is a simple electronic circuit that exhibits this property.

A latching relay uses a solenoid to actuate a ratcheting mechanism 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).

Hysteresis can be used when connecting arrays of elements such as nanoelectronics, electrochrome cells and memory effect devices using passive matrix addressing. 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. Thus, all rows can be addressed at the same time instead of individually.

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.

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.

Aerodynamics

In aerodynamics, hysteresis can be observed when decreasing the angle of attack of a wing after stall, regarding the lift and drag coefficients. The angle of attack at which the flow on top of the wing reattaches is generally lower than the angle of attack at which the flow separates during the increase of the angle of attack.

Backlash

Moving parts within machines, such as the components of a gear train, normally have a small gap between them, to allow movement and lubrication. As a consequence of this gap, any reversal in direction of a drive part will not be passed on immediately to the driven part. This unwanted delay is normally kept as small as practicable, and is usually called backlash. The amount of backlash will increase with time as the moving parts wear away.

In mechanics

Elastic hysteresis

Elastic hysteresis of an idealized rubber band. The area in the centre of the hysteresis loop is the energy dissipated due to internal friction.

In the elastic hysteresis of rubber, the area in the centre of a hysteresis loop is the energy dissipated due to material internal friction.

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

The effect can be demonstrated using 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 stretch and get longer. As more weights are loaded onto it, the band will continue to stretch because the force the weights are exerting on the band is increasing. When each weight is taken off, or unloaded, the band will contract 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 contracts less, resulting 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 the figure.

In terms of force, the rubber band was harder to stretch when it was being loaded than when it was being unloaded. In terms of time, when the band is unloaded, the effect (the length) lagged behind the cause (the force of the weights) because the length has not yet reached the value it had for the same weight during the loading part of the cycle. In terms of energy, more energy was required during the loading than the unloading, the excess energy being dissipated as thermal energy.

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.

When the intrinsic hysteresis of rubber is being measured, the material can be considered to behave like a gas. When a rubber band is stretched it heats up, and if it is suddenly released, it cools down perceptibly. These effects correspond to 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 can be measured only if the rubber band is thermally isolated.

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 made use of elastomer suspension, as did the original Mini car.

The primary cause of rolling resistance when a body (such as a ball, tire, or wheel) rolls on a surface is hysteresis. This is attributed to the viscoelastic characteristics of the material of the rolling body.

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.

Bubble shape hysteresis

The equilibrium shapes of bubbles expanding and contracting on capillaries (blunt needles) can exhibit hysteresis depending on the relative magnitude of the maximum capillary pressure to ambient pressure, and the relative magnitude of the bubble volume at the maximum capillary pressure to the dead volume in the system. The bubble shape hysteresis is a consequence of gas compressibility, which causes the bubbles to behave differently across expansion and contraction. During expansion, bubbles undergo large non equilibrium jumps in volume, while during contraction the bubbles are more stable and undergo a relatively smaller jump in volume resulting in an asymmetry across expansion and contraction. The bubble shape hysteresis is qualitatively similar to the adsorption hysteresis, and as in the contact angle hysteresis, the interfacial properties play an important role in bubble shape hysteresis.

The existence of the bubble shape hysteresis has important consequences in interfacial rheology experiments involving bubbles. As a result of the hysteresis, not all sizes of the bubbles can be formed on a capillary. Further the gas compressibility causing the hysteresis leads to unintended complications in the phase relation between the applied changes in interfacial area to the expected interfacial stresses. These difficulties can be avoided by designing experimental systems to avoid the bubble shape hysteresis.

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.^[19]

In materials

Magnetic hysteresis

Main article: Magnetic hysteresis

 

Theoretical model of magnetization m against magnetic field h. Starting at the origin, the upward curve is the initial magnetization curve. The downward curve after saturation, along with the lower return curve, form the main loop. The intercepts h_c and m_rs are the coercivity and saturation remanence.

When an external magnetic field is applied to a ferromagnetic material such as iron, the atomic domains align themselves with it. Even when the field is removed, part of the alignment will be retained: the material has become magnetized. Once magnetized, the magnet will stay magnetized indefinitely. To demagnetize it requires heat or 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 field strength H and magnetization M is not linear in such materials. If a magnet is demagnetized (H = M = 0) and the relationship between H and M is plotted for increasing levels of field strength, M follows the initial magnetization curve. This curve increases rapidly at first and then approaches an asymptote called magnetic saturation. If the magnetic field is now reduced monotonically, M follows a different curve. At zero field strength, the magnetization is offset from the origin by an amount called the remanence. If the H-M relationship is plotted for all strengths of applied magnetic field the result is a hysteresis loop called the main loop. The width of the middle section is twice the coercivity of the material.

A closer look at a magnetization curve generally reveals a series of small, random jumps in magnetization called Barkhausen jumps. This effect is due to crystallographic defects such as dislocations.

Magnetic hysteresis loops are not exclusive to materials with ferromagnetic ordering. Other magnetic orderings, such as spin glass ordering, also exhibit this phenomenon.

Physical origin

Main article: Ferromagnetism

The phenomenon of hysteresis in ferromagnetic materials is the result of two effects: rotation of magnetization and changes in size or number of magnetic domains. In general, the magnetization varies (in direction but not magnitude) across a magnet, but in sufficiently small magnets, it does not. In these single-domain magnets, the magnetization responds to a magnetic field by rotating. Single-domain magnets are used wherever a strong, stable magnetization is needed (for example, magnetic recording).

Larger magnets are divided into regions called domains. Across each domain, the magnetization does not vary; but between domains are relatively thin domain walls in which the direction of magnetization rotates from the direction of one domain to another. If the magnetic field changes, the walls move, changing the relative sizes of the domains. Because the domains are not magnetized in the same direction, the magnetic moment per unit volume is smaller than it would be in a single-domain magnet; but domain walls involve rotation of only a small part of the magnetization, so it is much easier to change the magnetic moment. The magnetization can also change by addition or subtraction of domains (called nucleation and denucleation).

Magnetic hysteresis models

The most known empirical models in hysteresis are Preisach and Jiles-Atherton models. These models allow an accurate modeling of the hysteresis loop and are widely used in the industry. However, these models lose the connection with thermodynamics and the energy consistency is not ensured. A more recent model, with a more consistent thermodynamical foundation, is the vectorial incremental nonconservative consistent hysteresis (VINCH) model of Lavet et al. (2011)

Applications

Main article: Magnet § Common uses

There are a great variety of applications of the hysteresis in ferromagnets. Many of these make use of their ability to retain a memory, for example magnetic tape, hard disks, and credit cards. In these applications, hard magnets (high coercivity) like iron are desirable so the memory is not easily erased.

Magnetically soft (low coercivity) iron is used for the cores in electromagnets. The low coercivity reduces that energy loss associated with hysteresis. The low energy loss during a hysteresis loop is also the reason why soft iron is used for transformer cores and electric motors.

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⁻² or C·m). The mechanism, an organization of the polarization into domains, is similar to that of magnetic hysteresis.

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 (185 °F) and solidifies from 32 to 40 °C (90 to 104 °F). 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

Main article: Cell biology

Hysteresis in cell biology often follows bistable systems where the same input state can lead to two different, stable outputs. Where bistability can lead to digital, switch-like outputs from the continuous inputs of chemical concentrations and activities, hysteresis makes these systems more resistant to noise. These systems are often characterized by higher values of the input required to switch into a particular state as compared to the input required to stay in the state, allowing for a transition that is not continuously reversible, and thus less susceptible to noise.

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.

Biochemical systems can also show hysteresis-like output when slowly varying states that are not directly monitored are involved, as in the case of the cell cycle arrest in yeast exposed to mating pheromone. Here, the duration of cell cycle arrest depends not only on the final level of input Fus3, but also on the previously achieved Fus3 levels. This effect is achieved due to the slower time scales involved in the transcription of intermediate Far1, such that the total Far1 activity reaches its equilibrium value slowly, and for transient changes in Fus3 concentration, the response of the system depends on the Far1 concentration achieved with the transient value. Experiments in this type of hysteresis benefit from the ability to change the concentration of the inputs with time. The mechanisms are often elucidated by allowing independent control of the concentration of the key intermediate, for instance, by using an inducible promoter.

Main article: Chromatin

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.

Main article: Morphogen

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.

Immunology

There is some evidence that T cells exhibit hysteresis in that it takes a lower signal threshold to activate T cells that have been previously activated. Ras GTPase activation is required for downstream effector functions of activated T cells. Triggering of the T cell receptor induces high levels of Ras activation, which results in higher levels of GTP-bound (active) Ras at the cell surface. Since higher levels of active Ras have accumulated at the cell surface in T cells that have been previously stimulated by strong engagement of the T cell receptor, weaker subsequent T cell receptor signals received shortly afterwards will deliver the same level of activation due to the presence of higher levels of already activated Ras as compared to a naïve cell.

Neuroscience

See also: Refractory period (physiology)

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 (Δvolume/Δpressure) is due to the additional energy required to overcome surface tension forces 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.

Voice and speech physiology

A hysteresis effect may be observed in voicing onset versus offset. The threshold value of the subglottal pressure required to start the vocal fold vibration is lower than the threshold value at which the vibration stops, when other parameters are kept constant. In utterances of vowel-voiceless consonant-vowel sequences during speech, the intraoral pressure is lower at the voice onset of the second vowel compared to the voice offset of the first vowel, the oral airflow is lower, the transglottal pressure is larger and the glottal width is smaller.

Ecology and epidemiology

Hysteresis is a commonly encountered phenomenon in ecology and epidemiology, where the observed equilibrium of a system can not be predicted solely based on environmental variables, but also requires knowledge of the system's past history. Notable examples include the theory of spruce budworm outbreaks and behavioral-effects on disease transmission.

It is commonly examined in relation to critical transitions between ecosystem or community types in which dominant competitors or entire landscapes can change in a largely irreversible fashion.

In ocean and climate science

Complex ocean and climate models rely on the principle.

In economics

Main article: Hysteresis (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.

When some negative shock reduces employment in a company or industry, fewer employed workers then remain. 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: the unemployment becomes permanently higher after negative shocks.

Permanently higher unemployment

The idea of hysteresis is used extensively in the area of labor economics, specifically with reference to the unemployment rate. According to theories based on hysteresis, severe economic downturns (recession) and/or persistent stagnation (slow demand growth, usually after a recession) cause unemployed individuals to lose their job skills (commonly developed on the job) or to find that their skills have become obsolete, or become demotivated, disillusioned or depressed or lose job-seeking skills. In addition, employers may use time spent in unemployment as a screening tool, i.e., to weed out less desired employees in hiring decisions. Then, in times of an economic upturn, recovery, or "boom", the affected workers will not share in the prosperity, remaining unemployed for long periods (e.g., over 52 weeks). This makes unemployment "structural", i.e., extremely difficult to reduce simply by increasing the aggregate demand for products and labor without causing increased inflation. That is, it is possible that a ratchet effect in unemployment rates exists, so a short-term rise in unemployment rates tends to persist. For example, traditional anti-inflationary policy (the use of recession to fight inflation) leads to a permanently higher "natural" rate of unemployment (more scientifically known as the NAIRU). This occurs first because inflationary expectations are "sticky" downward due to wage and price rigidities (and so adapt slowly over time rather than being approximately correct as in theories of rational expectations) and second because labor markets do not clear instantly in response to unemployment.

The existence of hysteresis has been put forward as a possible explanation for the persistently high unemployment of many economies in the 1990s. Hysteresis has been invoked by Olivier Blanchard among others to explain the differences in long run unemployment rates between Europe and the United States. Labor market reform (usually meaning institutional change promoting more flexible wages, firing, and hiring) or strong demand-side economic growth may not therefore reduce this pool of long-term unemployed. Thus, specific targeted training programs are presented as a possible policy solution. However, the hysteresis hypothesis suggests such training programs are aided by persistently high demand for products (perhaps with incomes policies to avoid increased inflation), which reduces the transition costs out of unemployment and into paid employment easier.

Additional considerations

Models of hysteresis

Each subject that involves hysteresis has models that are specific to the subject. In addition, there are hysteretic 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 linear superposition of square loops called non-ideal relays. Many complex models of hysteresis arise from the simple parallel connection, or superposition, of elementary carriers of hysteresis termed hysterons.

A simple and intuitive parametric description of various hysteresis loops may be found in the Lapshin model. Along with the smooth loops, substitution of trapezoidal, triangular or rectangular pulses instead of the harmonic functions allows piecewise-linear hysteresis loops frequently used in discrete automatics to be built in the model. There are implementations of the hysteresis loop model in Mathcad and in R programming language.

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 versatility 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, and soil liquefaction 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 magnetorheological dampers, base isolation devices for buildings and other kinds of damping devices; it has also been used in the modelling and analysis of structures built of reinforced concrete, steel, masonry and timber. The most important extension of Bouc-Wen Model was carried out by Baber and Noori and later by Noori and co-workers. That extended model, named, BWBN, can reproduce the complex shear pinching or slip-lock phenomenon that earlier model could not reproduce. BWBN model has been widely used in a wide spectrum of applications and have been incorporated in several software codes such as OpenSees.

Energy

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