Invisible hand

From Wikipedia, the free encyclopedia

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

The invisible hand describes the unintended social benefits of an individual's self-interested actions, a concept that was first introduced by Adam Smith in The Theory of Moral Sentiments, written in 1759, invoking it in reference to income distribution.

By the time he wrote The Wealth of Nations in 1776, Smith had studied the economic models of the French Physiocrats for many years, and in this work the invisible hand is more directly linked to production, to the employment of capital in support of domestic industry. The only use of "invisible hand" found in The Wealth of Nations is in Book IV, Chapter II, "Of Restraints upon the Importation from foreign Countries of such Goods as can be produced at Home." The exact phrase is used just three times in Smith's writings.

Smith may have come up with the two meanings of the phrase from Richard Cantillon who developed both economic applications in his model of the isolated estate.

The idea of trade and market exchange automatically channeling self-interest toward socially desirable ends is a central justification for the laissez-faire economic philosophy, which lies behind neoclassical economics. In this sense, the central disagreement between economic ideologies can be viewed as a disagreement about how powerful the "invisible hand" is. In alternative models, forces which were nascent during Smith's lifetime, such as large-scale industry, finance, and advertising, reduce its effectiveness.

Interpretations of the term have been generalized beyond the usage by Smith.

Adam Smith

The Theory of Moral Sentiments

The first appearance of the invisible hand in Smith occurs in The Theory of Moral Sentiments (1759) in Part IV, Chapter 1, where he describes a selfish landlord as being led by an invisible hand to distribute his harvest to those who work for him:

The proud and unfeeling landlord views his extensive fields, and without a thought for the wants of his brethren, in imagination consumes himself the whole harvest ... [Yet] the capacity of his stomach bears no proportion to the immensity of his desires ... the rest he will be obliged to distribute among those, who prepare, in the nicest manner, that little which he himself makes use of, among those who fit up the palace in which this little is to be consumed, among those who provide and keep in order all the different baubles and trinkets which are employed in the economy of greatness; all of whom thus derive from his luxury and caprice, that share of the necessaries of life, which they would in vain have expected from his humanity or his justice...The rich...are led by an invisible hand to make nearly the same distribution of the necessaries of life, which would have been made, had the earth been divided into equal portions among all its inhabitants, and thus without intending it, without knowing it, advance the interest of the society, and afford means to the multiplication of the species. When Providence divided the earth among a few lordly masters, it neither forgot nor abandoned those who seemed to have been left out in the partition.

Elsewhere in The Theory of Moral Sentiments, Smith has described the desire of men to be respected by the members of the community in which they live, and the desire of men to feel that they are honorable beings.

The Wealth of Nations

Adam Smith uses the metaphor in Book IV, Chapter II, paragraph IX of The Wealth of Nations.

But the annual revenue of every society is always precisely equal to the exchangeable value of the whole annual produce of its industry, or rather is precisely the same thing with that exchangeable value. As every individual, therefore, endeavours as much as he can both to employ his capital in the support of domestic industry, and so to direct that industry that its produce may be of the greatest value, every individual necessarily labours to render the annual revenue of the society as great as he can. He generally, indeed, neither intends to promote the public interest, nor knows how much he is promoting it. By preferring the support of domestic to that of foreign industry, he intends only his own security; and by directing that industry in such a manner as its produce may be of the greatest value, he intends only his own gain, and he is in this, as in many other cases, led by an invisible hand to promote an end which was no part of his intention. Nor is it always the worse for the society that it was not part of it. By pursuing his own interest he frequently promotes that of the society more effectually than when he really intends to promote it. I have never known much good done by those who affected to trade for the public good. It is an affectation, indeed, not very common among merchants, and very few words need be employed in dissuading them from it.

Using the invisible hand metaphor, Smith was trying to present how an individual exchanging money in their own self-interest unintentionally impacts the economy as a whole. In other words, there is something that binds self-interest, along with public interest, so that individuals who pursue their own interests will inevitably benefit society as a whole.

Other uses of the phrase by Smith

Only in The History of Astronomy (written before 1758) Smith speaks of the invisible hand, to which ignorants refer to explain natural phenomena otherwise unexplainable:

Fire burns, and water refreshes; heavy bodies descend, and lighter substances fly upwards, by the necessity of their own nature; nor was the invisible hand of Jupiter ever apprehended to be employed in those matters.

In The Theory of Moral Sentiments (1759) and in The Wealth of Nations (1776) Adam Smith speaks of an invisible hand, never of the invisible hand. In The Theory of Moral Sentiments Smith uses the concept to sustain a "trickling down" theory, a concept also used in neoclassical development theory: The gluttony of the rich serves to feed the poor.

The rich … consume little more than the poor, and in spite of their natural selfishness and rapacity, though they mean only their own conveniency, though the sole end which they propose from the labours of all the thousands whom they employ, be the gratification of their own vain and insatiable desires, they divide with the poor the produce of all their improvements. They are led by an invisible hand [emphasis added] to make nearly the same distribution of the necessaries of life, which would have been made, had the earth been divided into equal portions among all its inhabitants, and thus without intending it, without knowing it, advance the interest of the society, and afford means to the multiplication of the species. When Providence divided the earth among a few lordly masters, it neither forgot nor abandoned those who seemed to have been left out in the partition. These last too enjoy their share of all that it produces. In what constitutes the real happiness of human life, they are in no respect inferior to those who would seem so much above them. In ease of body and peace of mind, all the different ranks of life are nearly upon a level, and the beggar, who suns himself by the side of the highway, possesses that security which kings are fighting for.

Smith's visit to France and his acquaintance to the French Économistes (known as Physiocrats) changed his views from micro-economic optimisation to macro-economic growth as the end of Political Economy. So the landlord's gluttony in The Theory of Moral Sentiments is denounced in the Wealth of Nations as unproductive labour. Walker, the first president (1885 to 92) of the American Economic Association, concurred:

The domestic servant … is not employed as a means to his master's profit. His master's income is not due in any part to his employment; on the contrary, that income is first acquired … and in the amount of the income is determined whether the servant shall be employed or not, while to the full extent of that employment the income is diminished. As Adam Smith expresses it "a man grows rich by employing a multitude of manufacturers; he grows poor by maintaining a multitude of menial servants."

Smith's theoretical U-turn from a micro-economical to a macro-economical view is not reflected in The Wealth of Nations. Large parts of this book are retaken from Smith's lectures before his visit to France. So one must distinguish in The Wealth of Nations a micro-economical and a macro-economical Adam Smith. Whether Smith's quotation of an invisible hand in the middle of his work is a micro-economical statement or a macro-economical statement condemning monopolies and government interferences as in the case of tariffs and patents is debatable.

Economist's interpretation

The concept of the "invisible hand" is nearly always generalized beyond Smith's original uses. The phrase was not popular among economists before the twentieth century; Alfred Marshall never used it in his Principles of Economics textbook and neither does William Stanley Jevons in his Theory of Political Economy. Paul Samuelson cites it in his Economics textbook in 1948:

Even Adam Smith, the canny Scot whose monumental book, "The Wealth of Nations" (1776) , represents the beginning of modern economics or political economy-even he was so thrilled by the recognition of an order in the economic system that he proclaimed the mystical principle of the "invisible hand": that each individual in pursuing his own selfish good was led, as if by an invisible hand, to achieve the best good of all, so that any interference with free competition by government was almost certain to be injurious. This unguarded conclusion has done almost as much harm as good in the past century and a half, especially since too often it is all that some of our leading citizens remember, 30 years later, of their college course in economics.

In this interpretation, the theory is that the Invisible Hand states that if each consumer is allowed to choose freely what to buy and each producer is allowed to choose freely what to sell and how to produce it, the market will settle on a product distribution and prices that are beneficial to all the individual members of a community, and hence to the community as a whole. The reason for this is that self-interest drives actors to beneficial behavior in a case of serendipity. Efficient methods of production are adopted to maximize profits. Low prices are charged to maximize revenue through gain in market share by undercutting competitors. Investors invest in those industries most urgently needed to maximize returns, and withdraw capital from those less efficient in creating value. All these effects take place dynamically and automatically.

Since Smith's time, this concept has been further incorporated into economic theory. Léon Walras developed a four-equation general equilibrium model that concludes that individual self-interest operating in a competitive market place produces the unique conditions under which a society's total utility is maximized. Vilfredo Pareto used an Edgeworth box contact line to illustrate a similar social optimality. Ludwig von Mises, in Human Action uses the expression "the invisible hand of Providence", referring to Marx's period, to mean evolutionary meliorism. He did not mean this as a criticism, since he held that secular reasoning leads to similar conclusions. Milton Friedman, a Nobel Memorial Prize winner in economics, called Smith's Invisible Hand "the possibility of cooperation without coercion." Kaushik Basu has called the First Welfare Theorem the Invisible Hand Theorem.

Some economists question the integrity of how the term "invisible hand" is currently used. Gavin Kennedy, Professor Emeritus at Heriot-Watt University in Edinburgh, Scotland, argues that its current use in modern economic thinking as a symbol of free market capitalism is not reconcilable with the rather modest and indeterminate manner in which it was employed by Smith. In response to Kennedy, Daniel Klein argues that reconciliation is legitimate. Moreover, even if Smith did not intend the term "invisible hand" to be used in the current manner, its serviceability as such should not be rendered ineffective. In conclusion of their exchange, Kennedy insists that Smith's intentions are of utmost importance to the current debate, which is one of Smith's association with the term "invisible hand". If the term is to be used as a symbol of liberty and economic coordination as it has been in the modern era, Kennedy argues that it should exist as a construct completely separate from Adam Smith since there is little evidence that Smith imputed any significance onto the term, much less the meanings given it at present.

The former Drummond Professor of Political Economy at Oxford, D. H. MacGregor, argued that:

The one case in which he referred to the ‘invisible hand’ was that in which private persons preferred the home trade to the foreign trade, and he held that such preference was in the national interest, since it replaced two domestic capitals while the foreign trade replaced only one. The argument of the two capitals was a bad one, since it is the amount of capital that matters, not its subdivision; but the invisible sanction was given to a Protectionist idea, not for defence but for employment. It is not surprising that Smith was often quoted in Parliament in support of Protection. His background, like ours today, was private enterprise; but any dogma of non-intervention by government has to make heavy weather in The Wealth of Nations.

Harvard economist Stephen Marglin argues that while the "invisible hand" is the "most enduring phrase in Smith's entire work", it is "also the most misunderstood."

Economists have taken this passage to be the first step in the cumulative effort of mainstream economics to prove that a competitive economy provides the largest possible economic pie (the so-called first welfare theorem, which demonstrates the Pareto optimality of a competitive regime). But Smith, it is evident from the context, was making a much narrower argument, namely, that the interests of businessmen in the security of their capital would lead them to invest in the domestic economy even at the sacrifice of somewhat higher returns that might be obtainable from foreign investment. . . .

David Ricardo . . . echoed Smith . . . [but] Smith's argument is at best incomplete, for it leaves out the role of foreigners' investment in the domestic economy. It would have to be shown that the gain to the British capital stock from the preference of British investors for Britain is greater than the loss to Britain from the preference of Dutch investors for the Netherlands and French investors for France."

According to Emma Rothschild, Smith was actually being ironic in his use of the term. Warren Samuels described it as "a means of relating modern high theory to Adam Smith and, as such, an interesting example in the development of language."

Understood as a metaphor

Smith uses the metaphor in the context of an argument against protectionism and government regulation of markets, but it is based on very broad principles developed by Bernard Mandeville, Bishop Butler, Lord Shaftesbury, and Francis Hutcheson. In general, the term "invisible hand" can apply to any individual action that has unplanned, unintended consequences, particularly those that arise from actions not orchestrated by a central command, and that have an observable, patterned effect on the community.

Bernard Mandeville argued that private vices are actually public benefits. In The Fable of the Bees (1714), he laments that the "bees of social virtue are buzzing in Man's bonnet": that civilized man has stigmatized his private appetites and the result is the retardation of the common good.

Bishop Butler argued that pursuing the public good was the best way of advancing one's own good since the two were necessarily identical.

Lord Shaftesbury turned the convergence of public and private good around, claiming that acting in accordance with one's self-interest produces socially beneficial results. An underlying unifying force that Shaftesbury called the "Will of Nature" maintains equilibrium, congruency, and harmony. This force, to operate freely, requires the individual pursuit of rational self-interest, and the preservation and advancement of the self.

Francis Hutcheson also accepted this convergence between public and private interest, but he attributed the mechanism, not to rational self-interest, but to personal intuition, which he called a "moral sense". Smith developed his own version of this general principle in which six psychological motives combine in each individual to produce the common good. In The Theory of Moral Sentiments, vol. II, page 316, he says, "By acting according to the dictates of our moral faculties, we necessarily pursue the most effective means for promoting the happiness of mankind."

Contrary to common misconceptions, Smith did not assert that all self-interested labour necessarily benefits society, or that all public goods are produced through self-interested labour. His proposal is merely that in a free market, people usually tend to produce goods desired by their neighbours. The tragedy of the commons is an example where self-interest tends to bring an unwanted result.

The invisible hand is traditionally understood as a concept in economics, but Robert Nozick argues in Anarchy, State and Utopia that substantively the same concept exists in a number of other areas of academic discourse under different names, notably Darwinian natural selection. In turn, Daniel Dennett argues in Darwin's Dangerous Idea that this represents a "universal acid" that may be applied to a number of seemingly disparate areas of philosophical inquiry (consciousness and free will in particular), a hypothesis known as Universal Darwinism. However, positing an economy guided by this principle as ideal may amount to Social Darwinism, which is also associated with champions of laissez-faire capitalism.

Tawney's interpretation

Christian socialist R. H. Tawney saw Smith as putting a name on an older idea:

If preachers have not yet overtly identified themselves with the view of the natural man, expressed by an eighteenth-century writer in the words, trade is one thing and religion is another, they imply a not very different conclusion by their silence as to the possibility of collisions between them. The characteristic doctrine was one, in fact, which left little room for religious teaching as to economic morality, because it anticipated the theory, later epitomized by Adam Smith in his famous reference to the invisible hand, which saw in economic self-interest the operation of a providential plan... The existing order, except insofar as the short-sighted enactments of Governments interfered with it, was the natural order, and the order established by nature was the order established by God. Most educated men, in the middle of the [eighteenth] century, would have found their philosophy expressed in the lines of Pope:

Thus God and Nature formed the general frame,

And bade self-love and social be the same.

Naturally, again, such an attitude precluded a critical examination of institutions, and left as the sphere of Christian charity only those parts of life that could be reserved for philanthropy, precisely because they fell outside that larger area of normal human relations, in which the promptings of self-interest provided an all-sufficient motive and rule of conduct. (Religion and the Rise of Capitalism, pp. 191–192.)

Criticisms

Joseph E. Stiglitz

The Nobel Prize-winning economist Joseph E. Stiglitz, says: "the reason that the invisible hand often seems invisible is that it is often not there."^] Stiglitz explains his position:

Adam Smith, the father of modern economics, is often cited as arguing for the "invisible hand" and free markets: firms, in the pursuit of profits, are led, as if by an invisible hand, to do what is best for the world. But unlike his followers, Adam Smith was aware of some of the limitations of free markets, and research since then has further clarified why free markets, by themselves, often do not lead to what is best. As I put it in my new book, Making Globalization Work, the reason that the invisible hand often seems invisible is that it is often not there. Whenever there are "externalities"—where the actions of an individual have impacts on others for which they do not pay, or for which they are not compensated—markets will not work well. Some of the important instances have long understood environmental externalities. Markets, by themselves, produce too much pollution. Markets, by themselves, also produce too little basic research. (The government was responsible for financing most of the important scientific breakthroughs, including the internet and the first telegraph line, and many bio-tech advances.) But recent research has shown that these externalities are pervasive, whenever there is imperfect information or imperfect risk markets—that is always. Government plays an important role in banking and securities regulation, and a host of other areas: some regulation is required to make markets work. Government is needed, almost all would agree, at a minimum to enforce contracts and property rights. The real debate today is about finding the right balance between the market and government (and the third "sector" – governmental non-profit organizations). Both are needed. They can each complement each other. This balance differs from time to time and place to place.

The preceding claim is based on Stiglitz's 1986 paper, "Externalities in Economies with Imperfect Information and Incomplete Markets", which describes a general methodology to deal with externalities and for calculating optimal corrective taxes in a general equilibrium context. In it he considers a model with households, firms and a government.

Households maximize a utility function ${\displaystyle u^{h}(x^{h},z^{h})}$, where ${\displaystyle x^{h}}$ is the consumption vector and ${\displaystyle z^{h}}$ are other variables affecting the utility of the household (e.g. pollution). The budget constraint is given by ${\displaystyle x_{1}^{h}+q\cdot {\bar {x}}^{h}\leq I^{h}+\sum a^{hf}\cdot \pi ^{f}}$, where q is a vector of prices, a^hf the fractional holding of household h in firm f, π^f the profit of firm f, I^h a lump sum government transfer to the household. The consumption vector can be split as ${\displaystyle x^{h}=\left(x_{1}^{h},{\bar {x}}^{h}\right)}$.

Firms maximize a profit ${\displaystyle \pi ^{f}=y_{1}^{f}+p\cdot {\bar {y}}_{1}}$, where y^f is a production vector and p is vector of producer prices, subject to ${\displaystyle y_{1}^{f}-G^{f}({\bar {y}}^{f},z^{f})\leq 0}$, G_f a production function and z^f are other variables affecting the firm. The production vector can be split as ${\displaystyle y^{f}=\left(y_{1}^{f},{\bar {y}}^{f}\right)}$.

The government receives a net income ${\displaystyle R=t\cdot {\bar {x}}-\sum I^{h}}$, where ${\displaystyle t=(q-p)}$ is a tax on the goods sold to households.

It can be shown that in general the resulting equilibrium is not efficient.

Proof

Noam Chomsky

Noam Chomsky suggests that Smith (and more specifically David Ricardo) sometimes used the phrase to refer to a "home bias" for investing domestically in opposition to offshore outsourcing production and neoliberalism.

Rather interestingly, these issues were foreseen by the great founders of modern economics, Adam Smith for example. He recognized and discussed what would happen to Britain if the masters adhered to the rules of sound economics – what's now called neoliberalism. He warned that if British manufacturers, merchants, and investors turned abroad, they might profit but England would suffer. However, he felt that this wouldn't happen because the masters would be guided by a home bias. So as if by an invisible hand England would be spared the ravages of economic rationality. That passage is pretty hard to miss. It's the only occurrence of the famous phrase "invisible hand" in Wealth of Nations, namely in a critique of what we call neoliberalism.

Stephen LeRoy

Stephen LeRoy, professor emeritus at the University of California, Santa Barbara, and a visiting scholar at the Federal Reserve Bank of San Francisco, offered a critique of the Invisible Hand, writing that "[T]he single most important proposition in economic theory, first stated by Adam Smith, is that competitive markets do a good job allocating resources. (...) The financial crisis has spurred a debate about the proper balance between markets and government and prompted some scholars to question whether the conditions assumed by Smith...are accurate for modern economies.

John D. Bishop

John D. Bishop, a professor who worked at Trent University, Peterborough, indicates that the invisible hand might be applied differently for merchants and manufacturers than how it’s applied with society. He wrote an article in 1995 titled "Adam Smith's Invisible Hand Argument", and in the article, he suggests that Adam Smith might be contradicting himself with the "Invisible Hand". He offers various critiques of the "Invisible Hand", and he writes that “the interest of business people are in fundamental conflict with the interest of society as a whole, and that business people pursue their personal goal at the expense of the public good”. Thus, Bishop indicates that the “business people” are in conflict with society over the same interests and that Adam Smith might be contradicting himself. According to Bishop, he also gives the impression that in Smith's book 'The Wealth of Nations,' there's a close saying that "the interest of merchants and manufacturers were fundamentally opposed of society in general, and they had an inherent tendency to deceive and oppress society while pursuing their own interests." Bishop also states that the "invisible hand argument applies only to investing capital in one's own country for a maximum profit." In other words, he suggests that the invisible hand applies to only the merchants and manufacturers and that they’re not the invisible force that moves the economy. However, Bishop mentions that the argument “does not apply to the pursuit of self-interest (…) in any area outside of economic activities.”

Nonlinear system

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Nonlinear_system#See_also 

In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists because most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems.

Typically, the behavior of a nonlinear system is described in mathematics by a nonlinear system of equations, which is a set of simultaneous equations in which the unknowns (or the unknown functions in the case of differential equations) appear as variables of a polynomial of degree higher than one or in the argument of a function which is not a polynomial of degree one. In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a linear combination of the unknown variables or functions that appear in them. Systems can be defined as nonlinear, regardless of whether known linear functions appear in the equations. In particular, a differential equation is linear if it is linear in terms of the unknown function and its derivatives, even if nonlinear in terms of the other variables appearing in it.

As nonlinear dynamical equations are difficult to solve, nonlinear systems are commonly approximated by linear equations (linearization). This works well up to some accuracy and some range for the input values, but some interesting phenomena such as solitons, chaos, and singularities are hidden by linearization. It follows that some aspects of the dynamic behavior of a nonlinear system can appear to be counterintuitive, unpredictable or even chaotic. Although such chaotic behavior may resemble random behavior, it is in fact not random. For example, some aspects of the weather are seen to be chaotic, where simple changes in one part of the system produce complex effects throughout. This nonlinearity is one of the reasons why accurate long-term forecasts are impossible with current technology.

Some authors use the term nonlinear science for the study of nonlinear systems. This term is disputed by others:

Using a term like nonlinear science is like referring to the bulk of zoology as the study of non-elephant animals.

— Stanislaw Ulam

Definition

In mathematics, a linear map (or linear function) ${\displaystyle f(x)}$ is one which satisfies both of the following properties:

• Additivity or superposition principle: ${\displaystyle \textstyle f(x+y)=f(x)+f(y);}$
• Homogeneity: ${\displaystyle \textstyle f(\alpha x)=\alpha f(x).}$

Additivity implies homogeneity for any rational α, and, for continuous functions, for any real α. For a complex α, homogeneity does not follow from additivity. For example, an antilinear map is additive but not homogeneous. The conditions of additivity and homogeneity are often combined in the superposition principle

${\displaystyle f(\alpha x+\beta y)=\alpha f(x)+\beta f(y)}$

An equation written as

${\displaystyle f(x)=C}$

is called linear if ${\displaystyle f(x)}$ is a linear map (as defined above) and nonlinear otherwise. The equation is called homogeneous if ${\displaystyle C=0}$.

The definition ${\displaystyle f(x)=C}$ is very general in that ${\displaystyle x}$ can be any sensible mathematical object (number, vector, function, etc.), and the function ${\displaystyle f(x)}$ can literally be any mapping, including integration or differentiation with associated constraints (such as boundary values). If ${\displaystyle f(x)}$ contains differentiation with respect to ${\displaystyle x}$, the result will be a differential equation.

Nonlinear algebraic equations

Nonlinear algebraic equations, which are also called polynomial equations, are defined by equating polynomials (of degree greater than one) to zero. For example,

${\displaystyle x^{2}+x-1=0\,.}$

For a single polynomial equation, root-finding algorithms can be used to find solutions to the equation (i.e., sets of values for the variables that satisfy the equation). However, systems of algebraic equations are more complicated; their study is one motivation for the field of algebraic geometry, a difficult branch of modern mathematics. It is even difficult to decide whether a given algebraic system has complex solutions (see Hilbert's Nullstellensatz). Nevertheless, in the case of the systems with a finite number of complex solutions, these systems of polynomial equations are now well understood and efficient methods exist for solving them.^[11]

Nonlinear recurrence relations

A nonlinear recurrence relation defines successive terms of a sequence as a nonlinear function of preceding terms. Examples of nonlinear recurrence relations are the logistic map and the relations that define the various Hofstadter sequences. Nonlinear discrete models that represent a wide class of nonlinear recurrence relationships include the NARMAX (Nonlinear Autoregressive Moving Average with eXogenous inputs) model and the related nonlinear system identification and analysis procedures. These approaches can be used to study a wide class of complex nonlinear behaviors in the time, frequency, and spatio-temporal domains.

Nonlinear differential equations

A system of differential equations is said to be nonlinear if it is not a linear system. Problems involving nonlinear differential equations are extremely diverse, and methods of solution or analysis are problem dependent. Examples of nonlinear differential equations are the Navier–Stokes equations in fluid dynamics and the Lotka–Volterra equations in biology.

One of the greatest difficulties of nonlinear problems is that it is not generally possible to combine known solutions into new solutions. In linear problems, for example, a family of linearly independent solutions can be used to construct general solutions through the superposition principle. A good example of this is one-dimensional heat transport with Dirichlet boundary conditions, the solution of which can be written as a time-dependent linear combination of sinusoids of differing frequencies; this makes solutions very flexible. It is often possible to find several very specific solutions to nonlinear equations, however the lack of a superposition principle prevents the construction of new solutions.

Ordinary differential equations

First order ordinary differential equations are often exactly solvable by separation of variables, especially for autonomous equations. For example, the nonlinear equation

${\displaystyle {\frac {du}{dx}}=-u^{2}}$

has ${\displaystyle u={\frac {1}{x+C}}}$ as a general solution (and also u = 0 as a particular solution, corresponding to the limit of the general solution when C tends to infinity). The equation is nonlinear because it may be written as

${\displaystyle {\frac {du}{dx}}+u^{2}=0}$

and the left-hand side of the equation is not a linear function of u and its derivatives. Note that if the u² term were replaced with u, the problem would be linear (the exponential decay problem).

Second and higher order ordinary differential equations (more generally, systems of nonlinear equations) rarely yield closed-form solutions, though implicit solutions and solutions involving nonelementary integrals are encountered.

Common methods for the qualitative analysis of nonlinear ordinary differential equations include:

• Examination of any conserved quantities, especially in Hamiltonian systems
• Examination of dissipative quantities (see Lyapunov function) analogous to conserved quantities
• Linearization via Taylor expansion
• Change of variables into something easier to study
• Bifurcation theory
• Perturbation methods (can be applied to algebraic equations too)

Partial differential equations

The most common basic approach to studying nonlinear partial differential equations is to change the variables (or otherwise transform the problem) so that the resulting problem is simpler (possibly even linear). Sometimes, the equation may be transformed into one or more ordinary differential equations, as seen in separation of variables, which is always useful whether or not the resulting ordinary differential equation(s) is solvable.

Another common (though less mathematical) tactic, often seen in fluid and heat mechanics, is to use scale analysis to simplify a general, natural equation in a certain specific boundary value problem. For example, the (very) nonlinear Navier-Stokes equations can be simplified into one linear partial differential equation in the case of transient, laminar, one dimensional flow in a circular pipe; the scale analysis provides conditions under which the flow is laminar and one dimensional and also yields the simplified equation.

Other methods include examining the characteristics and using the methods outlined above for ordinary differential equations.

Pendula

Illustration of a pendulum

 

Linearizations of a pendulum

A classic, extensively studied nonlinear problem is the dynamics of a pendulum under the influence of gravity. Using Lagrangian mechanics, it may be shown that the motion of a pendulum can be described by the dimensionless nonlinear equation

${\displaystyle {\frac {d^{2}\theta }{dt^{2}}}+\sin(\theta )=0}$

where gravity points "downwards" and ${\displaystyle \theta }$ is the angle the pendulum forms with its rest position, as shown in the figure at right. One approach to "solving" this equation is to use ${\displaystyle d\theta /dt}$ as an integrating factor, which would eventually yield

${\displaystyle \int {\frac {d\theta }{\sqrt {C_{0}+2\cos(\theta )}}}=t+C_{1}}$

which is an implicit solution involving an elliptic integral. This "solution" generally does not have many uses because most of the nature of the solution is hidden in the nonelementary integral (nonelementary unless ${\displaystyle C_{0}=2}$).

Another way to approach the problem is to linearize any nonlinearities (the sine function term in this case) at the various points of interest through Taylor expansions. For example, the linearization at ${\displaystyle \theta =0}$, called the small angle approximation, is

${\displaystyle {\frac {d^{2}\theta }{dt^{2}}}+\theta =0}$

since ${\displaystyle \sin(\theta )\approx \theta }$ for ${\displaystyle \theta \approx 0}$. This is a simple harmonic oscillator corresponding to oscillations of the pendulum near the bottom of its path. Another linearization would be at ${\displaystyle \theta =\pi }$, corresponding to the pendulum being straight up:

${\displaystyle {\frac {d^{2}\theta }{dt^{2}}}+\pi -\theta =0}$

since ${\displaystyle \sin(\theta )\approx \pi -\theta }$ for ${\displaystyle \theta \approx \pi }$. The solution to this problem involves hyperbolic sinusoids, and note that unlike the small angle approximation, this approximation is unstable, meaning that ${\displaystyle |\theta |}$ will usually grow without limit, though bounded solutions are possible. This corresponds to the difficulty of balancing a pendulum upright, it is literally an unstable state.

One more interesting linearization is possible around ${\displaystyle \theta =\pi /2}$, around which ${\displaystyle \sin(\theta )\approx 1}$:

${\displaystyle {\frac {d^{2}\theta }{dt^{2}}}+1=0.}$

This corresponds to a free fall problem. A very useful qualitative picture of the pendulum's dynamics may be obtained by piecing together such linearizations, as seen in the figure at right. Other techniques may be used to find (exact) phase portraits and approximate periods.

Types of nonlinear dynamic behaviors

• Amplitude death – any oscillations present in the system cease due to some kind of interaction with other system or feedback by the same system
• Chaos – values of a system cannot be predicted indefinitely far into the future, and fluctuations are aperiodic
• Multistability – the presence of two or more stable states
• Solitons – self-reinforcing solitary waves
• Limit cycles – asymptotic periodic orbits to which destabilized fixed points are attracted.
• Self-oscillations - feedback oscillations taking place in open dissipative physical systems.