Aleksandr Lyapunov

How to use this course

Every engineer who designs a control loop, every physicist who asks whether an orbit will last, and every statistician who quotes a normal approximation uses something Aleksandr Lyapunov proved. His name is on stability, on functions that certify it, on exponents that measure chaos, and on a version of the central limit theorem. He published little and slowly, wrote with a rigour unusual for his time, and spent the last third of his life on one hard problem in celestial mechanics.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. The order follows his career: rotating fluids (1882–1885), stability of motion (1888–1902), potential theory (1886–1898), probability (1900–1901), and the figures of equilibrium (1903–1918). Dates before February 1918 were recorded in Russia in the Julian calendar; this course uses the modern calendar unless stated.

Timeline

Yaroslavl and Nizhny Novgorod, 1857–1876

Aleksandr Mikhailovich Lyapunov was born in Yaroslavl on 6 June 1857. His father, Mikhail Vasilievich Lyapunov, was an astronomer who had worked at Kazan University and then directed the Demidov Lyceum in Yaroslavl. His mother was Sofia Aleksandrovna Shilipova. The family was gifted: his younger brother Sergei became a composer and pianist, and another brother, Boris, became a specialist in Slavic languages and a member of the Soviet Academy of Sciences.

Aleksandr was taught at home by his father, who died in 1868. His education then passed to an uncle, R. M. Sechenov, a brother of the physiologist Ivan Sechenov. The uncle’s daughter, Natalia Rafailovna, would become Lyapunov’s wife in 1886.

In 1870 his mother moved the family to Nizhny Novgorod, where Aleksandr entered the gymnasium. He graduated with distinction in 1876. The schooling was classical and heavy on languages, but by the end he had chosen mathematics and the natural sciences.

St Petersburg and Chebyshev, 1876–1885

In 1876 Lyapunov entered the Faculty of Physics and Mathematics at St Petersburg University. He began in the natural sciences and moved to mathematics within a month. The faculty was dominated by Pafnuty Chebyshev, whose lectures combined concrete problems with strict demands on proof. Andrey Markov, a year older, was also studying under Chebyshev; the two became Chebyshev’s best-known pupils and, much later, colleagues in the Academy.

Lyapunov graduated in 1880 and received a gold medal for an essay on hydrostatics. On the recommendation of the mechanician D. K. Bobylev he stayed at the university to prepare for a professorship, and in 1881 he published two papers on the equilibrium of heavy bodies in fluids.

Chebyshev’s question

In 1882 Chebyshev gave him a problem. A homogeneous fluid mass rotating as a rigid body can take the form of a Maclaurin spheroid or, at higher angular momentum, a Jacobi ellipsoid. Do other figures of equilibrium branch off these ellipsoids as the angular velocity changes? Lyapunov could not answer it then. He took up an easier question that turned out to be just as deep: are the ellipsoids stable? His master’s thesis, On the stability of ellipsoidal forms of equilibrium of a rotating liquid, was completed in 1884 and defended at St Petersburg in 1885. Henri Poincaré, working independently, published his own famous memoir on the same figures in 1885. Lyapunov returned to Chebyshev’s original question twenty years later (Lesson 10).

Kharkov, 1885–1902

In the autumn of 1885 Lyapunov became Privatdozent in mechanics at Kharkov University. Teaching took most of his time at first. He prepared his own lecture courses in mechanics. Over the following seventeen years Kharkov became the place where he did nearly all the work for which he is known today.

From 1888 he published papers on the stability of mechanical systems with finitely many degrees of freedom. They culminated in the 1892 monograph The general problem of the stability of motion, issued by the Kharkov Mathematical Society and defended that year as his doctoral dissertation at Moscow University, with the aerodynamicist Nikolai Zhukovsky among the examiners. In 1893 he became ordinary professor.

He was vice-president of the Kharkov Mathematical Society from 1891 to 1898, and from 1899 its president and the editor of its Communications. His students at Kharkov included Vladimir Steklov, who extended his potential theory and later became an academician, and Nikolai Saltykov, later a professor in Belgrade. In the late 1890s he turned to the Dirichlet problem, and around 1900 his lectures on probability led to his proof of the central limit theorem.

In 1901 the St Petersburg Academy of Sciences elected him an associate member early in the year and, at the end of it, an ordinary academician in applied mathematics. The chair had stood empty since Chebyshev’s death in 1894. Lyapunov left Kharkov for the capital in 1902.

Rotating fluids and the ellipsoids

For the general reader. A spinning ball of liquid held together by its own gravity flattens at the poles, the way the Earth does. Spin it faster and it flattens more, until at some point a lopsided, three-axis shape becomes possible. Lyapunov’s first research asked which of these shapes a real fluid planet could actually keep.

The Maclaurin spheroids

Consider a homogeneous incompressible fluid of density \(\rho\) rotating rigidly with angular velocity \(\Omega\). In the rotating frame, equilibrium requires the free surface to be a level set of the sum of the gravitational potential and the centrifugal potential \(-\tfrac12\Omega^2(x^2+y^2)\). Colin Maclaurin showed in 1742 that oblate spheroids of eccentricity \(e\) are figures of equilibrium.

The right side rises from 0 at \(e=0\) to a maximum of about \(0.4493\) at \(e\approx0.9300\), then falls back to 0 as \(e\to1\). So there is a maximum angular velocity for a spheroid of given density, and below it each \(\Omega\) corresponds to two spheroids, a moderately flattened one and a very flat one.

Jacobi’s ellipsoids and the bifurcation

In 1834 Carl Jacobi found that triaxial ellipsoids rotating about their shortest axis are also figures of equilibrium. The Jacobi family meets the Maclaurin family at a single spheroid, with \(e\approx0.8127\) and \(\Omega^2/(\pi G\rho)\approx0.3742\). This is the first bifurcation point along the Maclaurin sequence, the first place where a new family of shapes branches off.

Stability by energy

Kelvin and Tait had proposed a criterion: a figure of equilibrium is stable if it minimizes the appropriate energy (gravitational plus rotational, at fixed angular momentum) among neighbouring shapes. Lyapunov’s thesis examined this principle with a care no one had applied to it. He noted that the second variation of energy may vanish in some directions and that a correct sufficient condition must then involve higher variations. He showed that positivity of these higher variations does guarantee stability, and applied the criterion to both ellipsoid families. With dissipation present (so-called secular stability) the Maclaurin spheroids lose stability exactly at the Jacobi bifurcation, and the Jacobi ellipsoids take over as the stable branch.

What stability means

For the general reader. A pencil balanced on its tip is in equilibrium, but the slightest breath topples it. A marble in a bowl returns after a nudge. Before Lyapunov, “stable” had several loose meanings. His 1892 memoir fixed one precise definition and built a whole theory on it. The definition is still the one in every textbook.

Perturbed motion

Lyapunov considered a known motion of a mechanical system, the unperturbed motion, and asked what happens to motions that start nearby. Writing the deviation as \(x(t)\in\mathbb R^n\), the equations of perturbed motion take the form

so that the unperturbed motion becomes the equilibrium \(x=0\). This reduction is the first idea of the memoir: stability of any motion is stability of an equilibrium of a different system.

Three features distinguish this from earlier usage. The time interval is infinite. The perturbation is in the initial conditions only, not in the equations. And stability is a property of a single solution, not of the system as a whole. Lyapunov also considered stability with respect to some of the variables only, a notion that later became important in mechanics and control.

Two methods

Lyapunov divided his methods into two groups. The first method uses explicit information about solutions, usually series expansions or the linearized equations (Lesson 06). The second method, often called the direct method, decides stability without solving anything, using an auxiliary function (Lesson 07).

The first method: linearization and characteristic exponents

For the general reader. Near an equilibrium, most systems behave like their linear approximation. If every small disturbance of the linear system dies out exponentially, the nonlinear system is stable too. Lyapunov made this precise, found the cases where linearization says nothing, and introduced the growth rates now called Lyapunov exponents.

Proof of part 1. It is cleanest with the second method. Since \(A\) is a stable matrix, the integral \(P=\int_0^\infty e^{A^{\mathsf T}t}e^{At}\,dt\) converges and is symmetric positive definite, and \(A^{\mathsf T}P+PA=-I\) (Lesson 07). Let \(V(x)=x^{\mathsf T}Px\). Along solutions,

For \(|x|\) small, \(|g(x)|\le|x|/(4\|P\|)\), so \(\dot V\le-\tfrac12|x|^2\le-\tfrac{1}{2\|P\|}V\). Hence \(V\) decays exponentially, and so does \(|x|\).

Critical cases

If all eigenvalues have non-positive real part and some lie on the imaginary axis, linearization cannot decide. Lyapunov analysed in detail the case of a single zero eigenvalue and the case of a pair of purely imaginary eigenvalues, where stability depends on higher-order terms. In the second case he also proved the theorem now called the Lyapunov centre theorem: for an analytic Hamiltonian or conservative system whose linearization has purely imaginary eigenvalues \(\pm i\omega\) with no other eigenvalue an integer multiple of \(i\omega\), there is a one-parameter family of periodic orbits emanating from the equilibrium with periods tending to \(2\pi/\omega\).

Characteristic numbers

For time-dependent linear systems \(\dot x=A(t)x\) eigenvalues lose their meaning. Lyapunov attached to each function \(x(t)\) its characteristic number

Modern usage takes the opposite sign and calls \(\chi[x]=\limsup_{t\to\infty}\frac1t\log|x(t)|\) the Lyapunov exponent. A linear system has at most \(n\) distinct exponents. Lyapunov singled out regular systems, for which the exponents behave as eigenvalues do, and proved that for regular systems negative exponents of the linear part again imply stability of the nonlinear system. The general theory of exponents along trajectories of nonlinear systems, the multiplicative ergodic theorem of Valery Oseledets (1968), rests on these ideas.

The second method: Lyapunov functions

For the general reader. To know that a marble will stay in a bowl, you do not need to compute its path. It is enough that it always loses height. Lyapunov turned this observation into a general method: find a function that behaves like energy and never increases along motions, and stability follows.

Let \(\dot x=f(x)\) with \(f(0)=0\), and let \(V\) be a continuously differentiable function near 0 with \(V(0)=0\). Its derivative along solutions is \(\dot V(x)=\nabla V(x)\cdot f(x)\), computable from the equation alone. \(V\) is positive definite if \(V(x)>0\) for \(x\neq0\) near 0.

Proof of 1. Given \(\varepsilon\), let \(m=\min\{V(x):|x|=\varepsilon\}>0\). By continuity choose \(\delta<\varepsilon\) with \(V<m\) on \(|x|<\delta\). A solution starting there has \(V(x(t))\le V(x(t_0))<m\) for all \(t\), so it can never reach the sphere \(|x|=\varepsilon\). Proof of 2. \(V(x(t))\) decreases to a limit \(c\ge0\). If \(c>0\), the solution stays in a compact annulus where \(\dot V\le-\mu<0\), and \(V\) would become negative in finite time. So \(c=0\), and positive definiteness forces \(x(t)\to0\). The time-dependent versions require bounds of \(V\) from above and below by functions of \(|x|\) independent of \(t\), which Lyapunov formulated.

The Lyapunov equation

Then \(V(x)=x^{\mathsf T}Px\) is a quadratic Lyapunov function for \(\dot x=Ax\), with \(\dot V=-x^{\mathsf T}Qx\). The hero illustration on this page shows the level sets of such a \(V\) for a damped oscillator. In control theory this equation and its relatives (Riccati equations, linear matrix inequalities) are among the most used tools of the subject.

Later developments

When \(\dot V\le0\) but is not negative definite, as for mechanical systems with friction acting only on velocities, asymptotic stability can still follow if no complete solution stays in the set \(\{\dot V=0\}\) other than the equilibrium. This invariance principle is due to Barbashin and Krasovskii (1952) and to LaSalle (1960). Nikolai Chetaev generalized the instability theorem in 1934. Converse theorems, showing that a Lyapunov function exists whenever the equilibrium is asymptotically stable, were proved by José Massera and Jaroslav Kurzweil between 1949 and 1956.

Potential theory and the Dirichlet problem

For the general reader. Find the steady temperature inside a body when the temperature on its surface is prescribed. This is the Dirichlet problem. In the nineteenth century it was often solved by formulas that assumed far more smoothness than anyone checked. Lyapunov identified exactly which surfaces make the classical method work, and those surfaces now carry his name.

Layer potentials

Let \(S\subset\mathbb R^3\) be a closed surface bounding a domain \(D\), with outward unit normal \(n\), and let \(\mu\) be a continuous density on \(S\). The double-layer potential is

It is harmonic off \(S\). Carl Neumann and others had proposed to solve the Dirichlet problem by choosing \(\mu\) so that the boundary values of \(W\) match the data, which needs precise knowledge of how \(W\) behaves as \(x\) approaches \(S\).

The interior Dirichlet problem \(\Delta u=0\) in \(D\), \(u=\varphi\) on \(S\), then becomes the integral equation \(W(x_0)-2\pi\mu(x_0)=\varphi(x_0)\) for \(\mu\), which Ivar Fredholm solved in general in 1900 with his theory of integral equations. Lyapunov’s 1897–1898 work established the continuity and jump properties on which this rests, and gave conditions under which the solution has normal derivatives on the boundary. The analogous results for the single-layer potential \(V(x)=\int_S\mu(y)|x-y|^{-1}dS_y\) and its normal derivative complete the classical method for the Neumann problem. Steklov and later Nikolai Günter developed this programme in Russia.

The central limit theorem

For the general reader. Add up many small independent random effects, and the total is approximately normally distributed, whatever the effects look like individually. This is the central limit theorem. Chebyshev and Markov proved versions of it by comparing moments. Lyapunov found a different route, through Fourier transforms, that needs much weaker assumptions and became the standard method of the twentieth century.

Setting

Let \(X_1,X_2,\dots\) be independent random variables with means \(m_k\), finite variances \(\sigma_k^2\), and \(B_n=\sum_{k=1}^n\sigma_k^2\). Chebyshev had outlined a proof in 1887 by the method of moments, assuming all moments finite and bounded. Markov completed that proof in 1898. Lyapunov needed only a little more than two moments.

The 1900 memoir Sur une proposition de la théorie des probabilités treated the case of third moments, and the 1901 memoir Nouvelle forme du théorème sur la limite de probabilité extended it to \(2+\delta\). Lyapunov also bounded the error: for \(\delta=1\) the difference from \(\Phi\) is at most a constant times \(L_{n,1}\log(1/L_{n,1})\). The Berry–Esseen theorem of 1941–42 removed the logarithm.

Proof sketch with characteristic functions

Take \(m_k=0\) and \(0<\delta\le1\). Let \(\varphi_k(t)=\mathbb Ee^{itX_k}\). From the inequality \(|e^{iu}-1-iu+\tfrac{u^2}{2}|\le\min(u^2,|u|^3)\le|u|^{2+\delta}\),

Lyapunov’s condition implies \(\max_k\sigma_k^2/B_n\to0\) (by Lyapunov’s inequality \(\sigma_k^{2+\delta}\le\mathbb E|X_k|^{2+\delta}\)), so every factor is close to 1 and one may take logarithms. Summing, \(\sum_k\log\varphi_k(t/\sqrt{B_n})=-\tfrac{t^2}{2}+O(|t|^{2+\delta}L_{n,\delta})+o(1)\), so the characteristic function of the normalized sum tends to \(e^{-t^2/2}\). The continuity theorem of Paul Lévy, proved in the 1920s, now finishes the proof. Lyapunov, writing before it existed, derived the convergence of distributions from the characteristic functions directly, with explicit estimates.

Jarl Lindeberg in 1922 found a weaker sufficient condition, and William Feller in 1935 showed it is also necessary when the individual variances are negligible. Markov later showed that the method of moments could also reach Lyapunov’s conditions.

Figures of equilibrium and the pear

For the general reader. As a Jacobi ellipsoid spins faster, it reaches a point where a pear-shaped figure branches off. George Darwin hoped that this pear would pinch in two and explain how double stars and the Earth–Moon system formed. Lyapunov spent years on the calculation and concluded that the pear is unstable. He was right, and the theory of fission collapsed.

Chebyshev’s question answered

Between 1906 and 1914 Lyapunov published four long memoirs, Sur les figures d’équilibre peu différentes des ellipsoïdes d’une masse liquide homogène douée d’un mouvement de rotation. He proved rigorously that at each bifurcation point (“ellipsoid of bifurcation”) on the Maclaurin and Jacobi sequences a new family of equilibrium figures branches off, expressed as convergent series in a small parameter. Poincaré had found these points in 1885 using linearized equations, without proving that the new families exist. Lyapunov’s convergence proofs settled Chebyshev’s question of 1882.

The reduction method

The analytic core of his existence proofs is a method for nonlinear equations near a point where the linearization is not invertible. Erhard Schmidt developed it independently for integral equations in 1908.

When \(N\) is one-dimensional the problem reduces to a single scalar equation, the bifurcation equation. Its solutions near the trivial branch give the new family. In the fluid problem, \(x\) is the deformation of the ellipsoid’s surface and \(\lambda\) the angular velocity.

Stability of the pear

Poincaré’s 1885 memoir located the point on the Jacobi sequence where the pear-shaped family branches off, and argued, through an approximate energy computation, that the pear might be stable for a while. Darwin’s numerical work seemed to agree. Lyapunov’s calculation, carried to a higher order, showed the opposite. In 1911 Poincaré wrote that he was no longer sure of his earlier view. Lyapunov published his computations in the third memoir of the series (1912), but no one checked them at the time. In 1917 James Jeans confirmed that the pear-shaped figures are unstable and found the error in Darwin’s numerics. Fission of a homogeneous rotating fluid into a binary is therefore not a continuous process through stable equilibria.

The Academy years, 1902–1917

In St Petersburg Lyapunov devoted himself almost entirely to research. He returned to the rotating fluid. The memoirs of 1903–1904 treated the figures of celestial bodies and the Clairaut equation for slowly rotating, nearly spherical, inhomogeneous masses. The four parts on figures close to ellipsoids followed between 1906 and 1914, and a further study, of series of equilibrium figures for a fluid of variable density, was published only after his death (1925–1927). He wrote his major memoirs in French, which made them accessible abroad.

His 1892 monograph found readers outside Russia only gradually. A French translation by Édouard Davaux was published at Toulouse, and wider attention came only from the 1930s, when Soviet mechanicians took up his methods.

He attended the International Congress of Mathematicians in Rome in 1908. He was an editor of volumes XVIII and XIX of the first series of Leonhard Euler’s Opera omnia. The Accademia dei Lincei elected him a foreign member in 1909 and the Paris Academy of Sciences elected him in 1916. The universities of St Petersburg, Kharkov and Kazan made him an honorary member.

Odessa, death and legacy

In the summer of 1917 Lyapunov moved to Odessa with his wife, Natalia, who was seriously ill with tuberculosis. He began lecturing at the university there, and in 1918 gave a lecture on the shape of celestial bodies. He was himself losing his sight to cataracts. The revolution and civil war made life in Odessa precarious. His wife’s condition worsened through 1918, and she died on 31 October. That same day Lyapunov shot himself. He died three days later, on 3 November 1918, without regaining consciousness. He had asked to be buried with her.

Legacy

• Stability and control: Lyapunov stability, Lyapunov functions and the Lyapunov equation are the foundation of nonlinear control theory and the analysis of dynamical systems. Chetaev, Malkin and Krasovskii built a Soviet school on them; Kalman and others brought them into Western engineering around 1960.
• Chaos and ergodic theory: Lyapunov exponents measure sensitive dependence on initial conditions. The Oseledets theorem guarantees they exist almost everywhere for a measure-preserving system, and Pesin theory links them to entropy.
• Probability: the Fourier method he introduced became the standard approach to limit theorems, and the Lyapunov condition is still the one most often checked in practice.
• Analysis: Lyapunov surfaces, the Lyapunov centre theorem and the Lyapunov–Schmidt reduction are standard tools in potential theory, Hamiltonian dynamics and bifurcation theory.
• Honours: a lunar crater bears his name. The Russian Academy of Sciences awards a Lyapunov Gold Medal for work in mathematics and mechanics.

Not every Lyapunov in mathematics is Aleksandr Mikhailovich. The theorem that the range of a non-atomic vector measure is convex is due to Aleksei Andreevich Lyapunov (1911–1973), a pioneer of Soviet cybernetics.

Connections

People linked to Aleksandr Lyapunov across this collection. A name without a link has no page here yet. See Aleksandr Lyapunov in the network of all 337 lives →

Teachers and advisors

• Pafnuty Chebyshev 1876-1885

Students

• Nikolai Saltykov 1905
• Vladimir Steklov

Influenced by

• Carl Gustav Jacob Jacobi
• Colin Maclaurin
• Sofia Kovalevskaya
• William Thomson, Lord Kelvin

Influenced

• James Jeans
• Jarl Lindeberg
• Nikolai Chetaev

Colleagues and collaborators

• Andrey Markov
• Nikolai Zhukovsky

Rivals

• Henri Poincaré
• James Jeans
• George Darwin

Family

• Sergei Lyapunov

Also mentioned in this life

• Stefan Banach
• Subrahmanyan Chandrasekhar
• Alexis Clairaut
• Abraham de Moivre
• Peter Gustav Lejeune Dirichlet
• Leonhard Euler
• Joseph Fourier
• Ivar Fredholm
• Carl Friedrich Gauss
• Johannes Kepler
• Pierre-Simon Laplace
• Paul Lévy
• Peter Guthrie Tait

Fields

• differential equations 13 lives
• probability theory 23 lives
• potential theory 9 lives
• mathematical physics 45 lives
• celestial mechanics 16 lives

On CoursesHub

• ODEs / PDEs / Green Functions · Physical Mathematics
• Classical Mechanics

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Aleksandr Mikhailovich Lyapunov, MacTutor History of Mathematics.
2. Lyapunov, Aleksandr Mikhailovich, Dictionary of Scientific Biography (via MacTutor).
3. Aleksandr Lyapunov, Wikipedia.
4. Aleksandr Lyapunov, Mathematics Genealogy Project.
5. V. I. Smirnov, Biography of A. M. Lyapunov, translated in Int. J. Control 55 (1992), centenary issue.

Mathematics

1. A. M. Lyapunov, The general problem of the stability of motion, Kharkov, 1892. French translation by É. Davaux, Ann. Fac. Sci. Toulouse; reprinted as Annals of Mathematics Studies 17, Princeton, 1947. English translation by A. T. Fuller, Int. J. Control 55 (1992) and Taylor & Francis, 1992.
2. A. M. Lyapunov, Sur une proposition de la théorie des probabilités, Bull. Acad. Imp. Sci. St.-Pétersbourg (5) 13 (1900), 359–386.
3. A. M. Lyapunov, Nouvelle forme du théorème sur la limite de probabilité, Mém. Acad. Imp. Sci. St.-Pétersbourg (8) 12 (1901), 1–24.
4. A. M. Lyapunov, Sur certaines questions qui se rattachent au problème de Dirichlet, J. Math. Pures Appl. (5) 4 (1898).
5. Lyapunov theorem, Encyclopedia of Mathematics.
6. H. Fischer, A History of the Central Limit Theorem, Springer, 2011.
7. From classical to modern central limit theorems, arXiv:2405.19828.
8. The dawning of the theory of equilibrium figures, arXiv:1409.3858.
9. S. Chandrasekhar, Ellipsoidal Figures of Equilibrium, Yale University Press, 1969.
10. H. K. Khalil, Nonlinear Systems, 3rd ed., Prentice Hall, 2002.
11. L. Barreira and Ya. Pesin, Lyapunov Exponents and Smooth Ergodic Theory, AMS, 2002.