L.P. Shilnikov in 2010
On 26th Dec 2011, the world lost one of the true pioneers of nonlinear
dynamical systems theory; Leonid Pavlovich Shilnikov (known simply as
LP to his friends). He died of cancer, at home in Nizhny Novgorod
surrounded by his family, nine days after passing his 77th birthday.
When I took first graduate course in nonlinear dynamics in the late
1980s, I learned of Lorenz attractors and Smale horeshoes. Both were
discovered in the early 1960s, and were pivotal to our current
geometric understanding of dynamical systems that feature chaotic
dynamics. But when the `chaos theory' revolution began in the West in
the 1970s it seems it was not clear how the horseshoe and the Lorenz
attractor relate to one another. However, unknown to us in the West,
at the same time as Ed Lorenz and Steve Smale made their
breakthroughs, a young mathematician in Gorky had made a remarkable
discovery which provides just such a link.
But whereas Lorenz and Smale received almost universal acclaim,
L.P. Shilnikov worked in a city deep in the USSR that was closed for
foreigners and he was not allowed foreign travel by the Soviet
authorities. It took two decades before his work gained the universal
recognition it deserved, especially in the West. As a result it would
seem that popular accounts on the historical development of `chaos
theory' often overlook LP's pivotal contributions.
L.P. Shilnikov in the 60's |
LP was a member of the now famous Andronov school of Russian
mathematicians and physicists who applied and extended many of
Poincaré and Liapunov's topological methods for analysing
dynamics. These people worked mostly in the Gorky Institute of
Physical-Technical Researches (GIFTI) which was founded by the
Moscow-trained physicist Aleksandr A. Andronov. In 1931, accompanied
by his mathematician wife Evgeniya Leontovich, Andronov moved to the
city of Nizhny Novgorod. Literally translated as ``lower new town''
and known to locals merely as Nizhny, it is an industrial city, the
third largest in Russia, situated some 300 miles East of Moscow. The
year after the Andronovs arrived, under Stalin's orders, Nizhny
Novgorod was renamed Gorky. And Gorky it remained until the fall of
the Soviet era, when, in 1990, like Leningrad, the city
reverted to its original name.
The Andronov school developed a comprehensive theory of nonlinear
oscillations for systems with two state variables. Their
results were collected in 1937 into the now classic monograph that
Andronov co-authored with Aleksandr Vitt and Semën Khaikin (Vitt's
name never appeared in the first edition of this work, he was a victim
of Stalin's purges and died in a Siberian labour camp in 1938. It
wasn't until 1966 that the work was fully translated into English, by
which time Vitt's name was finally given its rightful place among the
authors).
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By the time of his premature death in 1952, Andronov's School had
grown into a large and complex organisation. It had expanded in remit
too; to cover many areas of physics, mathematics and engineering in
which nonlinear oscillations were important. Andronov's widow, by then
known as Evgeniya Leontovich-Andronova, petitioned the Russian Academy
of Sciences to form a dedicated theory core, the `Institute of
Mathematics and Cybernetics'. Her wish was granted and she became head
of the Department of Differential Equations, with the explicit goal to
continue the tradition of classifying different kinds of nonlinear
oscillations using qualitative methods. In particular, she completed
and published work with her husband on classifying all cases of
dynamics near homoclinic loops of systems of differential equations on
a plane. The focus of her department's activity then switched to
studying how these ideas extended to higher-dimensional phase space.
Higher-dimensional generalisations of the Andronov-Leontovich theory
of homoclinic loops became the subject of Shilnikov PhD research
project. He switched to this topic from his earlier works (with Yuri
Neimark) on perturbation methods and piecewise-linear systems. These
were important themes in automatic control theory, which flourished in
the late 50s, but he found the field both boring and too
crowded. Homoclinic bifurcations were a different
matter. Andronov-Leontovich theory used topology of the plane heavily,
and extending the theory to make it free from
Poincaré-Bendixson-type arguments was a real challenge at the
time. The first results were more or less the same as in 2D; the
bifurcation of single, regular, isolated periodic orbits.
Then, in January 1963 shortly after defending his PhD at
GIFTI, the 28 year old L.P. Shilnikov made his key discovery.
He looked at homoclinic trajectories to a saddle-focus equilibrium in
three dimensions. Such points contain complex eigenvalues in their
linearisation and, if these have weaker real parts than the opposing
real eigenvalue, Shilnikov found that the corresponding homoclinic
loop implies chaotic dynamics. Specifically, he could prove
that the chaotic dynamics are governed by a Smale horseshoe. In fact,
not one Smale horseshoe; but an infinite number of different Smale
horseshoes (more precisely, a Bernoulli shift on infinitely many
symbols). This appears to be the first mathematically rigorous method
for generating chaos in dynamical systems that do not have an
underlying periodic forcing like the van der Pol oscillator. Over the
years, Shilnikov's mechanism of chaos has proven to be one of the most
robust and frequently occurring mechanisms chosen by nature.
The discovery came as a shock. Steve Smale's early ideas on the
horseshoe had already reached Gorky after Leontovich-Andronova
attended his talk at a conference in Kiev in 1961 (she remarked in
passing that Smale reminded her of Huckleberry Finn). However nobody
could expect that such dynamics, which contains infinitely many
different periodic motions, could be a necessary consequence of a
generic homoclinic bifurcation. Leontovich-Andronova recounted to
Shilnikov very much later her first reaction ``I immediately wanted to
say that this simply cannot be!''
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L.P. in the 80's
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Shilnikov's remarkable result was presented in rather short form in
the Doklady Akedemii Nauk SSSR in 1965, with the full results
and complete proof appearing in 1970. Within these few years Shilnikov
produced a string of papers which included the study of the dynamics
that is implied by Poincaré homoclinic tangles, the extension of
his saddle-focus result to arbitrary $n$-dimensional systems, and many
other kinds of homoclinic bifurcations. More and more results came
out. Leontovich-Andronova referred to him as `a Mozart' (quoting
Pushkin ``you, Mozart, are a god, and you don't know it'') such was
his capacity to discover more and more fascinating mechanisms that
generate complex dynamics. He began to attract the first of a
succession of talented PhD students; Nikolai Gavrilov,
Valentin Afraimovich, Lev Lerman, Vyacheslav Grines, Leonid Belyakov,
Vadim Bykov, Albert Morozov, Valery Lukyanov, Sergey Gonchenko,
Mikhail Malkin, Nikolai Roschin, Dmitry Turaev, Ilya Ovsyannikov,
Valery Biragov, Yuri Komlev, Igor Belykh, Mikhail Shashkov, Yan
Umansky, Oleg Sten'kin, Vladimir Gonchenko (Sergey's son) and LP's own
son Andrey.
Science was spoken openly and democratically in the Shilnikov seminars
in Gorky in the 1970s and 80s. Late in the evenings, fellow scientists
would come to LP's apartment and talk mathematics into the small
hours, chain smoking and tea drinking while he was pacing up and down
in his kitchen. One day, in 1976, the Moscow mathematician Yakov
Sinai gave a talk in the Shilnikov seminar. After the seminar, and as
they walked back to LPs home Sinai told him about the Lorenz
attractor. LP was fascinated. He saw straight away that what Lorenz
had observed also fit into his theory of homoclinic bifurcations. He
set his student Bykov, who had experience programming computers, on
the task of computing the homoclinic curves in the system.
At the same time with Bykov and his former student Afraimovich,
he showed
theoretically what lay behind the shape of the butterfly strange
attractor. By a bizarre twist of the Soviet era scientific life, their
paper had to lay for five years with the publisher. Eventually, in
the early 1980s, this work received wide dissemination within the
Soviet Union when LP included his new-found understanding of the
Lorenz attractor as a substantial appendix to the Russian translation
of a book on applications of the Hopf bifurcation by Marsden and
McCracken.
As LP's fame within Russian academia spread, he was barely tolerated
at his home University. His success at a young age broke the
mould. His great breakthrough was arrived at alone, and he did not add
the name of his Master's supervisor Neimark onto his key papers, as he
had done with his earliest work. Trumped up personal allegations
against LP surfaced. He was being watched. In 1970 he applied to
receive a Doctor of Science (DSc.) degree, a high honour and a
necessary requirement to obtain the status and salary equivalent to
`full professor' in the U.S. system. There was a four year delay
before his application was reviewed. He openly mentioned mistakes in
several of Neimark's papers, and Neimark took serious offence. The
jury was split, and he was turned down. After this setback, he
never tried again to obtain a DSc. Nor was he ever elected to be a
fellow of the Russian Academy of Sciences (despite winning their
prestigious Liapunov medal in 1998). It would seem that LP's open
scientific approach, interested only in the truth and never playing
political games, was to his own detriment.
Over the years, news of Shilnikov's work began to filter out slowly in
the West, following translation of the Doklady Akedemii into English.
As his fame spread, LP would receive invitations to give keynote talks
at international conferences. But he was never allowed to
go. Typically, the invitation letters would arrive already opened and
with the date of the conference having already passed.
His work gained further impact through the results of two PhD
students, working independently in the early 1980s. Paul Glendinning
was studying with Colin Sparrow at Cambridge. Simultaneously, Pierre
Gaspard was studying under Gregoire Nicolis at the Free University of
Brussels. Both were given the task of revisiting and understanding the
`obscure' Russian papers of one L.P. Shilnikov. With the aid of
computers and modern graphics, both were able to depict the geometry
of periodic orbits close to a saddle-focus homoclinic orbit as
discovered by LP in 1963. When Glendinning and Gaspard became aware
of each other's work, it was arranged that their key findings would be
published back to back in the same journal, the rather unlikely
Journal of Statistical Physics (in 1984 there were yet to be any
specialist journals of nonlinear dynamics and chaos). At around
the same time the Frenchman Charles Tresser published related
results in the Annals of the Henri Poincaré Institute. In the next few
years this particular Shilnikov mechanism was found to explain the
source of complex dynamics in a wide variety of different physical
systems; in chemistry, in fluid mechanics, in laser instabilities, and
in optical pulse propagation.
LP with D. Turaev and S. Gonchenko, Berlin 2004
Everything changed in Gorky after Glasnost, Perestroika, and the fall
of the iron curtain. In 1990 LP was finally allowed to visit the West.
He was invited by Neal Abraham in the US to attend a conference on
nonlinear optics. Abraham took the trouble to fly to Moscow to fetch
LP. Then, in 1991, a conference was organised in Brussels in
Shilnikov's honour. He was accompanied by his former students
Lev Lerman, Vadim Bykov and his son Andrey.
There was a real sense of `East meets West'. Perhaps what was most
interesting though - and I was there - was the eclectic range of
applications that were presented in which homoclinic bifurcations to a
saddle-focus provided the key to what was observed, both in
mathematical models and in experiments. There were talks on
chemistry, fluid mechanics, neuroscience, combustion, lasers, and even
astrophysics. LP was hailed as a hero and, in the evenings, much vodka
was drunk in his honour.
LP though, while greatly touched, expressed surprise that most of the
talks were on applications, and that there had been seemingly little
new mathematical development of homoclinic bifurcation theory in the
West. While this was partially true, and indeed many of the latest
theoretical developments had been due to Shilnikov's colleagues and
students, theoretical activity was taking place elsewhere, not
represented at the conference. For example, Xiao-Biao Lin in the US
and Bjorn Sandstede in Germany were developing complementary methods
for analysing Shilnikov-related phenomena in other kinds of evolution
models such as functional and partial differential equations.
LP at work in Andrey's house in 2010
Perhaps LPs comments stemmed from an incident which Paul Glendinning
recently recounted to me that occurred during one late-night vodka
drinking session. During the evening Paul, Colin Sparrow and a few
other Western colleagues engaged in scientific discussion with their
Russian counterparts. Given the lack of a common language, they turned
to drawing pictures interspersed with occasional mathematical symbols
to describe different cases of homoclinic bifurcations that they were
aware of. Both East and West rather proudly wrote next to each of the
diagrams the year in which that particular case had first been
analysed by scientists within their respective spheres of
influence. To the chagrin of the Western Europeans, it would seem that
time after time, the Russian's had got there first; sometimes, many
years in advance of the Western rediscovery. Paul Glendinning can
recall only one solitary case for which, arguably, Western scholars
had scooped those of Shilnikov's school.
In the last 20 years of his life L.P. Shilnikov received many plaudits
for his work. He wrote more than 200 scientific publications. Many of
the fruits of his work are made accessible in the two-volume book
Methods of qualitative Theory in Nonlinear Dynamics. Parts I, II
published with Andrey, Dimitry Turaev and Leon Chua in 1998 and 2001.
LP with his wife Lyudmila in 2010
LP lived out his final years in Nizhny, with his wife Lyudmila, close
to their daughter and her grown-up family. His son Andrey, by now a tenured
mathematical neuroscientist at Georgia State University, was a
frequent visitor. LP continued to publish original research. He
travelled freely, and further conferences were held in his honour to
mark both his 70th and 75th birthdays. Following a few health scares,
he gave up smoking and until his final battle with cancer, lived a
happy and fulfilled existence, with fishing being his second passion
after mathematics. One of LP's former students Valery
Biragov, who had become a priest and changed his name to Hegumen Vassian,
gave communion to LP the day before he died and also conducted
his funeral service.
For a more detailed biographical sketch of LP Shilnikov including a
range a photographs of the man and more specifics on his mathematical
achievements see the volume that appeared to mark his 75th birthday:
L.P.Shilnikov-75 Special issue of Regular and Chaotic Dynamics, 2010, Vol. 15, Nos 2-3
See also a family
video taken at LPs 77th birthday, 17th Dec 2011.
Alan Champneys, Bristol, UK.
Alan thanks Andrey Shilnikov and Dimitry Turaev for their comments.
Photos courtesy of Andrey Shilnikov
Handling editor: Jens Rademacher