User:GINOUX Jean-Marc/Proposed/Poincaré's limit cycles

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The concept of limit cycle was introduced by Henri Poincaré (1854-1912) in his seminal works On curves defined by a differential equation in 1882. Three years later, the french engineer Henry Léauté (1847-1916) was among the first to apply it to solve a problem of control of hydraulic machines. Until now historiography considered that Andronov was the first to establish in 1929 a correspondence between the periodic solution of self-sustaining dissipative systems and Poincaré’s limit cycles. However, the recent discovery of a series of lectures on wireless telegraphy given in spring 1908 by Poincaré (Ginoux et al., 2010) proves that he had already made applications of this concept twenty years before Andronov.


Henri Poincaré and the concept of limit cycle

In Chapter VI of his second memoir On curves defined by a differential equation, Poincaré (1882, p. 261) presented the Theory of limit cycles. By using the Theory of consequents that he had previously exposed, he demonstrated the existence of a new kind of closed curves that he called limit cycle. Poincaré (1884, p. 24) defined accurately this concept in the notice of his scientific works:

"I call thus closed curves that satisfy our differential equation and whose other curves defined by the same equation are approaching asymptotically without never reaching them [...] Let's suppose that we have drawn a limit cycle; it is obvious that the moving point (periodic orbit) we were talking about will never cross it and will always remain within the cycle, or always on the outside. It is true that the limit cycles are in general transcendental curves that can not be drawn accurately. But one can often draw two closed algebraic curves, concentric to each other, defining a sort of ring, so we can distinguish three regions in the plane, inside the ring, the ring region and outside the ring. Suppose we have demonstrated in some way that the limit cycle is in the annular region, it is certain then that if our moving point is inside the ring, it will never go to outside the ring. We can therefore, despite the instability of this moving point, assign upper limits to its coordinates."

He introduced then the Theory of the limit rings and the notion of topographic system. In the following Chapter VII, Poincaré (1882, p. 274) showed the first mathematical example of limit cycle. This was two simultaneous differential equations of first order and first degree defined as follows:

\[ \tag{1} \dfrac{dx}{x\left( {x^2+y^2-1} \right)-y\left( {x^2+y^2+1} \right)}=\dfrac{dy}{y\left( {x^2+y^2-1} \right)+x\left( {x^2+y^2+1} \right)} \]

Of course this system had been constructed ad hoc by Poincaré to illustrate his point and did not correspond, at that time to a physical reality. However, it enabled to highlight the existence of an invariant closed curve in the sense of Darboux (1878) which is simply the algebraic limit cycle of equation (see Fig. 1)\[ x^2+y^2=1 \].

However, as immediately noticed by Poincaré (1882, p. 283)

"When the limit cycles are not algebraic, a full discussion is obviously impossible, since we can never find, in finite terms, the equation of limit cycles."

This assertion of Poincaré will be verified a few decades later, particularly in France by Jules Haag (1943, 1944) who applied an asymptotic approach, introduced by Poincaré (1892-93-99 in Celestial Mechanics, for studying the limit cycle of Van der Pol's relaxation oscillator. This method will be then used in U.S.A. by Flanders and Stoker (1946), in Great Britain by Mary Lucy Cartwright and John Littlewood (1945, 1947, 1948, 1949) and, in U.S.S.R. by Anatoly Alekseevich Dorodnitsyn (1947). Later, Yuri Mitroploskii (1964) and Kenzi Odani (1995) have shown that the limit cycle of Van der Pol equation is not algeabraic.

Figure 1: Poincaré's first limit cycle (Poincaré, 1882, p. 280).

In Fig. 1, Poincaré has plotted the limit cycle of corresponding to the system of differential equations (1) and the trajectory curves describing a moving point which can never cross it and which will always remain within the cycle, or always on the outside (Poincaré, 1921, p. 58).

Thus, there did not appear until the discovery of his Lectures on wireless telegraphy (Poincaré, 1908, 1909) that Poincaré has used the concept of limit cycle to describe a practical application. One of the first of them appeared to be the one made by Henry Léauté.

Henry Léauté and the control of hydraulic machines

In 1885, the french engineer Henry Léauté (1847-1916) published a long memoir entitled Sur les oscillations à longue période dans les machines actionnées par des moteurs hydrauliques et sur les moyens de prévenir ces oscillations (On the long-period oscillations in machines driven by hydraulic motors and how to prevent these oscillations, Léauté, 1885). At that time, the engineers in hydraulic were faced to a difficult problem in floodgate systems. Indeed, the motor used for opening or closing the valve produced by feedback oscillations causing a very damaging beat phenomenon. These oscillations with a period of about ten seconds were then called long-period oscillations. To solve this problem Léauté made use of the concept of limit cycle developed three years before by Henri Poincaré (1882, p. 261) unfortunately without recognizing the correspondence he had just established between Science and Technology.

The diagram used by Léauté (see Fig. 2), involving position and velocity, was in exact correspondence with the phase plane representation of Poincaré. It is important to note that in the Russian edition as well as in the Anglo-Saxon edition of the book of Andronov, Witt and Khaikin (1959, 1966), appears a reference to Léauté (1885) in a footnote of the second paragraph of the first chapter entitled The concept of phase plane.

"The phase plane was first used for the study of the dynamics of oscillatory systems by Léauté [172], who investigated the operation of a certain automatic control equipment by constructing on the phase plane of this equipment the integral curves and limit cycles (without giving to them this name; he was apparently not aware of the work by Poincaré published a little earlier [108], where the limit cycles first appeared in mathematical literature). Afterwards the remarkable works by Léauté were, unfortunately, almost completely forgotten." (Andronov et al. 1966, p. 4)

However, it seems that this note on Léauté is an addition of N. A. Zheleztsov who greatly modified and increased the original edition of Andronov and Khaikin [1937] which does not contain any reference to the works of Léauté. This does not exclude the fact that Andronov was aware of them, since in his collected works, Léauté is mentioned many times (Andronov 1956, p. 307, p. 344, p. 510). Moreover, in an article entitled Self-oscillations of a simplified diagram containing an automatic variable pitch propeller Andronov, Bautin and Gorelik (1945) provided the ordinary differential equation corresponding to Léauté's problem. Finally, on the occasion of the centennial anniversary of the birth of Mandel'shtam (1879-1944) who was Andronov's Ph-D advisor, Gaponov-Grekhov and Rabinovich (1979, p. 599) recalled:

"Both Mandel’shtam himself and his students are characterized by an exceptionally strict attitude to the facts of history of science and to the accuracy of acknowledgement of priority that this implies. In particular, Mandel'shtam and Andronov believed until 1931 that they had been the first to juxtapose generation with limit cycles, but when they found that this had been done intuitively nearly simultaneously with the discovery of the limit cycles themselves, they took every opportunity to point this out: "... The following preliminary remark is necessary if we are to avoid distortion of historical perspective. Ten years before the discovery of radio, in a study of self-oscillations in an automatic-control device, the French engineer Léauté (1885) investigated the phase space of this device and draw integral curves and limit cycles for it (without applying those names to them: he was apparently unfamiliar with the paper that Poincare had published a bit earlier, in which limit cycles made their first appearance in mathematics). For reasons that we shall not discuss here, Léauté’s remarkable studies had been almost completely forgotten.""

If this extract confirms that Andronov had a perfect knowledge of the memoir of Léauté (1885) it leads nevertheless to another question. Why Mandel'shtam and his students, who seemed to be very concerned about the priority of discoveries in History of Science, didn't they found the article of Poincaré (1908)?

To solve this problem of control in machines driven by hydraulic motors, Léauté proposed to represent the evolution of the system in the phase plane having for abscissa the valve opening and for ordinate the corresponding speed of the machine. The curve presented on Fig. 2 transcribes the evolution of the oscillations. In the second paragraph of Chapter IV, entitled Properties of the closed cycle Léauté described the characteristics of a cycle which is nothing else but a limit cycle of Poincaré.

Figure 2: Léauté's limit cycle (Léauté, 1885, p. 10).

He considered what he called successive cycles corresponding to integral curves of the differential equation characterizing the evolution of the oscillations in this problem. He explained that these successive cycles can not cross the closed cycle and concluded:

"This explains the phenomena resulting from the existence of the closed cycle and gives an account of the production of long-period oscillations. The whole question is reduced to the determination of the conditions for which there will be no closed cycle. (Léauté 1885, p. 78)"

Although Léauté, who seeks conditions insuring that there is no closed cycle, is faced with a problem opposite to the one which will be studied by all those who, after him, will use the same approach, he carries on a correspondence between the periodic solution of a self-sustained oscillator and limit cycles, of the same type as the one which Poincaré (1908) will establish twenty years later".

Poincaré's forgotten conferences on Wireless Telegraphy

On July \( 4^{th} \), 1902 Poincaré became Professor of Theoretical Electricity at the École Supérieure des Postes et Télégraphes (today Sup'Télecom) in Paris where he taught until 1910. The director of this school, Édouard Éstaunié (1862-1942), also asked him to give a series of conferences every two years. In 1908, Poincaré chose as the subject: wireless telegraphy. The text of his lectures was first published weekly in the journal La Lumière Électrique (Poincaré, 1908) before being edited as a book the yera after (Poincaré, 1909).

In the fifth and last part of these lectures entitled: Télégraphie dirigée : oscillations entretenues (Directive telegraphy: maintained oscillations) Poincaré stated a necessary condition for the establishment of a stable regime of maintained oscillations in the singing arc (a forunner device of the triode used in Wireless Telegraphy). More precisely, he demonstrated the existence, in the phase plane, of a stable limit cycle.

The singing arc equation

Figure 3: Maintained oscillations in the singing arc (Poincaré, 1908, p. 390).

Starting from the following diagram (See Fig. 3), Poincaré (1908, p. 390) explained that this circuit consists of an Electro Motive Force (E.M.F.) of direct current E, a resistance R and a self, and in parallel, a singing arc and another self L and a capacitor. In order to provide the differential equation modeling the maintained oscillations he calls $x$ the capacitor charge and i the current in the external circuit. Thus, the intensity in the branch (ABCD) comprising the capacitor of capacity 1/H may be written\[ x' = \frac{dx}{dt} \]. The current intensity \( i_{a}\) in the branch (AFED) comprising the singing may be written while using Kirchoff's law\[ i_{a} = i + x'\]. Then, Poincaré established the following second order nonlinear differential equation (2) for the maintained oscillations in the singing arc:

\[ \tag{2} Lx'' + \rho x' + \varphi \left( i + x' \right) + H x = 0 \]

He specified that the term \( \rho x' \) corresponds to the internal resistance of the self and various damping while the term \( \varphi \left( i + x' \right) \) represents the E.M.F. of the arc which is related to the intensity by a function, unknown at that time. The main problem of equation (2) is that it depends on two variables x and i. So, it is necessary for Poincaré to get rid of i. By neglecting the external self and while equaling the tension in all branches of the circuit he found that:

\[ \tag{3} R i + \varphi \left( i + x' \right) = E \]

He explained that if the function \( \varphi \) was known, equation (3) would provide a relation between i and \( x' \) or between \( i + x' \) and x and then the variable i could be eliminated in the equation (2). Thus, he made the assumption that there exists a function relating i and \( x'\). Then, he directly replaced in equation (2) \( \varphi \left( i + x' \right) \) by \( \theta \left( x' \right) \) and wrote:

\[ \tag{4} Lx'' + \rho x' + \theta \left( x' \right) + H x = 0 \]

Stability condition for maintained oscillations and limit cycles

Then, Poincaré established, twenty years before Andronov (1929), that the stability of the periodic solution of equation (4) depends on the existence of a closed curve, i.e. a stable limit cycle in the phase plane. By using the variable changes he has introduced in his famous memoirs entitled: On the curves defined by differential equation Poincaré (1886, p. 168):

\[ x' = \frac{dx}{dt} = y \quad \mbox{;} \quad dt = \frac{dx}{dy} = y \quad \mbox{;} \quad x'' = \frac{dy}{dt} = \frac{ydy}{dx} \]

Thus, equation (4) becomes:

\[ \tag{5} Ly\frac{dy}{dx} + \rho y + \theta \left( y \right) + H x = 0 \]

Poincaré (1908, p. 390) stated then that: "Maintained oscillations correspond to closed curves if there exist any".

and he gave the following representation for the solution of equation (5):

Figure 4: Closed curve solution of Eq. (5) (Poincaré, 1908, p. 390).

Let's notice that this closed curve is only a metaphor of the solution since Poincaré do not use any graphical integration method such as isoclines. Moreover, the main purpose of this representation is to specify the sense of rotation of the trajectory curve which is a preliminary necessary condition to the establishment of the following proof involving the Green-Ostrogradsky theorem.

Then, Poincaré explained that if \( y = 0 \) then \( dy/dx \) is infinite and so, the curve admits vertical tangents. Moreover, if x decreases \( x'\), i.e. y is negative. He concluded that the trajectory curves turns in the direction indicated by the arrow (see Fig. 4) and wrote:

"Stability condition. - Let's consider another non-closed curve satisfying the differential equation, it will be a kind of spiral curve approaching indefinitely near the closed curve (so called limit cycle). If the closed curve represents a stable regime, by following the spiral in the direction of the arrow one should be brought back to the closed curve, and provided that this condition is fulfilled the closed curve will represent a stable regime of maintained waves and will give rise to a solution of this problem".

Then, it clearly appears that the closed curve which represents a stable regime of maintained oscillations is nothing else but a limit cycle as Poincaré has defined it in his own works (Poincaré, 1886, p. 30). But this, first giant step is not sufficient to prove the stability of the oscillating regime. Poincaré had to demonstrate now that the periodic solution of equation (4) (the closed curve) corresponds to a stable limit cycle.

Possibility condition of the problem: stability of limit cycles

In the following part of his lectures, Poincaré gave what he calls a condition de possibilité du problème. In fact, he established a condition of stability of the periodic solution of equation (4), i.e. a condition of stability of the limit cycle under the form of inequality. After multiplying equation (5) by \( x'dt \) Poincaré integrated it over one period while taking into account that the first and fourth term are vanishing since they correspond to the conservative part of this nonlinear equation (It is easy to show that\[ \int {Lydy} + \int {Hxdx} = \frac{1}{2}Ly^2 + \frac{1}{2}Hx^2 = 0 \]). He found:

\[ \tag{6} \rho \int {{x}'^2dt} +\int {\theta \left( {{x}'} \right){x}'dt} = 0 \]

Then, he explained that since the first term is quadratic, the second one must be negative in order to satisfy this equality. So, he stated that the oscillating regime is stable iff:

\[ \tag{7} \int {\theta \left( {{x}'} \right){x}'dt} < 0 \]

In fact, Poincaré's approach is identical to that will be used by Alfred Liénard (1869-1958) twenty years later in his famous paper.

Liénard's works on maintained oscillations

In 1928, the french engineer Alfred Liénard (1928) investigated the condition of existence and uniqueness of the solution of a nonlinear differential equation considered as a generalization of the Van der Pol (1926) equation for the triode oscillations and todays refered as Liénard's equation.

\[ \tag{8} \frac{d^2x}{dt^2} + \omega f\left( x \right)\frac{dx}{dt} + \omega^2x = 0 \]

Figure 5: Characteristic of oscillation of Eq. (8) (Liénard, 1928, p. 904).

where \( F\left( x \right) = \int_0^x f\left( x \right)dx \) (where \( F\left( x \right) \) is necessarily an odd function) represents the characteristic of oscillation of the nonlinear device considered. In the case of the singing arc it corresponds to its E.M.F. Then, let's note that Eq. (4) and (8) are completely analogous. By considering that \( F\left( x \right) \) is first negative from \( 0 \) to a certain value \( X_0 \) of \( x \) and is then increasing for \( x \geqslant X_0 \) (see Fig. 5), Liénard states that Eq. (8) admits a unique and stable periodic solution. It is also important to notice that neither Liénard (1928) nor Van der Pol (1926) have recognized that the periodic solution of their equation is a Poincaré's limit cycle. In order to state his result Liénard introduced a special transformation leading to the following system in which the original equation (8) is studied in the so-called Liénard's plane:

\[ \tag{9} \frac{dy}{dx} + \frac{x}{y - F\left( x \right)} = 0 \]

In order to establish a comparison between Poincaré's results and that of Liénard it is necessary to transform Eq. (5) into a dimensionless system. This can be easily done by using this variables changes\[ x \rightarrow \sqrt{L/H}x \], \( t \rightarrow \mu t \) and while posing\[\mu = 1/{\sqrt{LH}}\]. Then, starting from Eq. (5) and by neglecting the resistance \( \rho \) of the self we have:

\[ \tag{10} \frac{dx}{dt} = y \quad \mbox{;} \quad \frac{dy}{dt} = -x - \mu \theta \left( y \right) \]

It is also necessary to transform Eq. (9) by using this variables changes\[ x \rightarrow -y \], \( y \rightarrow x \) and the fact that \( F\left( x \right) \) is and odd function we have:

\[ \tag{11} \frac{dx}{dt} = y \quad \mbox{;} \quad \frac{dy}{dt} = -x + F \left( y \right) \]

In his paper, Liénard (1928, p. 906) states that a condition of existence and uniqueness of a stable periodic solution of Eq. (8) is that the following line integral evaluated along a cycle over a period:

\[ \tag{12} \int F\left( x \right)dy > 0 \]

Then, taking into account the previous variable changes, this condition may be rewritten as:

\[ \tag{13} \int F\left( y \right)dx < 0 \]

By considering the system (10) and by posing \( \mu = 1 \), Poincaré's condition may be rewritten as:

\[ \tag{14} \int \theta \left( y \right)dx < 0 \]

So, it becomes obvious that both conditions (13) and (14) are identical. Nevertheless, two question arise: why Liénard does not mention Poincaré's works and why he does not recognize that his stable periodic solution is nothing else but a Poincaré's limit cycle?

Andronov's works on self-oscillations

At the end of the twenties, Aleksandr Aleksandrovich Andronov (1901-1952) started a Ph-D with Leonid Isaakovich Mandel'shtam who had great knowledge of Poincaré 's works and was deeply involved in Wireless Telegraphy and so, in nonlinear oscillations. In fact, the correspondence Andronov (1929) established in the famous note at the French Comptes Rendus was preceded by a short presentation of his Ph-D works (According to Boyco (1983, p. 30) Andronov's Ph-D thesis has not been preserved) at the sixth congress of Russian Physicists at Moscow between the \( 5^{th}\) and \(16^{th}\) August 1928. In this work Andronov (1928) presented the foundations of what would become the Theory of Nonlinear Oscillations and wrote (Andronov, 1928, p. 24):

The stable motions existing in devices capable of self-oscillations must always correspond to limit cycles.

On Monday, October \(14^{th}\), 1929 the French mathematician Jacques Hadamard (1865-1963) presented to the Academy of Sciences of Paris the famous note from Alexander Andronov entitled Poincaré limit cycles and the theory of self-sustaining oscillations.

Self-oscillations and limit cycles

In this note Andronov explained that such systems he called self-oscillators can be represented in the phase plane by two simultaneous differential equations:

\[ \tag{15} \frac{dx}{dt} = P\left( x, y \right) \quad \mbox{;} \quad \frac{dy}{dt} = Q\left( x, y \right) \]

and Andronov (1929, p. 560) stated that:

"It may easily be shown that, to periodic motions satisfying these conditions, there correspond, in the $xy$ plane, isolated closed curves, approached in spiral fashion by neighboring solutions from the interior or the exterior (for increasing t). As a result, self-oscillations arising in systems characterized by equations of type (15) correspond mathematically to stable Poincaré's limit cycles."

Stability condition of limit cycles

The next step for Andronov was to show that the periodic solution, i.e. the limit cycle is stable. To this purpose he considered the following system, where \( \mu \) is a real parameter, as an example:

\[ \tag{16} \frac{dx}{dt} = y + \mu f\left( x, y; \mu \right) \quad \mbox{;} \quad \frac{dy}{dt} = -x + \mu g\left( x, y; \mu \right) \]

He explained that for \( \mu = 0 \) the solution of this system is\[ x = R \cos\left( t \right) \], \( y = - R \sin\left( t \right) \) as it is obvious to check. This enables him to introduce an unusual (Unusual since it corresponds to a clockwise rotation and not to the classical counter clockwise trigonometric rotation. But, it corresponds exactly with the rotation direction of the trajectory curve such as Poincaré had established it. See Fig. 4) variables changes in polar coordinates. Then, by using the methods introduced by Poincaré (1892, tome I, p. 89) he stated that for sufficiently small \( \mu \neq 0 \)‚ the xy plane contains only isolated closed curves, near to circles with radii defined by the equation:

\[ \tag{17} \int_0^{2\pi} \left[ f\left(R \cos \xi, - R \sin \xi ; 0 \right) \cos \xi - g\left(R \cos \xi, - R \sin \xi ; 0 \right) \sin \xi \right] d\xi = 0 \]

Andronov provides a stability condition for the steady-state motion, i.e. for the limit cycle:

\[ \tag{18} \int_0^{2\pi} \left[ f_x \left(R \cos \xi, - R \sin \xi ; 0 \right) \cos \xi + g_y \left(R \cos \xi, - R \sin \xi ; 0 \right) \sin \xi \right] d\xi < 0 \]

In fact, this condition is based on the use of characteristic exponents introduced by Poincaré in his so-called New Methods on Celestial Mechanics (1892, tome I, p. 161) and after by Lyapounov 1892, 1907) in his famous textbook General Problem of Stability of the Motion. That's the reason why Andronov will call later the stability condition (18): stability in the sense of Lyapounov or Lyapounov stability. Although, condition (17) is strictly equivalent to that of Poincaré (6) and so, to that of Poincaré (14) and Liénard (13), condition (18) is much more general.

Nevertheless, the same question arises: Was Andronov informed of these forgotten lectures of Poincaré (1908)?


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