1 Limit Cycles and Bifurcations Oscillations are one of the most important phenomena that occur in dynamical systems. A system oscillates when it has a nontrivial periodic solution ( ) ( )t T x t+ = 0t

2

, x for some . In a phase portrait an oscillation or periodic solution looks like a closed curve.

0T > Example 1.3: Van der Pol Oscillator

( )1 2

22 1 11

x x

x x x== +

&& x

In the case 0 = we have a continuum of periodic solutions, while in the second case

0 there is only one. An isolated periodic orbit is called a limit cycle. Example 1.4: A stable and an unstable limit cycle

1.1 Existence of Periodic Orbits Periodic orbits in the plane are special in that they divide the plane into a region inside the orbit and a region outside it. This makes it possible to obtain criteria for detecting the presence or absence of periodic orbits for second-order systems, which have no generalizations to higher order systems.

Theorem 1.1: (Poincar-Bendixson Criterion) Consider the system ( )x f x=& and let M be a closed bounded subset of the plane such that: M contains no equilibrium points, or contains only one equilibrium point such that the Jacobian matrix [ ]/f x at this point has eigenvalues with positive real parts. Every trajectory starting in M remains in M for all future time. Then M contains a periodic orbit of ( )x f x=& . Theorem 1.2: (Negative Pointcar-Bendixson Criterion) If, on a simply connected region

of the plane, the expression D 1 1 2/ / 2f x f x + is not identically zero and does not change sign, then the system ( )x f x=& has no periodic orbits lying entirely in . D Proof On any orbit ( )x f x=& , we have 1 2 2/dx dx f f1/= . Therefore, on any closed orbit , we have ( ) ( )2 1 2 1 1 1 2 2, ,f x x dx f x x dx

= 0 This implies, by Greens theorem, that

1 2 1 21 2

0S

f f dx dxx x

+ = where S is the interior of . If 1 1 2 2/ /f x f x 0 + > (or 0< ) on , then we cannot find a region such that the last equality holds. Hence, there can be no closed orbits entirely in .

DS D

D

1.2 Bifurcations The qualitative behavior of a second-order system is determined by the pattern of its equilibrium points and periodic orbits, as well as by their stability properties. One issue of practical importance is whether the system maintains its qualitative behavior under infinitesimally small perturbations. When it does so, the system is said to be structurally stable. In particular, we are interested in perturbations that will change the equilibrium points or periodic orbits of the system or change their stability properties.

Example 1.5: We consider the system

21 1

2 2

x xx x

= =

&&

1. If 0 > we have two equilibrium points ( )1,2 , 0Q = 2. If 0 = we have the degenerated case ( )0,0 3. If 0 < we have no equilibrium

Theorem 1.3: (Hopf Bifurcations) Consider the system ),( 21 xxfx =& , where is a parameter. Let ( be an equilibrium point for any value of the parameter)0,0 . Let c be such that ( )0,0feig jx = . If the real parts of the eigenvalues 1,2 of /f x are such that ( )1,2Re 0ddx > and the origin is stable for c = then the origin remain stable for small perturbations of there is a 2 such that the origin is unstable surrounded by a stable limit cycle for

2c < < .

1 Limit Cycles and Bifurcations1.1 Existence of Periodic Orbits1.2 Bifurcations