Lyapunov stability theory

Sebastian Seung

9.641 Lecture 10: October 18, 2000

1  Dynamical systems theory: definitions

• One general class of dynamical system includes differential equations of the form

  
  

  dx dt = F(x,t)

  where x is a vector and F is a vector-valued function.

• Here we will consider autonomous dynamical systems, in which there is no explicit dependence of F on time:

  
  

  dx dt = F(x)

• x^* is said to be an equilibrium point (fixed point, stationary point, steady state) if x(t) = x^* implies equality for all future time.
• There are several mathematical definitions of the term ``stability.'' The one due to Lyapunov is most useful here.

  • The equilibrium state x = 0 is stable if, for any e > 0, there exists d > 0, such that |x(0)| < d implies |x(t)| < e for all t ³ 0.
  • In other words, the system trajectory can be kept arbitrarily close to the origin by starting sufficiently close to it.
  • An equilibrium state is unstable if the above condition is not satisfied. It's a bit tricky to negate the quantifiers, but here goes. There exists at least one e such that for every d > 0, there exists a trajectory with |x(0)| < d and |x(t)| ³ e for some t. For a linear system, instability is equivalent to blowing up. In a nonlinear system, this is not the case.
• An equilibrium point 0 is asymptotically stable if it is stable, and in addition there exists d > 0 such that |x(0)| < d implies that x(t)® 0 as t®¥.
• An equilibrium point which is Lyapunov stable but not asymptotically stable is called marginally stable.
• If asymptotic stability holds for any initial condition, the equilibrium point is said to be globally asymptotically stable. Note that for linear networks, asymptotic stability and global asymptotic stability are equivalent, so the distinction between local and global stability is not necessary.

2  Symmetric linear networks

We saw already that the stability of linear networks can be characterized in terms of eigenvalues of the weight matrix. But this doesn't generalize to nonlinear systems. Here is another way of deriving sufficient conditions for stability.

• Suppose that the connections are symmetric W_ij = W_ji, and define the quadratic energy function

  
  

  E = - 1 2 å ij  Wijxi xj - å i  bi xi + 1 2 å i  xi2 (1) = 1 2 xT(I-W)x - bTx (2)

• Then the linear network is a gradient dynamics

  
  

  dxi dt = - ¶E ¶xi

  Symmetry is important here.

• Since [(E)\dot] £ 0, E is nonincreasing. Equality holds only at an equilibrium point. As long as E is concave up, we get convergence to minimum.

3  Symmetric nonlinear networks

• Is [(x)\dot]_i + x_i = f(å_j W_ijx_j + b_i) a gradient dynamics? Unfortunately not, as we can check by equality of mixed partials.
• Suppose that L(x) is a scalar function of the state x, has continuous partial derivatives, is lower bounded, radially unbounded, and [(L)\dot] £ 0 with equality only at equilibrium states of the dynamics. Then the set of minima of L is globally asymptotically stable, except for the set of initial conditions that are unstable equilibria.
• Instead, define a Lyapunov function

  
  

  L = - 1 2 å ij  Wijxi xj - å i  bi xi + å i  F(xi)

  where F¢ º f. Note that this reduces to quadratic energy function for linear network if f(x) = x. This can be used to show that the network dynamics converges to fixed points.

• Lemma: f(a)-f(b) has same sign as a-b, if f is monotonic increasing.
• Calculate [(L)\dot] and show that it is nonpositive, and vanishes only at fixed points. This means that L is nonincreasing. So starting from almost all initial conditions, the network converges to local minima of L.
• Geometric interpretation. Minus the gradient of L is within 90 degrees of the velocity.
• It can be shown that the previous nonlinear network is equivalent to

  
  

  . y   i  + yi = å j  Wijf(yj) + bi

  Through what Grossberg calls the S-S transform. This is only true when the time constants are uniform.

4  Asymmetric networks

5  Why should we care about stability?

Qualitative dynamics is important: steady state, limit cycle, chaos.

The Lyapunov function gives us a computational interpretation of network dynamics as optimization.

Most importantly, the Lyapunov function will be important for understanding Hebbian learning.