University:
University of Michigan
Course:
MATH 216 | Introduction to Differential Equations
Academic year:

2024
Views:

360

Pages:

1
Author:

Tatum B

Structural Stability Reading. §9.1, §9.3/503-504 Definition. A system of ordinary differential equations is called structurally stable when all nearby systems have the same qualitative behavior. The precise notions of nearby and same behavior may vary with the context. §A. Structural stability of linear systems. Theorem 1. If X ′ = A2×2 X is a two by two real system of constant coefficient linear homogeneous equations which is a proper nodal sink, proper nodal source, spiral sink, spiral source, or saddle, then for all nearby A the type of the equilibrium is the same. Discussion. One sensitive case is when there are purely imaginary eigenvalues (including the possibility λ = 0). Then, nearby eigenvalues need not be purely imaginary any may lie in either the left of right hand half planes. The second sensitive case is a repeated nonzero real eigenvalue where for most nearby matrices there will be two nearly equal but distinct real eigenvalues. Definition. For an n × n system X ′ = An×n X system of homogeneous linear constant coefficient equations, the equilibrium 0 is a sink (resp. source) when the eigenvalues of A all have strictly negative (resp. positive) real part. For sinks all solutions decay exponentially to zero. For sources the solutions grow exponentially. Theorem 2. If the n × n system X ′ = An×n X is a sink or a source, then the same is true for all nearby A. §B. Structural Stability of one dimensional autonomous systems. Theorem 3. If f (x) is a smooth real valued function such that i. The equation f (x) = 0 has only a finite number of roots. ii. At each root x, f ′ (x) 6= 0. iii. lim inf |x|→∞ |f (x)| > 0. Then the ordinary differential equation x′ = f (x) is structurally stable in the sense that there is an ε > 0 so that if < ε sup f (x) − f˜(x) + f ′ (x) − f˜′ (x) x∈R then the phase diagram of x′ = f˜(x) is qualitatively identical to that of x′ = f (x). Discussion. These are the only systems with a finite number of equilibria which are structurally stable. The exceptional cases are equilibria where f ′ (x) = 0 and functions which approach arbitrarily close to 0 for |x| tending to infinitiy. In either case there are arbitrarily small perturbations which change the number of equilibria. Draw pictures to convince yourself of this.

Structural Stability

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