Differential equations with random parameters have gained significant prominence in recent years due to their importance in mathematical modeling and data assimilation. In many cases, random ordinary differential equations (RODEs) are studied by using Monte Carlo methods or by direct numerical simulation techniques using polynomial chaos (PC), i.e., by a series expansion of the random parameters in combination with forward integration. Here we take a dynamical systems viewpoint and focus on the invariant sets of differential equations such as steady states, stable/unstable manifolds, periodic orbits, and heteroclinic orbits. We employ PC to compute representations of all these different types of invariant sets for RODEs. This allows us to obtain fast sampling, geometric visualization of distributional properties of invariants sets, and uncertainty quantification of dynamical output such as periods or locations of orbits. We apply our techniques to a predator-prey model, where we compute steady states and stable/unstable manifolds. We also include several benchmarks to illustrate the numerical efficiency of adaptively chosen PC depending upon the random input. Then we employ the methods for the Lorenz system, obtaining computational PC representations of periodic orbits, stable/unstable manifolds, and heteroclinic orbits.


  1. invariant manifold
  2. periodic orbit
  3. heteroclinic orbit
  4. Lorenz system
  5. polynomial chaos
  6. random differential equation

MSC codes

  1. 37H10
  2. 34F05
  3. 60H35
  4. 41A58
  5. 65C50

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Information & Authors


Published In

cover image SIAM Journal on Applied Dynamical Systems
SIAM Journal on Applied Dynamical Systems
Pages: 577 - 618
ISSN (online): 1536-0040


Submitted: 28 December 2018
Accepted: 22 January 2020
Published online: 17 March 2020


  1. invariant manifold
  2. periodic orbit
  3. heteroclinic orbit
  4. Lorenz system
  5. polynomial chaos
  6. random differential equation

MSC codes

  1. 37H10
  2. 34F05
  3. 60H35
  4. 41A58
  5. 65C50



Funding Information

Volkswagen Foundation https://doi.org/10.13039/501100001663

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