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The Lorenz system as a gradient-like system

Jeremy P Parker

Abstract

We formulate, for continuous-time dynamical systems, a sufficient condition to be a gradient-like system, i.e. that all bounded trajectories approach stationary points and therefore that periodic orbits, chaotic attractors, etc. do not exist. This condition is based upon the existence of an auxiliary function defined over the state space of the system, in a way analogous to a Lyapunov function for the stability of an equilibrium. For polynomial systems, Lyapunov functions can be found computationally by using sum-of-squares optimisation. We demonstrate this method by finding such an auxiliary function for the Lorenz system. We are able to show that the system is gradient-like for $0\leqρ\leq12$ when $σ=10$ and $β=8/3$, significantly extending previous results. The results are rigorously validated by a novel procedure: First, an approximate numerical solution is found using finite-precision floating-point sum-of-squares optimisation. We then prove that there exists an exact solution close to this using interval arithmetic.

The Lorenz system as a gradient-like system

Abstract

We formulate, for continuous-time dynamical systems, a sufficient condition to be a gradient-like system, i.e. that all bounded trajectories approach stationary points and therefore that periodic orbits, chaotic attractors, etc. do not exist. This condition is based upon the existence of an auxiliary function defined over the state space of the system, in a way analogous to a Lyapunov function for the stability of an equilibrium. For polynomial systems, Lyapunov functions can be found computationally by using sum-of-squares optimisation. We demonstrate this method by finding such an auxiliary function for the Lorenz system. We are able to show that the system is gradient-like for when and , significantly extending previous results. The results are rigorously validated by a novel procedure: First, an approximate numerical solution is found using finite-precision floating-point sum-of-squares optimisation. We then prove that there exists an exact solution close to this using interval arithmetic.
Paper Structure (10 sections, 9 theorems, 33 equations, 2 figures, 2 tables)

This paper contains 10 sections, 9 theorems, 33 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Mathematical results
  3. Application to the Lorenz system
  4. Sum-of-squares relaxation
  5. Validation with interval arithmetic
  6. A semi-definite program as a linear system
  7. Interval arithmetic
  8. The full validation for the Lorenz system
  9. Conclusion
  10. An auxiliary function method for maps

Key Result

Lemma 1

Let $g:\mathbb{R}^n\to\mathbb{R}_{\geq0}$ be a continuous, nonnegative function defined over state space. Suppose there exists a continuously differentiable function $V:\mathbb{R}^n\to\mathbb{R}$ which satisfies for all $x\in \mathbb{R}^n$. Then $g(y)=0$ for all limit points $y$.

Figures (2)

  • Figure 1: Dual objective value reported by the SDP solver for a naïve implementation of \ref{['thm:lorenz']} (with $M=16$). A value close to zero indicates that the SDP is likely to be feasible, i.e. that the Lorenz system is indeed gradient-like at that value of $\rho$.
  • Figure 2: Bifurcation diagram (black, left axis) and SOS results (colours) for the Hénon map at $b=0.3$. For each choice of polynomial degree $d$ and map iterate $k$, the right vertical axis shows the dual objective value reported by MOSEK. A value close to zero indicates that the problem is likely to be feasible.

Theorems & Definitions (16)

  • Definition 1
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • ...and 6 more