[Paper Review] Wright type delay differential equations with negative Schwarzian

Eduardo Liz, Manuel Pinto+3 more|ArXiv.org|Nov 27, 2001

Stability and Controllability of Differential Equations11 references4 citations

TL;DR

This paper establishes that the classical 3/2 stability condition for the Wright equation extends to a broad class of delay differential equations with decreasing or unimodal nonlinearities that are smooth, satisfy negative feedback and boundedness, and have everywhere negative Schwarzian derivative. The key result proves global asymptotic stability of the zero equilibrium under these generalized conditions, resolving a long-standing conjecture in the context of nonlinear delay equations with non-exponential nonlinearities.

ABSTRACT

We prove that the well-known 3/2 stability condition established for the Wright equation (WE) still holds if the nonlinearity $p(\exp(-x)-1)$ in WE is replaced by a decreasing or unimodal smooth function f with $f'(0)<0$ satisfying the standard negative feedback and below boundedness conditions and having everywhere negative Schwarz derivative.

Motivation & Objective

• To extend the well-known 3/2 stability condition of the Wright equation to a broader class of delay differential equations with non-exponential nonlinearities.
• To investigate whether the negative Schwarzian derivative condition, common in one-dimensional dynamics, ensures global stability in scalar delay-differential equations.
• To resolve the Wright conjecture in a generalized setting by proving global attractivity of the zero equilibrium under mild structural assumptions on the nonlinearity.
• To demonstrate that the 3/2 stability condition holds not only for the exponential nonlinearity in the original Wright equation but also for smooth, decreasing or unimodal functions with negative Schwarzian derivative.

Proposed method

• The authors analyze the scalar delay-differential equation $ x'(t) = f(x(t-1)) $ under conditions (H1)–(H3), where $ f $ is $ C^3 $, satisfies negative feedback and boundedness, and has negative Schwarzian derivative.
• They use a variational approach along the equilibrium $ x=0 $, studying the linearized equation $ x'(t) = f'(0)x(t-1) $, and analyze the stability threshold $ -f'(0) < \pi/2 $.
• The proof relies on constructing bounds on the range of solutions using the Schwarzian derivative and properties of the function $ r(x) $, which acts as a comparison function for $ f(x) $.
• They define auxiliary functions $ R(x) $, $ A(x) $, and $ B(x) $ to estimate solution extrema and derive inequalities involving $ m $ and $ M $, the infimum and supremum of a solution's range.
• By contradiction, they show that assuming $ m < 0 < M $ leads to inconsistencies under the 3/2 condition, using rational and linear approximations of $ f $ near zero.
• The argument is extended to cases where $ f''(0) > 0 $, $ f''(0) < 0 $, and $ f''(0) = 0 $, using symmetry and transformation techniques to reduce to the positive second derivative case.

Experimental results

Research questions

• RQ1 Does the 3/2 stability condition for the Wright equation hold for nonlinearities other than the exponential form $ f(x) = p(e^{-x} - 1) $?
• RQ2 Can the negative Schwarzian derivative condition be used to ensure global asymptotic stability in delay differential equations with non-monotone nonlinearities?
• RQ3 Is the global attractivity of the zero equilibrium preserved when $ f $ is unimodal and satisfies $ (Sf)(x) < 0 $, even if it has an inflection point or a local extremum?
• RQ4 Can the classical 3/2 stability result be generalized to nonlinearities that are not strictly decreasing, provided they satisfy the negative Schwarzian condition and standard feedback and boundedness constraints?

Key findings

• The 3/2 stability condition holds for all $ C^3 $ nonlinearities $ f $ satisfying (H1)–(H3), including unimodal and non-monotone functions with negative Schwarzian derivative.
• Global asymptotic stability of the zero equilibrium is established for all $ f $ with $ f'(0) \in [-1.5, 0) $, provided $ f $ satisfies the negative Schwarzian derivative condition.
• The proof shows that assuming $ m < 0 < M $ leads to a contradiction under the 3/2 condition, thereby proving global attractivity.
• The result generalizes Wright’s original theorem beyond exponential nonlinearities, as the proof does not rely on the specific exponential form of $ f $, but on the negative Schwarzian derivative and smoothness.
• The method applies uniformly across cases with $ f''(0) > 0 $, $ f''(0) < 0 $, and $ f''(0) = 0 $, using symmetry and transformation to reduce to the positive second derivative case.
• The negative Schwarzian derivative condition is shown to be both natural and necessary for such global stability results, as it appears in many biological models like Mackey-Glass and Nicholson equations.