On a Gradation for Asymptotic Stability

I. INTRODUCTION

• Classical Hopf classification does not distinguish convergence laws within the attracting-equilibrium regime, although vanishing linear damping can leave cubic stabilization and t^-1/2 decay.With λ < 0, the linear radial term yields exponential convergence; at λ = 0 and α < 0, the cubic term stabilizes the origin algebraically.Source: §I · p. 1
• At a Bautin point, the vanishing cubic term exposes distinct stable rates: t^-1/2 when β1 = 0 and β2 < 0, and t^-1/4 when β1 = β2 = 0.The first nonzero stabilizing coefficient determines the rate; Figure 1 illustrates the separation between degrees 0, 2, and 4.Source: §I · p. 1, Figure 1 · p. 1
• These intermediate algebraic regimes are structural properties of the vector field but are not explicitly organized by classical qualitative stability terminology.The framework also gives a nonsmooth converse characterization: a degree-m estimate is equivalent to a continuous Lyapunov certificate with matching degree-m dissipation.Source: §I · pp. 1–2
• The framework assigns degree 0 to exponential convergence and degree m > 0 to an admissible algebraic decay bound of order O(t^-1/m).An upper Lyapunov dissipation inequality certifies an admissible degree; a matching lower inequality on a forward-invariant set certifies exactness by ruling out faster degrees.Source: §I · pp. 1–2

II. RELATED WORK

• Classical uniform asymptotic-stability characterizations use class-KL trajectory estimates and Lyapunov functions, including converse results for time-varying systems.The paper's objective is to distinguish asymptotically stable equilibria by their convergence rates.Source: §II · p. 2
• Power-type Lyapunov inequalities link dissipation exponents to finite-time, exponential, or algebraic decay, while this framework identifies the smallest admissible decay order.The framework treats that order as a stability parameter rather than fixing the exponent in advance.Source: §II · p. 2
• Unlike rational stability's prescribed integer p, this framework lets a real degree m vary and defines stability degree as the smallest admissible order, when attained.Matching Lyapunov lower bounds certify that this boundary is exact.Source: §II · p. 2
• Unlike Jagt and Peet [16], who quantify rate and gain for a prescribed decay family, this framework classifies the boundary of admissible decay orders and applies to time-varying systems.Their rate-performance framework is developed for autonomous ODEs.Source: §II · p. 2
• Algebraically decaying trajectories share a zero classical Lyapunov exponent, whereas the proposed degree distinguishes their algebraic decay orders.For such trajectories, t^-1 log ∥x(t)∥ tends to zero.Source: §II · p. 2

III. DEGREE OF STABILITY

• Degree 0 corresponds to exponential decay, whereas degree m > 0 corresponds to algebraic decay of order t^-1/m.The comparison family interpolates between exponential and polynomial convergence.Source: §III · p. 2
• Uniform degree-m estimates imply local uniform asymptotic stability, making the rate-indexed classes refinements of asymptotic stability.The comparison envelopes decay to zero and bound trajectories uniformly in initial time.Source: §III · p. 3
• A degree-m estimate yields an O(t^-1/m) trajectory bound; equivalently, algebraic exponent β corresponds to m = 1/β, with larger m indicating slower convergence.The estimate is specified locally around the origin, with constants that may depend on initial time unless the estimate is uniform.Source: §III · pp. 2–3
• The lower admissible degree is the infimum of admissible orders, and degree-m stability requires that this infimum be attained.Uniform admissible orders form a subset of all admissible orders, so the lower uniform degree cannot be smaller when both exist.Source: §III · p. 3
• The lower admissible degree need not be attained: a rate proportional to (log t)t^-1/m* admits every degree q > m* but not m*.This gives D = (m*, ∞), despite a finite infimum.Source: §III · p. 3
• An asymptotically stable origin may have no finite degree estimate; unstable origins receive no stability degree.The paper describes m = ∞ as a formal label for convergence slower than every algebraic rate.Source: §III · p. 3
• For smooth scalar autonomous systems with a finite nonzero Taylor leading term, attained degrees are even integers; flat systems are not covered.For leading order k > 1, the degree is k - 1, while k = 1 gives degree 0.Source: §III · p. 3

IV. LYAPUNOV CHARACTERIZATION

• The Lyapunov characterization section develops direct certificates, illustrates them on normal forms, vector systems, and time-varying systems, and establishes a nonsmooth converse.The converse concerns admissible degree-m estimates.Source: §IV · p. 3

A. Direct Lyapunov Method for Degree-m Stability

• An upper Lyapunov dissipation inequality certifies an admissible degree, while a matching lower inequality on a forward-invariant set accumulating at the origin certifies the exact degree.For the exact-degree conclusion, the lower inequality rules out all smaller admissible degrees.Source: A. Direct Lyapunov Method for Degree-m… · pp. 3–4
• Theorem 4.1’s upper condition yields a uniform degree-m estimate; at m = 0, it gives uniform degree-0 stability.The theorem assumes a continuously differentiable Lyapunov function and positive constants on a neighborhood of the origin.Source: A. Direct Lyapunov Method for Degree-m… · pp. 3–4
• The exact-degree criterion for m > 0 additionally requires a forward-invariant set containing nonzero points arbitrarily close to the origin and a matching lower inequality there.The theorem concludes m = min Du under this additional hypothesis.Source: A. Direct Lyapunov Method for Degree-m… · p. 4
• The theorem extends to nonuniform estimates by allowing Lyapunov bounds and dissipation constants to depend on initial time.The upper inequality certifies admissibility, and the matching lower inequality on an invariant set yields the minimum degree.Source: A. Direct Lyapunov Method for Degree-m… · p. 4
• Quadratic scaling V(t,x) ≍ ||x||^2 avoids regularity issues that can arise when using ||x||^p for arbitrary p.The resulting dissipation exponent 1 + m/2 extends the exponential condition recovered at m = 0.Source: A. Direct Lyapunov Method for Degree-m… · p. 4

Hopf normal form

• At the critical Hopf parameter, vanishing linear radial damping changes the degree from 0 to 2, corresponding to decay of order (t−t0)^−1/2.When λ < 0, the degree is 0; when λ = 0 and α < 0, the degree is 2.Source: B. Examples · p. 4

Bautin normal form

• The Bautin normal form distinguishes degrees 0, 2, and 4 as linear and then cubic radial stabilization vanish.The degree-4 case corresponds to decay of order (t−t0)^−1/4.Source: B. Examples · pp. 4–5

Fractional-degree example

• On a curved invariant manifold, the example certifies exact degree 3/2, while transverse deviations decay exponentially.The set S = {y = 0} is invariant and carries the slow algebraic dynamics.Source: B. Examples · p. 5

Time-varying example

• In the time-varying example, rotation terms cancel in the Lyapunov derivative, yielding exact degree 2 and decay of order (t−t0)^−1/2.The derivative is bounded using norm inequalities rather than an exact scalar radial normal form.Source: B. Examples · p. 5

Numerical verification

• Numerical trajectories for Examples 4.4–4.7 closely match their predicted reference decay rates.The figure compares solid ∥x(t)∥ curves with dashed reference rates.Source: Figure 2 · p. 5, B. Examples · p. 5

Converse construction

• The converse Lyapunov function is constructed from the flow by a supremum–integral formula, but need not be smooth.The resulting derivative condition is therefore stated using the upper Dini derivative along solutions.Source: C. Converse Lyapunov Characterization · pp. 5–6

Converse characterization

• A uniform degree-m estimate holds if and only if a continuous Lyapunov function satisfies matching degree-m bounds and dissipation measured by the upper Dini derivative.The theorem characterizes membership in D_u, not exactness.Source: C. Converse Lyapunov Characterization · p. 5

Critical Hopf illustration

• The critical Hopf example's converse construction yields a quadratic Lyapunov function with matching degree-2 dissipation.The constructed function is V(x) = C∥x∥².Source: C. Converse Lyapunov Characterization · p. 6

Time-varying illustration

• The construction gives a uniform degree-2 certificate and its matching lower bound; for the associated expression, the supremum is attained at τ = 0.Source: C. Converse Lyapunov Characterization · p. 6

• The degree-based framework places exponential and algebraic convergence rates on a common scale for local asymptotic stability.Source: §V · p. 6
• The paper develops Lyapunov certificates for admissible and exact degrees, plus a nonsmooth converse characterization of uniform admissible degrees.Source: §V · p. 6
• Future work will investigate degree computation from leading vector-field terms, smooth converse certificates, and computational certification methods.Source: §V · p. 6

• For m = 0, the dissipation inequality gives a uniform degree-0 estimate and recovers the classical Lyapunov criterion for exponential stability.Source: Appendix I · p. 7
• The upper Lyapunov dissipation inequality yields a uniform degree-m estimate by comparison, establishing admissibility.The proof first keeps trajectories in the region where the Lyapunov bounds hold.Source: Appendix I · pp. 7–8
• A matching lower inequality on a forward-invariant set rules out all degrees below m, proving exactness.The proof excludes q = 0 and then every 0 < q < m.Source: Appendix I · pp. 8–10
• The resulting origin is uniformly degree-m stable.Source: Appendix I · p. 10

• Assuming (9)–(10), scalar comparison and forward-time estimates establish the uniform degree-m estimate.Source: APPENDIX II PROOF OF THEOREM 4.8 · pp. 10–11
• Conversely, starting from a uniform degree-m estimate, the proof constructs a Lyapunov function and establishes its bounds and continuity.Source: APPENDIX II PROOF OF THEOREM 4.8 · pp. 12–14
• The constructed function's Dini derivative inequality establishes the remaining condition and completes the converse proof.Source: APPENDIX II PROOF OF THEOREM 4.8 · p. 14