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Explicit scalar quadratic descent and its forward domain

Statement

Let $a > 0$ and $x_{0} \in R$ . A solution means a differentiable real-valued function on an interval containing $0$ that satisfies $\dot{x} = - a x^{2}$ and $x (0) = x_{0}$ . On the largest open interval containing $0$ on which $1 + a x_{0} t \neq 0$ , the unique solution is $x (t) = x_{0} / (1 + a x_{0} t)$ . If $x_{0} \geq 0$ , it exists for all $t \geq 0$ and tends to zero as $t \to \infty$ . If $x_{0} < 0$ , its maximal forward interval is $[0, - 1 / (a x_{0}))$ .

For finitely many independent coordinates ${\dot{x}}_{i} = - a_{i} x_{i}^{2}$ , where $n \geq 1$ and every $a_{i} > 0$ , every initial point in $(0, \infty)^{n}$ has a forward solution converging to the origin. For vector solutions, convergence to the origin as $t \to \infty$ means that for every $ε > 0$ there is $T \geq 0$ such that $∥ x (t) ∥_{2} < ε$ for all $t > T$ , where $∥ z ∥_{2} = (\sum_{i = 1}^{n} z_{i}^{2})^{1 / 2}$ is the Euclidean norm. Here the attraction region of the origin is the set of initial points in $R^{n}$ whose unique solutions exist for every $t \geq 0$ and converge to the origin as $t \to \infty$ . Assuming countable choice, the attraction region contains the positive-Lebesgue-measure box $(0, 1)^{n}$ . This attraction conclusion is a dynamical statement and does not require a second-order classification of the origin.

Facts & Assumptions

Given: The coefficients, initial values and independent equations in the Statement; countable choice is assumed for the measure assertion.

[F1]

A solution of an initial-value problem is a differentiable function on an interval containing the initial time and satisfying the equation and initial condition (First-order systems, initial value problems, and solutions on intervals).

[F2]

A continuous vector field that is locally Lipschitz in its state variable has a unique local solution; any two solutions through the same initial data agree on every common interval containing the initial time (Local Lipschitz continuity in the state variable, locally uniform in time and parameters, Picard-Lindelöf local existence and uniqueness for first-order systems).

[F3]

The sum, product, quotient and chain rules apply to differentiable real functions where the relevant denominator is nonzero (Sum, product, quotient, and chain rules from difference quotients).

[F4]

Under countable choice, an $n$ -dimensional open box is Lebesgue measurable and its measure is the product of its side lengths (The Axiom of Countable Choice (AC_ω), A box in ℝⁿ with parameters aᵢ≤ bᵢ is Lebesgue measurable of measure ∏_i<n(bᵢ-aᵢ), whichever of its faces are included).

[F5]

Local Lipschitz continuity in the state variable, locally uniform in time, means that every compact time-state cylinder in the open domain has one finite state-Lipschitz constant (Local Lipschitz continuity in the state variable, locally uniform in time and parameters).

[F6]

Convergence as $t \to + \infty$ uses the real $ε$ -threshold condition. A one-sided infinite limit at $T$ is the infinite-limit condition on the restriction to $t < T$ or $t > T$ , with $T$ a limit point of that restricted domain (Limits at +∞ and -∞, and infinite limits at a point, The left and right limits of f at c, as limits of the restrictions of f to A ∩ (-∞, c) and A ∩ (c, ∞)).

[F7]

A function from a nondegenerate real interval to $R^{n}$ is differentiable if and only if all its components are differentiable, and its derivative has those component derivatives; this includes domain-relative endpoint derivatives (The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral).

[F8]

For $z \in R^{n}$ , the Euclidean norm is $∥ z ∥_{2} = (\sum_{i = 1}^{n} z_{i}^{2})^{1 / 2}$ , so it is nonnegative and $∥ z ∥_{2}^{2} = \sum_{i = 1}^{n} z_{i}^{2}$ ; it is a norm inducing the Euclidean metric (The Euclidean inner product ⟨ x,y⟩ = ∑_k<n xₖ yₖ on ℝⁿ, Cauchy-Schwarz |⟨ x,y⟩| ≤ ‖ x‖₂‖ y‖₂ with its equality case, the triangle inequality for ‖·‖₂, the parallelogram law and polarisation).

Proof

Given: The coefficients, initial values and independent equations in the Statement; countable choice is assumed for the measure assertion.

Proof technique: direct.

Goal: Establish the explicit solutions, their maximal intervals and the attraction assertion.

1.1givenF1F2F3F5

The field $f (x) = - a x^{2}$ is continuous and locally Lipschitz: for $∣ x ∣, ∣ y ∣ \leq R$ , $∣ f (x) - f (y) ∣ \leq 2 a R ∣ x - y ∣$ . Thus [F2] applies. On the stated open interval the quotient rule gives $\frac{d}{d t} \frac{x_{0}}{1 + a x_{0} t} = - \frac{a x_{0}^{2}}{(1 + a x_{0} t)^{2}} = - a {(\frac{x_{0}}{1 + a x_{0} t})}^{2}$ , and the value at zero is $x_{0}$ . It is therefore a solution by [F1], and [F2] makes every other solution agree with it on their common interval containing zero.

2.1givenF2F3F6step 1.1

If $x_{0} = 0$ , the formula is identically zero. If $x_{0} > 0$ , its denominator is at least one for $t \geq 0$ and the quotient tends to zero. If $x_{0} < 0$ , put $T = - 1 / (a x_{0}) > 0$ ; the formula is defined for $0 \leq t < T$ and tends to $- \infty$ as $t ↑ T$ . A real-valued differentiable solution on an interval extending beyond $T$ would be continuous at $T$ and, by uniqueness, equal the formula for $0 \leq t < T$ , a contradiction. Hence its maximal forward interval is $[0, T)$ . The same pole argument at the finite endpoint also shows that the stated open interval is maximal when $x_{0} > 0$ ; for $x_{0} = 0$ it is all of $R$ .

3.1givenF1F7step 1.1step 2.1

Apply steps 1.1 and 2.1 to each coordinate with initial values $x_{i, 0} > 0$ . The intersection $I$ of their finitely many open solution intervals is an open interval containing $0$ and all $t \geq 0$ . On $I$ , put $x (t) = (x_{i} (t))_{i = 1}^{n}$ , with $x_{i} (t) = x_{i, 0} / (1 + a_{i} x_{i, 0} t)$ . By [F7] this is a differentiable vector function with $x^{'} (t) = (- a_{i} x_{i} (t)^{2})_{i = 1}^{n}$ , so [F1] makes it a solution of the vector initial-value problem. Conversely, [F7] makes every component of any other vector solution a scalar solution; scalar uniqueness from step 1.1 gives agreement on every common interval containing $0$ .

4.1givenF6F8step 2.1step 3.1algebra

Fix $ε > 0$ . For each $i$ , scalar convergence from step 2.1 and [F6] give $T_{i} \geq 0$ such that $∣ x_{i} (t) ∣ < ε / n$ for all $t > T_{i}$ . Choose these finitely many thresholds and take their maximum $T$ ; this finite choice uses no countable-choice assumption. For $t > T$ , [F8] gives $∥ x (t) ∥_{2}^{2} = \sum_{i = 1}^{n} x_{i} (t)^{2} < n (ε / n)^{2} = ε^{2} / n \leq ε^{2}$ . Since both the norm and $ε$ are nonnegative, $∥ x (t) ∥_{2} < ε$ . This is exactly the vector convergence condition in the Statement.

5.1givenF4step 3.1step 4.1∎

By the definition in the Statement, these forward solutions place $(0, 1)^{n}$ in the attraction region; the box has Lebesgue measure $1$ by [F4]. No second-order classification entered this argument.

Depends on

Used by · 1

Positive-measure attraction to a degenerate saddle

Dependency tree · 2 levels

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