A Note on Bifurcation of Equilibrium Forms of a Gas Balloon Parachute

Zgorzelska, Anita; Guze, Hanna

doi:10.2478/amsil-2025-0009

Full Article

1.

Mathematical model

Bifurcation theory is a powerful tool in studying deformations of elastic beams, shells or plates. Lots of works have been devoted to the study of bifurcation in elasticity theory, see for instance [1], [4], [5], [11], [12], [14] and references therein.

Our study was motivated by gas balloons. Precisely, we are interested in the behaviour of the part of a balloon that is called an envelope. Following the description in [8], the fabric of the envelope is flexible (elastic). It is composed of large vertical sections called gores. Each gore is made up of the same number of horizontal sections called panels. The panels and gores are held together by stitching and by heavy duty flexible load tapes which help support the weight of the balloon and minimize a strain on the fabric. The top part of the envelope consisting of one panel of each gore is named a parachute (see Fig. 1). The standard parachute possesses a circular deflation port – a crown ring that is closed off by a circular panel which is held sealed during a flight by a flexible hook-and-loop closure. Moreover, we can treat a parachute in a balloon envelope as a two-dimensional object, because its height is much smaller than the length of a top rim - a horizontal tape between the parachute and the rest of envelope. An equilibrium form of the gas balloon parachute is described in polar coordinates by a 2π- periodic C^m⁺²-smooth positive function r(θ), m ∈ ℕ ∪ {0}.

Let C^m(2π), m ∈ ℕ ∪ {0}, denote the Banach space of 2π-periodic C^m-smooth functions r(θ) with the standard norm ${‖ r ‖}_{m} = \sum_{k = 0}^{m} max_{0 \in [0, 2 π]} | r^{(k)} (θ) |,$ {\left\| r \right\|_m} = \sum\limits_{k = 0}^m {\mathop {{\rm{max\;}}}\limits_{0 \in [0,2\pi ]} } \left| {{r^{\left( k \right)}}\left( \theta \right)} \right|, where r⁽^k⁾(θ) denotes the k-th derivative of r(θ) and r⁽⁰⁾(θ) = r(θ). It is well known that C^m(2π) is continuously embedded into the Hilbert space L²(2π) with the scalar product $〈 f, g 〉 = \int_{0}^{2 π} f (θ) g (θ) d θ .$ \left\langle {f,\;g} \right\rangle = \int \nolimits_0^{2\pi } f\left( \theta \right)g\left( \theta \right)d\theta .

The total energy of the parachute is given by: (1.1) $E (r, α, β) = \int_{0}^{2 π} (\sqrt{r^{2} (θ) + {r^{'}}^{2} (θ)} + α r (θ)) d θ - β ln S,$ E\left( {r,\;\alpha ,\;\beta } \right) = \int \nolimits_0^{2\pi } (\sqrt {{r^2}\left( \theta \right) + {{r'}^2}\left( \theta \right)} + \alpha r\left( \theta \right))d\theta - \beta {\rm{\;ln\;}}S, where α > 0 is an elasticity coefficient of heavy duty load tapes, β > 0 is a physical parameter describing a compressed gas inside the gas balloon, and S denotes the area of parachute, i.e. $S = S (r) = \frac{1}{2} \int_{0}^{2 π} r^{2} (θ) d θ .$ S = S\left( r \right) = {1 \over 2}\mathop \int \nolimits_0^{2\pi } {r^2}\left( \theta \right)d\theta .

Gas balloons are inflated with a gas of lower molecular weight than the ambient atmosphere. The most popular gas here is helium. Let us point out that a similar model was investigated by the second author and J. Janczewska in [7] and [8]. As a result of conversations with J. Janczewska, the formula for the energy functional has been improved (simplified). The component of the energy functional corresponding to the energy of compresses gas inside the gas balloon depends only on one parameter. However the main conclusions concerning deformations of the parachute are the same.

It can be easily calculated that the Fréchet derivative of the energy functional E with respect to the variable r is of the form: $\begin{array}{l} E_{r}^{'} (r, α, β) h = \int_{0}^{2 π} \frac{r^{3} (θ) + 2 r (θ) {r^{'}}^{2} (θ) - r^{2} (θ) r^{″} (θ)}{{(r^{2} (θ) + r^{2} (θ))}^{3 / 2}} h (θ) d θ \\ + \int_{0}^{2 π} (α - \frac{β}{S} r (θ)) h (θ) d θ, \end{array}$ \eqalign{ E_r^\prime \left( {r,\alpha ,\beta } \right)h = & \int \nolimits_0^{2\pi } \frac{{r^3 \left( \theta \right) + 2r\left( \theta \right)r'^2 \left( \theta \right) - r^2 \left( \theta \right)r''\left( \theta \right)}}{{(r^2 \left( \theta \right) + r^2 \left( \theta \right))^{3/2} }}h\left( \theta \right)d\theta \cr \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, &+ \mathop \int \nolimits_0^{2\pi } \left( {\alpha - \frac{\beta }{S}r\left( \theta \right)} \right)h\left( \theta \right)d\theta , \cr} where α, β ∈ ℝ₊, r, h ∈ C^m⁺²(2π) and r(θ) > 0 for θ ∈ [0, 2π]. Critical points of the energy functional E(r, α, β) are 2π-periodic C^m⁺²-smooth positive solutions of the equation (1.2) $\frac{r^{3} (θ) + 2 r (θ) {r^{'}}^{2} (θ) - r^{2} (θ) r^{″} (θ)}{{(r^{2} (θ) + {r^{'}}^{2} (θ))}^{3 / 2}} + α - \frac{β}{S} r (θ) = 0 .$ {{{r^3}\left( \theta \right) + 2r\left( \theta \right){{r'}^2}\left( \theta \right) - {r^2}\left( \theta \right)r''\left( \theta \right)} \over {{{({r^2}\left( \theta \right) + {{r'}^2}\left( \theta \right))}^{3/2}}}} + \alpha - {\beta \over S}r\left( \theta \right) = 0. We are interested in radially symmetric solutions of the equation (1.2). Substituting r(θ) ≡ r into (1.2), we get an algebraic equation $1 + α - \frac{β}{r π} = 0$ 1 + \alpha - {\beta \over {r\pi }} = 0 with a solution given by (1.3) $R_{α} = \frac{β}{π (1 + α)},$ {R_\alpha } = {\beta \over {\pi \left( {1 + \alpha } \right)}}, which corresponds to a circular shape of the top rim of radius R_α. To sum up, for all β ∈ ℝ₊ there exists a family of radially symmetric solutions of the equation (1.2) given by $Γ^{β} = {(R_{α}, α) : α \in ℝ_{+}},$ \Gamma ^\beta = \left\{ {\left( {R_\alpha ,\alpha } \right):\alpha \in {\mathbb R}_ + } \right\}, where R_α is defined by (1.3).

2.

Symmetry-breaking bifurcation problem

We now want to find all values of the parameter α for which the radially symmetric solution R_α loses its stability. For this purpose we will study bifurcation from the set of radial solutions with respect to α.

Definition 2.1

(R_α, α) ∈ Γ^β is called a symmetry-breaking bifurcation point of the equation (1.2) with respect to the set Γ^β if there exists a branch of non-radially symmetric solutions (r(t), α(t)) of (1.2), depending on |t| < ε, with r(0) = R_α and α(0) = α.

Set $α_{k} = k^{2} - 1, k \geq 1 .$ {\alpha _k} = {k^2} - 1,\,\,\,\,\,\,\,\,\,\,\,\;k \ge 1.

Theorem 2.2

For each k ≥ 2, there exists a smooth family of non-radially symmetric solutions (r(t), α(t)) of (1.2), defined for |t| < ε, satisfying $r (t) (θ) = R_{α (t)} + t \cdot \frac{1}{\sqrt{π}} cos (k θ) + o (| t |) .$ r\left( t \right)\left( \theta \right) = {R_{\alpha \left( t \right)}} + t \cdot {1 \over {\sqrt \pi }}{\rm{\;cos\;}}\left( {k\theta } \right) + o\left( {\left| t \right|} \right). In particular, at t = 0 we have r(0) = R_{α_k} and α(0) = α_k. This implies that (R_{α_k}, α_k) ∈ Γ^β is a symmetry-breaking bifurcation point for the equation (1.2).

The proof of the above theorem relies on the Crandall-Rabinowitz theorem concerning simple bifurcation points (see [6]). We will apply the gradient (variational) version of this theorem, developed by A. Yu. Borisovich (see [2], [3]). To enhance clarity, let us state this theorem.

Theorem 2.3

Assume that H is a Hilbert space with a scalar product (·, ·)_H. Let X and Y be Banach spaces continuously embedded in H. Suppose that a C^r-operator F : X_δ(0)× ℝ_δ(α₀) → Y and a C^r+1-functional E : X_δ(0)× ℝ_δ(α₀) → ℝ, where r ≥ 2, satisfy the following conditions:

F (0, α) = 0 for α ∈ ℝ_δ(α₀),
dim ker $F_{x}^{'} (0, α_{0}) = 1$ F_x^\prime (0,\alpha _0 ) = 1 , $F_{x}^{'} (0, α_{0}) e = 0$ F_x^\prime (0,\alpha _0 )e = 0 , (e, e)_H = 1,
codim im $F_{x}^{'} (0, α_{0}) = 1$ F_x^\prime (0,\alpha _0 ) = 1 ,
$E_{x}^{'} (x, α) h = {(F (x, α), h)}_{H}$ E_x^\prime (x,\alpha )h = (F(x,\alpha ),h)_H for (x, α) ∈ X_δ(0) × ℝ_δ(α₀) and h ∈ X,
$E_{x x a}^{‴} (0, α_{0}) (e, e, 1) \neq 0$ E_{xxa}^{'''} (0,\alpha _0 )(e,e,1) \ne 0 .

Then (0, α₀) is a bifurcation point of the equation

(2.1)

F (x, α) = 0 .

F\left( {x,\;\alpha } \right) = 0.

In fact, the solution set of this equation in a certain neighborhood of (0, α₀) consists of the curve Γ₁ = {(0, α): α ∈ ℝ_δ(α₀)} and a C^r⁻²-curve Γ₂, intersecting only at (0, α₀). Moreover, if r ≥ 3, the curve Γ₂ can be parametrized by a variable t, |t| ≤ ε, as

Γ_{2} = {(x (t), α (t)) : t \in ℝ_{ε} (0)}, where x (0) = 0, α (0) = α_{0} and x^{'} (0) = e .

\Gamma _2 = \left\{ {\left( {x\left( t \right),\alpha \left( t \right)} \right):t \in {\mathbb R}_\varepsilon \left( 0 \right)} \right\}{\text{ , }}where{\text{ }}x\left( 0 \right) = 0,\,\,\,\alpha \left( 0 \right) = \alpha _0 \,\,{ and }\,\,x'\left( 0 \right) = e.

Let us introduce the symbol $C_{e}^{m} (2 π)$ C_e^m \left( {2\pi } \right) , m ∈ ℕ ∪ {0} be the subspace of C^m(2π) of even functions. Set $X = C_{e}^{m} (2 π) and Y = C^{m} (2 π), m \in ℕ \cup {0} .$ X = C_e^m \left( {2\pi } \right)\,\,\,\,{\text{and}}\,\,\,\,\,\,Y = C^m \left( {2\pi } \right){\text{,}}\,\,\,\,\,\,m \in {\mathbb N} \cup \left\{ 0 \right\}. Given any α₀ ∈ ℝ₊ take (R_α₀, α₀) ∈ Γ^β. Starting now, X_δ(0) and (ℝ₊)_δ(α₀) denote balls of radius δ around 0 in X and α₀ ∈ ℝ₊, correspondingly. For ϱ ∈ X_δ(0) and α ∈ (ℝ₊)_δ(α₀) define (2.2) $r (θ) = R_{α} + ϱ (θ) .$ r\left( \theta \right) = {R_\alpha } + \varrho \left( \theta \right). Note that r(θ) represents a small perturbation in X from R_α given by (1.3). Substituting (2.2) in (1.1), we get the new energy functional Ê given by $\hat{E} (ϱ, α, β) = \int_{0}^{2 π} (\sqrt{{(R_{α} + ϱ)}^{2} + ϱ^{' 2}} + α (R_{α} + ϱ)) d θ - β ln \hat{S},$ \hat E\left( {\varrho ,\;\alpha ,\;\beta } \right) = \int \nolimits_0^{2\pi } \left( {\sqrt {{{({R_\alpha } + \varrho )}^2} + {\varrho ^{'2}}} + \alpha \left( {{R_\alpha } + \varrho } \right)} \right)d\theta - \beta {\rm{\;ln\;}}\hat S, where ϱ ∈ X_δ(0), α ∈ (ℝ₊)_δ(α₀) and $\hat{S} = \hat{S} (ϱ, α) = \frac{1}{2} \int_{0}^{2 π} {(R_{α} + ϱ)}^{2} d θ .$ \hat S = \hat S\left( {\varrho ,\;\alpha } \right) = {1 \over 2}\mathop \int \nolimits_0^{2\pi } {({R_\alpha } + \varrho )^2}d\theta . Furthermore, the Fréchet derivative of Ê with respect to ϱ is expressed by $\begin{array}{l} {\hat{E}}_{ϱ}^{'} (ϱ, α, β) h = \int_{0}^{2 π} \frac{{(R_{α} + ϱ)}^{3} + 2 (R_{α} + ϱ) ϱ^{' 2} - {(R_{α} + ϱ)}^{2} ϱ^{″}}{{({(R_{α} + ϱ)}^{2} + ϱ'^{2})}^{3 / 2}} h d θ \\ + \int_{0}^{2 π} (α - \frac{β}{\hat{S}} (R_{α} + ϱ)) h d θ . \end{array}$ \eqalign{ & \hat E_\varrho ^\prime\left( {\varrho ,\;\alpha ,\;\beta } \right)h = \mathop \int \nolimits_0^{2\pi } {{{{({R_\alpha } + \varrho )}^3} + 2\left( {{R_\alpha } + \varrho } \right){\varrho ^{'2}} - {{({R_\alpha } + \varrho )}^2}\varrho ''} \over {{{({{({R_\alpha } + \varrho )}^2} + \varrho {'^2})}^{3/2}}}}hd\theta \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \mathop \int \nolimits_0^{2\pi } \left( {\alpha - {\beta \over {\hat S}}\left( {{R_\alpha } + \varrho } \right)} \right)hd\theta . \cr} Let us define the mapping F̂ : X_δ(0) × (ℝ ₊)_δ(α₀) → Y by the formula $\hat{F} (ϱ, α, β) = \frac{{(R_{α} + ϱ)}^{3} + 2 (R_{α} + ϱ) ϱ^{' 2} - {(R_{α} + ϱ)}^{2} ϱ^{″}}{{({(R_{α} + ϱ)}^{2} + ϱ'^{2})}^{3 / 2}} + α - \frac{β}{\hat{S}} (R_{α} + ϱ) .$ \hat F\left( {\varrho ,\;\alpha ,\;\beta } \right) = {{{{({R_\alpha } + \varrho )}^3} + 2\left( {{R_\alpha } + \varrho } \right){\varrho ^{'2}} - {{({R_\alpha } + \varrho )}^2}\varrho ''} \over {{{({{({R_\alpha } + \varrho )}^2} + \varrho {'^2})}^{3/2}}}} + \alpha - {\beta \over {\hat S}}\left( {{R_\alpha } + \varrho } \right). Of course F̂ is smooth. It is easy to notice the following

Lemma 2.4

The mapping F̂ is the variational gradient of Ê with respect to the inner product in L²(2π), i.e., ${\hat{E}}_{ϱ}^{'} (ϱ, α, β) h = \int_{0}^{2 π} \hat{F} (ϱ, α, β) h d θ = 〈 \hat{F} (ϱ, α, β), h 〉$ \hat E_\varrho ^\prime\left( {\varrho ,\;\alpha ,\;\beta } \right)h = \mathop \int \nolimits_0^{2\pi } \hat F\left( {\varrho ,\;\alpha ,\;\beta } \right)hd\theta = \left\langle {\hat F\left( {\varrho ,\;\alpha ,\;\beta } \right),\;h} \right\rangle fol all ϱ ∈ X_δ(0), h ∈ X and α ∈ (ℝ₊)_δ(α₀).

Consider the equation of the form (2.3) $\hat{F} (ϱ, α, β) = 0 .$ \hat F\left( {\varrho ,\;\alpha ,\;\beta } \right) = 0. The equation (2.3) has a trivial family of solutions ${\hat{Γ}}^{β} = {(0, α) \in X \times ℝ_{+} : α \in {(ℝ_{+})}_{δ} (α_{0})} .$ \hat \Gamma ^\beta = \left\{ {\left( {0,\alpha } \right) \in X \times {\mathbb R}_ + :\alpha \in ({\mathbb R}_ + )_\delta \left( {\alpha _0 } \right)} \right\}. In order to prove the existence of a symmetry-breaking bifurcation branch of solutions of the equation (1.2) at (R_{α_k}, α_k), we will investigate bifurcation from the set of trivial solutions ${\hat{Γ}}^{β}$ \hat \Gamma^\beta of the equation (2.3).

Lemma 2.5

For each α ∈ (ℝ₊)_δ(α₀), ${\hat{F}}_{ϱ}^{'} (0, α, β) : X \to Y$ \hat F_\varrho ^\prime {\rm{(0,}}\alpha {\rm{,}}\beta {\rm{) : }}X \to Y given by the formula (2.4) ${\hat{F}}_{ϱ}^{'} (0, α, β) h = - \frac{1}{R_{α}} h^{″} - β (\frac{1}{π R_{α}^{2}} h - \frac{1}{π^{2} R_{α}^{2}} \int_{0}^{2 π} h d θ)$ {\hat F}_\varrho ^\prime\left( {0,\;\alpha ,\;\beta } \right)h = - {1 \over {{R_\alpha }}}h'' - \beta \left( {{1 \over {\pi R_\alpha ^2}}h - {1 \over {{\pi ^2}R_\alpha ^2}}\mathop \int \nolimits_0^{2\pi } hd\theta } \right) is a Fredholm map of index 0.

The proof is similar to that in [8]. It is sufficient to show that ${\hat{F}}_{ϱ}^{'} (0, α, β)$ \hat F_\varrho ^\prime (0,\alpha ,\beta ) is the sum of a Fredholm map of index 0 and a completely continuous map. According to the implicit function theorem, a necessary condition for bifurcation from the trivial solutions of (2.3) at (0, α₀) is that dim ker ${\hat{F}}_{ϱ}^{'} (0, α_{0}, β) > 0$ \hat F_\varrho ^\prime (0,\alpha _0 ,\beta ) > 0 . To determine the critical values of the bifurcation parameter, we need to solve the equation (2.5) ${\hat{F}}_{ϱ}^{'} (0, α, β) h = 0$ {\hat F}_\varrho ^\prime\left( {0,\;\alpha ,\;\beta } \right)h = 0 considering two additional constraints (2.6) $\int_{0}^{2 π} h (θ) cos θ d θ = 0$ \mathop \int \nolimits_0^{2\pi } h\left( \theta \right){\rm{\;cos\;}}\theta d\theta = 0 and (2.7) $\int_{0}^{2 π} h (θ) d θ = 0 .$ \mathop \int \nolimits_0^{2\pi } h\left( \theta \right)d\theta = 0. The evenness of h(θ) and the condition (2.6) prevent any displacement of the mass center of the parachute. Additionally, conditions (2.6) and (2.7) rule out h(θ) = cos(θ) and h(θ) = const ≠ 0. Consequently, assumption (2.7) leads to a loss of radial symmetry. Furthermore, assumption (2.7) reduces equation (2.5) to $- \frac{1}{R_{α}} h^{″} - β \frac{1}{π R_{α}^{2}} h = 0 .$ - {1 \over {{R_\alpha }}}h'' - \beta {1 \over {\pi R_\alpha ^2}}h = 0. We choose the bifurcation mode $e_{k} (θ) = \frac{1}{\sqrt{π}} cos (k θ)$ {e_k}\left( \theta \right) = {1 \over {\sqrt \pi }}{\rm{\;cos\;}}\left( {k\theta } \right) for k ≥ 2. Using formula (1.3), we finally obtain $α = α_{k} = k^{2} - 1, k \geq 2 .$ \alpha = {\alpha _k} = {k^2} - 1,\;\,\,\,\,\,\,\,k \ge 2.

Now, it is enough to show that ${\hat{E}}_{ϱ ϱ α}^{‴} (0, α_{k}, β) e_{k} e_{k} \neq 0 .$ \hat E_{_{\varrho \varrho \alpha }}^{'''}\left( {0,\;{\alpha _k},\;\beta } \right){e_k}{e_k} \ne 0. It follows from Lemma 2.4 that ${\hat{E}}_{ϱ ϱ}^{″} (0, α, β) h g = 〈 {\hat{F}}_{ϱ}^{'} (0, α, β) h, g 〉 .$ \hat E_{^{\varrho \varrho }}^{''}\left( {0,\;\alpha ,\;\beta } \right)hg = \left\langle {\hat F_\varrho ^\prime\left( {0,\;\alpha ,\;\beta } \right)h,\;g} \right\rangle . Applying (2.4) and substituting h = g = e_k, we obtain (2.8) ${\hat{E}}_{ϱ ϱ}^{″} (0, α, β) e_{k} e_{k} = \frac{1}{R_{α}} (k^{2} - 1 - α) .$ \hat E_{\varrho \varrho }^{''}\left( {0,\;\alpha ,\;\beta } \right){e_k}{e_k} = {1 \over {{R_\alpha }}}\left( {{k^2} - 1 - \alpha } \right). Finally, differentiating (2.8) with respect to α we have ${\hat{E}}_{ϱ ϱ α}^{‴} (0, α, β) e_{k} e_{k} = - \frac{π}{β} (k^{2} - 2 - 2 α),$ \hat E_{\varrho \varrho \alpha }^{'''}\left( {0,\;\alpha ,\;\beta } \right){e_k}{e_k} = - {\pi \over \beta }\left( {{k^2} - 2 - 2\alpha } \right), and thus ${\hat{E}}_{ϱ ϱ α}^{‴} (0, α_{k}, β) e_{k} e_{k} = - \frac{π}{β} k^{2} < 0, k \geq 2 .$ \hat E_{\varrho \varrho \alpha }^{'''}\left( {0,\;{\alpha _k},\;\beta } \right){e_k}{e_k} = - {\pi \over \beta }{k^2} < 0,\;\,\,\,\,\,\,\,\,\,k \ge 2. It follows from Theorem 2.3 that (0, α_k) is a bifurcation point of (2.3) and the solution set of (2.3) in a neighbourhood of this point is the sum of ${\hat{Γ}}^{β}$ \hat \Gamma^\beta and a smooth curve (ϱ(t), α(t)), |t| < ε, such that ϱ(0) = 0, α(0) = α_k and $ϱ (θ) = \frac{t}{\sqrt{π}} cos (k θ) + o (| t |),$ \varrho \left( \theta \right) = {t \over {\sqrt \pi }}{\rm{\;cos\;}}\left( {k\theta } \right) + o\left( {\left| t \right|} \right), which, together with (2.2), proves Theorem 2.2.

3.

Subcritical behaviour of the parachute of gas balloon.

In the previous section, using the Crandall–Rabinowitz theorem, we proved the existence of a family of non-radially symmetric solutions of the equation (1.2) at the point (R_{α_k}, α_k), parameterized by a real parameter t ∈ (−ε, ε).

As the next step, our goal is to parameterize the non-radially symmetric branches of solutions of the equation (1.2) using the bifurcation parameter α. To achieve this, we will apply the Lyapunov–Schmidt finite-dimensional reduction and Sapronov’s key function method. For the convenience of the reader, we state this theorem.

Assume that the assumptions of Theorem 2.3 are satisfied. Let us consider the equation (3.1) $F (x, α) + (ξ - (x, e)_{H}) e = 0,$ F\left( {x,\;\alpha } \right) + (\xi - \left( {x,\;e{)_H}} \right)e = 0, where x ∈ X_δ(0), α ∈ ℝ_δ(α₀) and ξ ∈ ℝ. By the implicit function theorem there are open sets S ⊂ ℝ × ℝ_δ(α₀) and U ⊂ X_δ(0) and there exists a C^r-smooth function x̃ : S → U such that the solution set of (3.1) in a neighbourhood of (0, 0, α₀) ∈ X_δ(0) × ℝ × ℝ_δ(α₀) is the graph of x̃ and x̃(0, α₀) = 0.

Define Φ: S → ℝ by $Φ (ξ, α) = - E (\tilde{x} (ξ, α), α) + \frac{1}{2} {(ξ - {(\tilde{x} (ξ, α), e)}_{H})}^{2} .$ \Phi \left( {\xi ,\;\alpha } \right) = - E\left( {\tilde x\left( {\xi ,\;\alpha } \right),\;\alpha } \right) + {1 \over 2}{\left( {\xi - {{(\tilde x\left( {\xi ,\;\alpha } \right),\;e)}_H}} \right)^2}. Φ is called the key function. It is known that (0, α₀) ∈ X × ℝ is a bifurcation point of (2.1) if and only if (0, α₀) ∈ ℝ × ℝ is a bifurcation point of the equation Φ_ξ(ξ, α) = 0 (see [9], Proposition 2.3).

Theorem 3.1 (The key function method, [13]).

Under the conditions stated above:

(i)
If $Φ_{ξ ξ α}^{‴} (0, α_{0}) \neq 0$ \Phi _{\xi \xi \alpha }^{'''} (0,\alpha _0 ) \ne 0 then (0, α₀) ∈ X × ℝ is a bifurcation point of the equation (2.1) and the solution set of (2.1) in a small neighbourhood of (0, α₀) is a union of two branches: Γ₁ = {(0, α): α ∈ ℝ_δ(α₀)} and a C^r⁻²-curve Γ₂, intersecting only at (0, α₀).
(ii)
If $Φ_{ξ ξ α}^{‴} (0, α_{0}) \neq 0$ \Phi _{\xi \xi \alpha }^{'''} (0,\alpha _0 ) \ne 0 and $Φ_{ξ ξ ξ}^{‴} (0, α_{0}) \neq 0$ \Phi _{\xi \xi \xi }^{'''} (0,\alpha _0 ) \ne 0 then (0, α₀) ∈ X × ℝ is said to be a transcritical bifurcation point of (2.1), and the curve Γ₂ can be parametrized as follows: $Γ_{2} : x (α) = C (α - α_{0}) e + o (| α - α_{0} |), α \in (α_{0} - η, α_{0} + η),$ {\Gamma _2}:x\left( \alpha \right) = C\left( {\alpha - {\alpha _0}} \right)e + o\left( {\left| {\alpha - {\alpha _0}} \right|\;} \right),\,\,\,\,\,\,\,\alpha \in \left( {{\alpha _0} - \eta ,\;{\alpha _0} + \eta } \right), where $C = - 2 \frac{Φ_{ξ ξ α}^{‴} (0, α_{0})}{Φ_{ξ ξ ξ}^{‴} (0, α_{0})}$ C = - 2{{\Phi _{\xi \xi \alpha }^{'''}\left( {0,{\alpha _0}} \right)} \over {\Phi _{\xi \xi \xi }^{'''}\left( {0,{\alpha _0}} \right)}} and 0 < η ≤ δ (see Fig. 2(a)).
(iii)
Let $\begin{array}{l} Φ_{ξ ξ α}^{‴} (0, α_{0}) \neq 0, \\ Φ_{ξ ξ ξ}^{‴} (0, α_{0}) = 0, \\ Φ_{ξ ξ ξ ξ}^{″ ″} (0, α_{0}) \neq 0 . \end{array}$ \eqalign{ & \Phi _{\xi \xi \alpha }^{'''}\left( {0,\;{\alpha _0}} \right) \ne 0, \cr & \Phi _{\xi \xi \xi }^{'''}\left( {0,\;{\alpha _0}} \right) = 0, \cr & \Phi _{\xi \xi \xi \xi }^{''''}\left( {0,\;{\alpha _0}} \right) \ne 0. \cr}

Set $D = - 6 \frac{Φ_{ξ ξ α}^{‴} (0, α_{0})}{Φ_{ξ ξ ξ ξ}^{″ ″} (0, α_{0})} .$ D = - 6{{\Phi _{\xi \xi \alpha }^{'''}\left( {0,{\alpha _0}} \right)} \over {\Phi _{\xi \xi \xi \xi }^{''''}\left( {0,{\alpha _0}} \right)}}.

If D < 0 then (0, α₀) ∈ X × ℝ is said to be a subcritical bifurcation point of (2.1), and the curve Γ₂ can be parametrized as follows: $Γ_{2} : x^{\pm} (α) = \pm \sqrt{| D |} {(α_{0} - α)}^{\frac{1}{2}} e + o (| α - α_{0} |^{\frac{1}{2}}), τ \in (α_{0} - η, α_{0}],$ {\Gamma _2}:{x^ \pm }\left( \alpha \right) = \pm \sqrt {\left| D \right|} {({\alpha _0} - \alpha )^{{1 \over 2}}}e + o\left( {|\alpha - {\alpha _0}{|^{{1 \over 2}}}} \right),\,\,\,\,\,\,\,\tau \in \left( {{\alpha _0} - \eta ,\;{\alpha _0}} \right], where 0 < η ≤ δ (see Fig. 2(b)).

If D > 0 then (0, α₀) ∈ X × ℝ is said to be a postcritical bifurcation point of (2.1), and the curve Γ₂ can be parametrized as follows: $Γ_{2} : x^{\pm} (α) = \pm \sqrt{D} {(α - α_{0})}^{\frac{1}{2}} e + o (| α - α_{0} |^{\frac{1}{2}}), α \in [α_{0}, α_{0} + η),$ {\Gamma _2}:{x^ \pm }\left( \alpha \right) = \pm \sqrt D {(\alpha - {\alpha _0})^{{1 \over 2}}}e + o(|\alpha - {\alpha _0}{|^{{1 \over 2}}}),\;\,\,\,\,\,\,\,\alpha \in \left[ {{\alpha _0},\;{\alpha _0} + \eta } \right), where 0 < η ≤ δ (see Fig. 2(c)).

Fact 3.2 (see [10]).

The first few terms of the Taylor series of the key function Φ(ξ, α) at the point (0, α₀) are given by the following formulae: $\begin{matrix} Φ (0, α_{0}) & = & - E (0, α_{0}), \\ Φ_{ξ}^{'} (0, α_{0}) & = & 0, \\ Φ_{ξ ξ}^{″} (0, α_{0}) & = & 0, \\ Φ_{ξ α \dots α}^{(1 + . k)} (0, α_{0}) & = & 0 for all k = 1, 2, \dots, \\ Φ_{ξ ξ α}^{‴} (0, α_{0}) & = & - E_{x x α}^{‴} (0, α_{0}) e e, \\ Φ_{ξ ξ ξ}^{‴} (0, α_{0}) & = & - E_{x x x}^{‴} (0, α_{0}) e e e, \\ Φ_{ξ ξ ξ ξ}^{″″} (0, α_{0}) & = & - E_{x x x x}^{″″} (0, α_{0}) e e e e - 3 E_{x x x}^{‴} (0, α_{0}) e e h, \end{matrix}$ \matrix{ {\Phi \left( {0,\;{\alpha _0}} \right)} = { - E\left( {0,\;{\alpha _0}} \right),} \cr {\Phi _\xi ^\prime\left( {0,\;{\alpha _0}} \right)} = {0,} \cr {\Phi _{\xi \xi }^{''}\left( {0,\;{\alpha _0}} \right)} = {0,} \cr {\Phi _{\xi \alpha \ldots \alpha }^{\left( {1 + .k} \right)}\left( {0,\;{\alpha _0}} \right)} = {0\,for\,\,all{\rm{ }}k = 1,2, \ldots ,} \cr {\Phi _{\xi \xi \alpha }^{'''}\left( {0,\;{\alpha _0}} \right)} = { - E_{xx\alpha }^{'''}\left( {0,\;{\alpha _0}} \right)ee,} \cr {\Phi _{\xi \xi \xi }^{'''}\left( {0,\;{\alpha _0}} \right)} = { - E_{xxx}^{'''}\left( {0,\;{\alpha _0}} \right)eee,} \cr {\Phi _{\xi \xi \xi \xi }^{''''}\left( {0,\;{\alpha _0}} \right)} = { - E_{xxxx}^{''''}\left( {0,\;{\alpha _0}} \right)eeee - 3E_{xxx}^{'''}\left( {0,\;{\alpha _0}} \right)eeh,} \cr } where h is a unique solution of the equation $F_{x}^{'} (0, α_{0}) h - {(h, e)}_{H} e = - F_{x x}^{″} (0, α_{0}) e e .$ F_x^\prime\left( {0,\;{\alpha _0}} \right)h - {(h,\;e)_H}e = - F_{xx}^{''}\left( {0,\;{\alpha _0}} \right)ee.

Theorem 3.3

Let α_k = k² − 1, k ≥ 2 be a critical value of bifurcation parameter α ∈ ℝ₊. Then (0, α_k) ∈ X × ℝ₊ is a subcritical bifurcation point of the equation (2.3).

Proof

Fix k ≥ 2. Consider the key function corresponding to the equation (2.3) in a neighbourhood of (0, α_k). According to Theorem 3.1 we have to show that $Φ_{ξ ξ ξ}^{‴} (0, α_{k}) = 0$ \Phi _{\xi \xi \xi }^{'''} (0,\alpha _k ) = 0 and $Φ_{ξ ξ α}^{‴} (0, α_{k})$ \Phi _{\xi \xi \alpha }^{'''} (0,\alpha _k ) . $Φ_{ξ ξ ξ ξ}^{″ ″} (0, α_{k}) > 0$ \Phi _{\xi \xi \xi \xi }^{''''} (0,\alpha _k ) > 0 . For this purpose we will use Fact 3.2. From the previous section, we have ${\hat{E}}_{ϱ ϱ α}^{‴} (0, α_{k}, β) e_{k} e_{k} = - \frac{π}{β} k^{2} < 0 .$ \hat E_{\varrho \varrho \alpha }^{'''}\left( {0,\;{\alpha _k},\;\beta } \right){e_k}{e_k} = - {\pi \over \beta }{k^2} < 0. Using Fact 3.2 it follows that $Φ_{ξ ξ α}^{‴} (0, α_{k}) = \frac{π}{β} k^{2} > 0 .$ \Phi _{\xi \xi \alpha }^{'''}\left( {0,\;{\alpha _k}} \right) = {\pi \over \beta }{k^2} > 0. A straightforward calculation gives ${\hat{F}}_{ϱ ϱ}^{″} (0, α_{k}, β) e_{k} e_{k} = \frac{k^{2}}{R_{α_{k}}^{2}} (\frac{1}{π} {sin}^{2} (k θ) - \frac{2}{π} {cos}^{2} (k θ) + \frac{1}{π}) .$ \hat F_{\varrho \varrho }^{''}\left( {0,\;{\alpha _k},\;\beta } \right){e_k}{e_k} = {{{k^2}} \over {R_{{\alpha _k}}^2}}\left( {{1 \over \pi }{\rm{si}}{{\rm{n}}^2}\left( {k\theta } \right) - {2 \over \pi }{\rm{co}}{{\rm{s}}^2}\left( {k\theta } \right) + {1 \over \pi }} \right). According to Lemma 2.4, we obtain ${\hat{E}}_{ϱ ϱ ϱ}^{‴} (0, α_{k}, β) e_{k} e_{k} e_{k} = 〈 {\hat{F}}_{ϱ ϱ}^{″} (0, α_{k}, β) e_{k} e_{k}, e_{k} 〉 = 0 .$ \hat E_{\varrho \varrho \varrho }^{'''}\left( {0,\;{\alpha _k},\;\beta } \right){e_k}{e_k}{e_k} = \left\langle {\hat F_{\varrho \varrho }^{''}\left( {0,\;{\alpha _k},\;\beta } \right){e_k}{e_k},\;{e_k}} \right\rangle = 0. Combining this with Fact 3.2, we conclude that $Φ_{ξ ξ ξ}^{‴} (0, α_{k}) = 0$ \Phi _{\xi \xi \xi }^{'''} (0,\alpha _k ) = 0 . We check at once that $h_{k} (θ) = \frac{1}{R_{α_{k}} π} {cos}^{2} (k θ) - \frac{1}{R_{α_{k}} π}$ {h_k}\left( \theta \right) = {1 \over {{R_{{\alpha _k}}}\pi }}{\rm{co}}{{\rm{s}}^2}\left( {k\theta } \right) - {1 \over {{R_{{\alpha _k}}}\pi }} is a unique solution of the equation ${\hat{F}}_{ϱ}^{'} (0, α_{k}, β) h - 〈 h, e_{k} 〉 e_{k} = - {\hat{F}}_{ϱ ϱ}^{″} (0, α_{k}, β) e_{k} e_{k} .$ \hat F_\varrho ^\prime\left( {0,\;{\alpha _k},\;\beta } \right)h - \langle h,\;{e_k}\rangle {e_k} = - \hat F_{\varrho \varrho }^{''}\left( {0,\;{\alpha _k},\;\beta } \right){e_k}{e_k}. It follows that ${\hat{E}}_{ϱ ϱ ϱ}^{‴} (0, α_{k}, β) e_{k} e_{k} h_{k} = 〈 {\hat{F}}_{ϱ ϱ}^{″} (0, α_{k}, β) e_{k} e_{k}, h_{k} 〉 = - \frac{7 k^{2}}{4 R_{α_{k}}^{3} π} .$ \hat E_{\varrho \varrho \varrho }^{'''}\left( {0,\;{\alpha _k},\;\beta } \right){e_k}{e_k}{h_k} = \left\langle {\hat F_{\varrho \varrho }^{''}\left( {0,\;{\alpha _k},\;\beta } \right){e_k}{e_k},\;{h_k}} \right\rangle = - {{7{k^2}} \over {4R_{{\alpha _k}}^3\pi }}. Finally, we obtain ${\hat{E}}_{ϱ ϱ ϱ ϱ}^{″″} (0, α_{k}, β) e_{k} e_{k} e_{k} e_{k} = - \frac{9 k^{4}}{4 R_{α_{k}}^{3} π},$ \hat E_{\varrho \varrho \varrho \varrho }^{''''}\left( {0,\;{\alpha _k},\;\beta } \right){e_k}{e_k}{e_k}{e_k} = - {{9{k^4}} \over {4R_{{\alpha _k}}^3\pi }}, and as a result, using Fact 3.2, we get $Φ_{ξ ξ ξ ξ}^{″″} (0, α_{k}) = \frac{k^{2}}{4 R_{α_{k}}^{3} π} (9 k^{2} + 21) > 0 .$ \Phi _{\xi \xi \xi \xi }^{''''}\left( {0,\;{\alpha _k}} \right) = {{{k^2}} \over {4R_{{\alpha _k}}^3\pi }}\left( {9{k^2} + 21} \right) > 0. Hence $D = - 6 \frac{Φ_{ξ ξ α}^{‴} (0, α_{k})}{Φ_{ξ ξ ξ ξ}^{″″} (0, α_{k})} = - \frac{8 R_{α_{k}}^{3} π^{2}}{β (3 k^{2} + 7)} < 0 .$ D = - 6{{\Phi _{\xi \xi \alpha }^{'''}\left( {0,{\alpha _k}} \right)} \over {\Phi _{\xi \xi \xi \xi }^{''''}\left( {0,{\alpha _k}} \right)}} = - {{8R_{{\alpha _k}}^3{\pi ^2}} \over {\beta \left( {3{k^2} + 7} \right)}} < 0. According to Theorem 3.1 we conclude that (0, α_k) is a subcritical bifurcation point of the equation (2.3). Furthermore, in a small neighbourhood of this point, the solution set consists of the trivial branch ${\hat{Γ}}^{β}$ \hat \Gamma^\beta and a C^∞-curve ${\hat{Γ}}_{2}^{β}$ \hat \Gamma _2^\beta , which is parametrized as follows: ${\hat{Γ}}_{2}^{β} : ϱ^{\pm} (α) = \pm \sqrt{| D |} {(α_{k} - α)}^{\frac{1}{2}} e_{k} + o ({| α - α_{k} |}^{\frac{1}{2}}) for α \in (α_{k} - η, α_{k}],$ \hat \Gamma _2^\beta :{\varrho ^ \pm }\left( \alpha \right) = \pm \sqrt {\left| D \right|} {({\alpha _k} - \alpha )^{{1 \over 2}}}{e_k} + o\left( {{{\left| {\alpha - {\alpha _k}} \right|}^{{1 \over 2}}}} \right)\,\,\,\,\,\,{\rm{for}}\,\,\,\,\,\,\,\alpha \in ({\alpha _k} - \eta ,\;{\alpha _k}], where 0 < η ≤ δ.

4.

Graphical representation of the parachute of gas balloon in the neighbourhood of bifurcation points

We have shown that for the elasticity parameter values α_k = k² − 1, k ∈ ℕ, k ≥ 2, non-radially symmetric solutions (r(t), α(t)) of our problem appear. Using the simplified formula $r (t) (θ) \approx R_{α_{k}} + t \cdot \frac{1}{\sqrt{π}} cos (k θ)$ r\left( t \right)\left( \theta \right) \approx {R_{{\alpha _k}}} + t \cdot {1 \over {\sqrt \pi }}{\rm{\;cos\;}}\left( {k\theta } \right) and the Mathematica software, we present below how the top rim of a parachute may behave for k = 3, 4, 7, 8, 9, 12.

A Note on Bifurcation of Equilibrium Forms of a Gas Balloon Parachute

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