Delayed Feedback Control and Bifurcation Analysis of an Autonomy System

and Applied Analysis 3 Lemma 2. For the polynomial equation (10), one has the following results. (i) If r < 0, then (10) has at least one positive root. (ii) If r ≥ 0 and 󳵻 = p − 3q ≤ 0, then (10) has no positive roots. (iii) If r ≥ 0 and󳵻 = p−3q > 0, then (10) has positive roots if and only if z 1 = (1/3)(−p +√󳵻) > 0 and h(z 1 ) ≤ 0. Suppose that (10) has positive roots, without loss of generality; we assume that it has three positive roots, defined by z 1 , z 2 , and z 3 , respectively. Then (9) has three positive roots ω 1 = √z1, ω2 = √z2, ω3 = √z3. (12) Thus, we have cosωτ = b 1 ω 2 (ω 2 − a 1 ) − (a 2 ω 2 − a 0 ) (b 2 ω 2 − b 0 ) (b 2 ω − b 0 ) 2 + b 1 ω . (13) If we denote τ (j) k = 1 ω k {arccos( b 1 ω 2 (ω 2 − a 1 ) − (a 2 ω 2 − a 0 ) (b 2 ω 2 − b 0 ) (b 2 ω − b 0 ) 2 + b 1 ω ) +2jπ, k = 1, 2, 3; j = 0, 1, . . . } , (14) then ±iω k is a pair of purely imaginary roots of (9) with τ(j) k . Define τ 0 = τ (0) k 0 = min k∈{1,2,3} {τ (0) k } , ω 0 = ω k 0 . (15) Note that when τ = 0, (5) becomes λ 3 + (a 2 + b 2 ) λ 2 + (a 1 + b 1 ) λ + a 0 + b 0 = 0. (16) Therefore, applying Lemmas 1 and 2 to (5), we obtain the following lemma. Lemma 3. For the third-degree transcendental equation (5), one has the following results. (i) If r ≥ 0 and 󳵻 = p − 3q ⩽ 0, then all roots with positive real parts of (5) have the same sum to those of the polynomial equation (16) for all τ ≥ 0. (ii) If either r < 0 or r ≥ 0 and 󳵻 = p − 3q > 0, z 1 > 0, and h(z 1 ) < 0, then all roots with positive real parts of (5) have the same sum to those of the polynomial equation (16) for τ ∈ [0, τ 0 ). Let λ(τ) = α(τ) + iω(τ) be the root of (9) near τ = τ(j) k satisfying α (τ (j) k ) = 0, ω (τ (j) k ) = ω k . (17) Then it is easy to verify the following transversality condition. Lemma 4. Suppose that z k = ω 2 k and h(z k ) ̸ = 0, where h(z) is defined by (15). Then d(Reλ(τ(j) k ))/dτ ̸ = 0, and d(Reλ(τ(j) k ))/ dτ and h(z k ) have the same sign. Now, we study the characteristic equation (5) of system (4). Applying Lemmas 3 and 4 to (5), we have the following theorem. Theorem 5. Let τ(j) k and ω 0 , τ 0 be defined by (14) and (15), respectively. Then consider the following. (i) If r ≥ 0 and 󳵻 = p − 3q ≤ 0, then all roots with positive real parts of (5) have the same sum to those of the polynomial equation (16) for all τ ≥ 0. (ii) If either r < 0 or r ≥ 0 and 󳵻 = p − 3q > 0, z 1 > 0, and h(z 1 ) < 0, then h(z) has at least one positive root z k , and all roots with positive real parts of (5) have the same sum to those of the polynomial equation (16) for τ ∈ [0, τ 0 ). (iii) If the conditions of (ii) are satisfied and h(z k ) ̸ = 0, then system (2) exhibits Hopf bifurcation at the equilibrium E 0 for τ = τ(j) k . 3. Stability and Direction of Bifurcating Periodic Orbits In the previous section, we obtain the conditions underwhich family periodic solutions bifurcate from the equilibriumE 0 at the critical value of τ. As pointed by in Hassard et al. [17], it is interesting to determine the direction, stability, and period of these periodic solutions. Following the ideal of Hassard et al., we derive the explicit formulae for determining the properties of the Hopf bifurcation at the critical value of τ using the normal form and the centermanifold theory.Throughout this section, we always assume that system (2) undergoes Hopf bifurcations at the equilibrium E 0 for τ = τ k , and then ±iω k is corresponding to purely imaginary roots of the characteristic equation at the equilibrium E 0 . Letting τ = τt, τ = τ k + μ, system (2) is transformed into an FDE in C = C([−1, 0], R) as ?̇? = L μ (u t ) + f (μ, u t ) , (18) where u(t) = (x(t), y(t), z(t))T ∈ R and L μ : C → R, f : R × C → R are given, respectively, by L μ (φ) = (τ k + μ)( a −1 0 1 K − b −1 a 0 −8 )( φ 1 (0) φ 2 (0) φ 3 (0) ) + (τ k + μ)( 0 0 0 0 −K 0 0 0 0 )( φ 1 (−1) φ 2 (−1) φ 3 (−1) ) , (19) f (τ, φ) = (τ k + μ)( 0 0 4φ 2 (0) φ 3 (0) ) . (20) 4 Abstract and Applied Analysis By the Riesz representation theorem, there exists a function η(θ, μ) of bounded variation for θ ∈ [−1, 0], such that