Projectile flight stability evaluation method based on Hopf bifurcation analysis

Abstract

The invention belongs to the technical field of nonlinear dynamics and system control, and particularly relates to a projectile flight stability evaluation method based on Hopf bifurcation analysis. Establishing a nonlinear angular motion model of the pulse correction projectile; based on a pulse correction projectile nonlinear angular motion model, selecting interested aerodynamic parameters in a projectile nonlinear free motion stage or control parameters in a controlled flight stage as bifurcation control variables; calculating a system bifurcation point according to the bifurcation control variable, and judging a system bifurcation type based on a central manifold theorem and a normativity theory; calculating a limit cycle amplitude of the system according to the approximate equilibrium equation of the system; and evaluating the stability. Based on Hopf bifurcation analysis, the flight stability of the projectile can be accurately evaluated, the phenomenon of nonlinear flight sudden change instability is effectively avoided, and solid support is provided for aerodynamic design and guidance control process research of the projectile.

Description

Technical Field

[0001] This invention belongs to the field of nonlinear dynamics and system control technology, specifically relating to a projectile flight stability evaluation method based on Hopf bifurcation analysis. Background Technology

[0002] Projectile flight stability is a key factor determining external ballistic performance and directly affects firing accuracy. For projectiles equipped with pulse correction mechanisms, good flight stability is a fundamental prerequisite for ensuring effective correction control. In practical engineering applications, projectiles often experience amplified attitude disturbances or even instability under high-altitude or high-speed flight conditions, leading to increased trajectory deviations and impacting combat effectiveness. Currently, the amplitude plane method and traditional linear stability theory are mainly used in engineering to evaluate projectile flight status. However, these methods have limited analytical capabilities for aircraft systems involving strong nonlinear coupling relationships and struggle to accurately identify nonlinear instability problems caused by changes in aerodynamic, structural, or control parameters.

[0003] Bifurcation theory, as a nonlinear dynamics analysis tool, possesses the ability to identify critical points in system stability and abrupt parameter responses. With the development of nonlinear control technology, bifurcation theory has been gradually applied to the stability analysis of aircraft systems, demonstrating significant practical value in the aerodynamic / control coupling analysis of aircraft, reentry vehicles, and missile / rocket platforms. By introducing bifurcation analysis methods, key design parameters affecting flight stability can be identified in the early stages of engineering, assisting in the correction of mechanism configurations, aerodynamic layout optimization, and control law design.

[0004] However, current engineering research on bifurcation analysis methods for the uncontrolled nonlinear free motion and controlled nonlinear angular motion phases of pulse-corrected missiles is still insufficient, lacking systematic and feasible analytical tools. In-depth modeling and analysis of the relationship between aerodynamic and control parameters and flight stability, and the establishment of a nonlinear stability assessment framework suitable for engineering applications, are of significant engineering importance for improving system design reliability and control accuracy. Summary of the Invention

[0005] To overcome the limitations of traditional linear methods in assessing projectile nonlinear instability in engineering practice, this invention proposes a projectile flight stability assessment method based on Hopf bifurcation analysis. On one hand, this method can be used to identify the impact of key aerodynamic parameters (i.e., aerodynamic parameters of interest) on the flight stability during the nonlinear free motion phase of the projectile. On the other hand, this method can be used to identify the impact of control parameters on flight stability during the controlled flight phase, and has the ability to quantitatively analyze changes in pulse correction mechanism parameters. The proposed method can provide effective engineering analysis tools and technical support for the flight stability design, aerodynamic layout optimization, and control strategy development of various aircraft.

[0006] The technical solution to achieve the objective of this invention is: a projectile flight stability evaluation method based on Hopf bifurcation analysis, comprising the following steps:

[0007] Step (1): Establish a nonlinear angular motion model for the pulse-corrected projectile;

[0008] Step (2): Based on the pulse-corrected projectile nonlinear angular motion model of step (1), select the aerodynamic parameters of interest in the nonlinear free motion stage of the projectile, or the control parameters μ in the controlled flight stage, as the bifurcation control variables.

[0009] Step (3): Calculate the bifurcation point of the system based on the bifurcation control variables in step (2), and determine the bifurcation type of the system based on the central manifold theorem and normality theory; calculate the limit cycle amplitude of the system based on the approximate equilibrium equation of the system.

[0010] Step (4): Stability assessment.

[0011] Furthermore, step (1) specifically includes the following steps:

[0012] Step (11): Let the system state variables be [δ1, δ2, ω]. η ,ω ζ ] T Where δ1 is the elevation angle of attack, δ2 is the directional angle of attack, and ω η and ω ζ These are the pitch and yaw angular velocities in the missile's axis coordinate system, respectively.

[0013] Step (12): Based on the theorem of motion of the center of mass and the differential equation of the oscillation angular velocity, the nonlinear angular motion equation of the projectile is obtained as follows:

[0014]

[0015] Step (13): Considering the aerodynamic forces and torques of the projectile under high angle of attack conditions, perform nonlinear aerodynamic modeling of the projectile's aerodynamic forces and torques; the power series expansion of the aerodynamic coefficients is expressed as follows:

[0016] c a =c a0 +c a2 δ 2 +... (2)

[0017] In the formula, c a0 c is the derivative of the linear coefficients of aerodynamic forces and moments. a2 It is the derivative of the cubic coefficient; in the polynomial model, only the total angle of attack δ is considered, and there exists a relationship cosδ=cosδ2cosδ1;

[0018] Step (14): Substitute equation (2) into the nonlinear angular motion equation (1), neglecting the gravity term, Magnus force, and Coriolis inertial force, and assume that during the projectile's flight... The nonlinear angular motion model of the pulse-corrected projectile is obtained as follows:

[0019]

[0020] Among them, the expressions for the aerodynamic force and torque acting on the projectile during uncontrolled free motion and controlled flight of the pulse correction projectile are shown in (4) and (5), respectively:

[0021]

[0022] Furthermore, step (3) specifically involves:

[0023] Step (31): Calculate the system equilibrium point;

[0024] Step (32): Calculate the bifurcation point of the system;

[0025] Step (33): Reduce the four-dimensional nonlinear system to a two-dimensional system according to the central manifold theorem;

[0026] Step (34): Calculate the amplitude and period of the system limit cycle.

[0027] Furthermore, step (31) specifically involves:

[0028] Setting the right side of the nonlinear angular motion model of the pulse correction projectile in step (14), i.e., equation (3), to 0, we obtain the unique equilibrium point O[0,0,0,0] of the uncontrolled free flight state of the pulse correction projectile dynamic system. T And the Jacobian matrix at the equilibrium point is:

[0029]

[0030] For the controlled flight phase with multiple equilibrium solutions, the system equilibrium point can be obtained by solving the Matcont algorithm.

[0031] Furthermore, step (32) specifically involves:

[0032] According to the Jacobian matrix (6) at the equilibrium point, when the control parameter μ changes, the system stability change is judged based on the Holwitz criterion of the four-dimensional system. The point where the system stability changes is the bifurcation point μ0, and the system eigenvalues ​​λ1, λ2, λ3, λ4 and eigenvectors ξ1, ξ2, ξ3, ξ4 at the bifurcation point are obtained.

[0033] Furthermore, step (33) specifically involves:

[0034] Introduce a non-singular linear transformation x = py, where y = [y1, y2, y3, y4]. T , where p is a square matrix formed by the real and imaginary parts of the eigenvectors corresponding to the second and fourth eigenvalues ​​of J; let u = [y1, y2] T v = [y3, y4] T Substituting into system equation (3), we get

[0035]

[0036] According to the central manifold theorem, v can be expressed as v = h(u), and h(0) = h′(0) = 0; substituting v = h(u) into the second term of equation (7), based on the chain rule for differentiation, we have:

[0037]

[0038] Substituting equation (8) into the first term of equation (7), we obtain the differential equation for h(u):

[0039]

[0040] Based on the required order of the central manifold, the equation is as follows:

[0041]

[0042] Substituting equation (10) into the differential equation (9) of h(u), we can obtain the corresponding a based on the coefficients of each term. ijk b ijk Then we get the expression v = h(u); substituting h(u) into the first term of expression (7), we get the reduced equation for the flow on the central manifold as follows:

[0043]

[0044] The reduced equation can be written in the following form:

[0045]

[0046] From canonical form theory, the third-order canonical form is:

[0047]

[0048] in,

[0049] Introducing polar coordinate transformations y1 = rcosθ and y2 = rsinθ, we obtain the following equations:

[0050]

[0051] Furthermore, step (34) specifically involves:

[0052] According to equation (14), the approximate equilibrium equation of the system is obtained: cμr + ar 3 =0;

[0053] Amplitude of the limit cycle The period T = 2π / ω, where ω is the frequency.

[0054] Furthermore, step (4) specifically involves:

[0055] When a≠0 and c≠0, the system exhibits a Hopf bifurcation at μ=0, and the bifurcation characteristics are as follows:

[0056] (4) When c>0 and a>0, the equilibrium point is unstable with respect to μ>0 and asymptotically stable with respect to μ<0. When μ<0, there is an unstable limit cycle.

[0057] (5) When c>0 and a<0, the equilibrium point is unstable with respect to μ>0 and asymptotically stable with respect to μ<0. When μ<0, there is an asymptotically stable limit cycle.

[0058] (6) When c<0 and a>0, the equilibrium point is asymptotically stable with μ>0 and unstable with μ<0. When μ<0, there is an unstable limit cycle.

[0059] (4) When c<0 and a<0, the equilibrium point is asymptotically stable with respect to μ>0 and unstable with respect to μ<0. When μ<0, there is an asymptotically stable limit cycle.

[0060] Compared with the prior art, the significant advantages of this invention are:

[0061] To address the instability issues that high-angle-of-attack flight motions of high-altitude tail-fin missiles and projectiles may exhibit under pulse force, which cannot be explained by existing linear analysis methods, a flight stability assessment method based on Hopf bifurcation analysis is proposed. Furthermore, this invention provides new ideas and methods for optimizing the aerodynamic layout and external ballistic parameters of pulse-corrected missiles. This method, based on Hopf bifurcation analysis, can accurately assess the flight stability of projectiles.

[0062] By using bifurcation analysis as an evaluation tool, it can be widely applied to solve problems in engineering practice, such as the flight stability design of pulse correction missiles, providing a reasonable basis for parameter design and effectively avoiding the phenomenon of nonlinear flight instability, thus providing solid support for the aerodynamic design and guidance control process research of missiles and rockets. Attached Figure Description

[0063] Figure 1 This is a flowchart of a projectile flight stability assessment method based on Hopf bifurcation analysis.

[0064] Figure 2It is a bifurcation diagram of angular motion under nonlinear static moment.

[0065] Figure 3 This is a schematic diagram of the angular motion limit cycle under nonlinear static moment.

[0066] Figure 4 This is a graph showing the change of the directional angle of attack δ2 equilibrium curve with Limp.

[0067] Figure 5 Different magnitudes of impulse I imp Schematic diagram of the location of the lower bifurcation point.

[0068] Figure 6 Different magnitudes of impulse I imp Schematic diagram of the limit cycle at the lower bifurcation point. Detailed Implementation

[0069] To illustrate the technical solution and objectives of this invention, the invention will be further described below with reference to the accompanying drawings and specific embodiments. All other embodiments obtained by those skilled in the art based on the embodiments of this invention without inventive effort should fall within the scope of protection of this invention.

[0070] This invention proposes a projectile flight stability evaluation method based on Hopf bifurcation analysis. The technical solution and implementation steps of this method are as follows.

[0071] Step (1): Establish a nonlinear angular motion model of the pulse correction projectile.

[0072] Step (2): Based on the model in step (1), select the key aerodynamic parameters of the projectile's nonlinear free motion stage, or the correction mechanism parameters, i.e., the control parameters, of the controlled flight stage as the bifurcation control variables, i.e., the bifurcation parameters.

[0073] Step (3): Based on the bifurcation control variables selected in step (2), calculate the bifurcation points of the system, and determine the bifurcation type of the system based on the central manifold theorem and normality theory. Calculate the amplitude of the system's limit cycle based on the system's approximate equilibrium equations.

[0074] Step (4) Analyze system stability. Based on the range of aerodynamic and control parameter variations under stable system conditions, provide a basis for the aerodynamic layout design or correction mechanism design of the pulse correction projectile.

[0075] The specific steps for constructing the nonlinear angular motion model of the pulse correction projectile in step (1) are as follows.

[0076] Step (11): Since the flight stability of the projectile is closely related to angular motion, and attitude angle changes are crucial in correction control, the system state variables are selected as [δ1, δ2, ω]. η ,ω ζ ]T δ1 is the elevation angle of attack, δ2 is the directional angle of attack, and ω η and ω ζ These represent the pitch and yaw angular velocities in the missile's axis coordinate system, respectively.

[0077] Step (12): According to the theorem of motion of the center of mass, and by simultaneously solving the differential equations of the oscillating angular velocity, the following set of equations of nonlinear angular motion of the projectile can be obtained.

[0078]

[0079] Step (13) considers the aerodynamic forces and moments of the projectile under high angle of attack conditions, and performs nonlinear aerodynamic modeling of the projectile's aerodynamic forces and moments. The power series expansion of the aerodynamic coefficients can be expressed as follows:

[0080] c a =c a0 +c a2 δ 2 +... (2)

[0081] In the formula, c a0 c is the derivative of the linear coefficients of the aforementioned aerodynamic forces and moments. a2 It is the derivative of its cubic coefficient. In the polynomial model, considering only the total angle of attack δ, there exists the relationship cosδ=cosδ2cosδ1.

[0082] Step (14): Consider substituting the nonlinear aerodynamic force shown in equation (2) into the nonlinear angular motion equation (1), neglecting the gravity term and the relatively small magnitudes of the Magnus force and Coriolis inertial force. Furthermore, during the projectile's flight, since the trajectory deviates slightly from the firing surface, at this time... The nonlinear angular motion model of the pulse-corrected projectile is obtained as follows:

[0083]

[0084] Among them, the expressions for the aerodynamic force and torque of the pulse correction projectile during uncontrolled free motion and controlled flight are shown in (4) and (5) respectively, and the relevant symbol definitions are consistent with those in "External Ballistics of Projectiles and Rockets".

[0085]

[0086] In step (3), the steps for determining the bifurcation point and bifurcation type of the system are as follows.

[0087] Step (31): Calculate the system equilibrium point. Setting the right side of equation (3) to 0, the unique equilibrium point O[0,0,0,0] of the pulse-corrected projectile dynamics system in uncontrolled free flight state can be obtained. T The Jacobian matrix of this system of equations at the equilibrium point is obtained as follows:

[0088]

[0089] For the controlled flight phase with multiple equilibrium solutions, Matcont can be used to solve it.

[0090] Step (32): Calculate the system bifurcation point. According to matrix (6), when the control parameter μ changes, the system stability is judged based on the Holwitz criterion for four-dimensional systems (judging the cases where the real part of the eigenvalue is greater than 0, equal to 0, or less than 0 respectively). The point μ0 where the system stability changes is the bifurcation point. At this time, the characteristic polynomial has a pair of purely imaginary eigenvalues ​​and a pair of eigenvalues ​​with negative real parts (when the real part is positive, the system is unstable, which is not a concern in actual practice). The eigenvalues ​​λ1, λ2, λ3, λ4 and the eigenvectors ξ1, ξ2, ξ3, ξ4 at the bifurcation point can be obtained.

[0091] Step (33): Reduce the dimensionality of the system according to the central manifold theorem. Introduce a non-singular linear transformation x = py, where y = [y1, y2, y3, y4]. T , where p is a square matrix composed of the real and imaginary parts of the eigenvectors corresponding to the second and fourth eigenvalues ​​of J. Let u = [y1, y2] T v = [y3, y4] T Substituting it into the system equations, we can obtain

[0092]

[0093] According to the central manifold theorem, v can be expressed as v = h(u), and h(0) = h′(0) = 0. Substituting v = h(u) into the second term of equation (7), based on the chain rule for differentiation, we have

[0094]

[0095] Substituting equation (8) into the first term of equation (7) and rearranging, we obtain the differential equation for h(u).

[0096]

[0097] Since it is difficult to solve h(u) exactly, the equation based on the required central manifold order is as follows:

[0098]

[0099] Substituting equation (10) into the differential equation (9) of h(u), we can obtain the corresponding a based on the coefficients of each term. ijk b ijk Then we can obtain the expression v = h(u). Substituting h(u) into the first term of expression (7), we can obtain the reduced equation for the flow on the central manifold as follows.

[0100]

[0101] Based on the above process, the original four-dimensional nonlinear system can be reduced to a two-dimensional system according to the central manifold theorem.

[0102] The reduced equation can be written in the following form

[0103]

[0104] From the theory of canonical forms, its third-order canonical form can be obtained as follows:

[0105]

[0106] in,

[0107] Introducing polar coordinate transformations y1 = rcosθ and y2 = rsinθ, we obtain the following equations.

[0108]

[0109] Step (34): Based on equation (14), the approximate equilibrium equation of the system, cμr + ar, is obtained. 3 =0. Then the expression cμr+ar 3 The non-zero solutions of = 0 determine the limit cycle of the system. The amplitude of the limit cycle The period T = 2π / ω.

[0110] Furthermore, in step (4), the system stability determination steps are as follows.

[0111] When a≠0 and c≠0, the system exhibits a Hopf bifurcation at μ=0, and the bifurcation characteristic is:

[0112] (7) When c>0 and a>0, the equilibrium point is unstable with respect to μ>0 and asymptotically stable with respect to μ<0. When μ<0, there is an unstable limit cycle.

[0113] (8) When c>0 and a<0, the equilibrium point is unstable with respect to μ>0 and asymptotically stable with respect to μ<0. When μ<0, there is an asymptotically stable limit cycle.

[0114] (9) When c<0 and a>0, the equilibrium point is asymptotically stable with respect to μ>0 and unstable with respect to μ<0. When μ<0, there is an unstable limit cycle.

[0115] When c<0 and a<0, the equilibrium point is asymptotically stable with respect to μ>0 and unstable with respect to μ<0. When μ<0, there exists an asymptotically stable limit cycle.

[0116] Example

[0117] See attached document Figure 1 , Figure 1This is a flowchart of a projectile flight stability assessment method based on Hopf bifurcation analysis.

[0118] Firstly, regarding the uncontrolled nonlinear free motion phase of the pulse-corrected projectile, static torque dominates among all aerodynamic torques. Using Hopf bifurcation analysis, it is assessed that under the influence of nonlinear static torque, the stability of the projectile gyroscope weakens, and an unstable limit cycle is generated in the angular motion. Setting the right side of equation (3) to 0, the unique equilibrium point O[0,0,0,0] of the uncontrolled free flight state of the pulse-corrected projectile dynamics system can be obtained. T Substituting the aerodynamic and structural parameters of the projectile at a certain point on its flight trajectory into the system equations, the Jacobian matrix of the projectile's dynamic system at the zero equilibrium point O can be calculated as follows:

[0119]

[0120] See attached document Figure 2 , Figure 2 This is a bifurcation diagram of angular motion under nonlinear static moment. Calculations show that m... z0 = -4.2487 is the bifurcation point of the system. The system bifurcates from the equilibrium point into an unstable limit cycle, and the system undergoes a subcritical Hopf bifurcation.

[0121] See attached document Figure 3 , Figure 3 This is a schematic diagram of the angular motion limit cycle under nonlinear static moment. It can be seen that as the derivative of the linear static moment coefficient increases, the amplitude of the limit cycle increases; when m... z0 When the angle is -2, the amplitude of the unstable limit cycle generated by the angular motion is approximately 15°. In nonlinear dynamics, the unstable limit cycle motion is essentially an unrealizable periodic motion. Corresponding to the physical laws of actual external ballistic motion, this means that under the action of nonlinear static torque, the gyro stability of the tail fin projectile gradually weakens.

[0122] Secondly, regarding the controlled flight phase of the pulse-corrected projectile, considering the significant impact of correction mechanism parameters such as pulse impulse magnitude and axial eccentricity on the controlled flight phase, additional correction mechanism parameters are selected as variables to analyze their influence on the system's bifurcation characteristics.

[0123] See attached document Figure 4 , Figure 4 The equilibrium curve of the direction of attack angle δ2 varies with L imp The graph shows the change in axial thrust eccentricity L. impAs a bifurcation parameter (positive when the corrector is installed before the projectile's center of mass), solid lines in the diagram represent stable equilibrium points, and dashed lines represent unstable equilibrium points. Hopf bifurcation points occurring in the system are marked with an asterisk. The closer the thrust mechanism is installed to the center of mass, the easier it is for the projectile's angular motion to remain stable. When the pulse corrector is installed beyond a certain range, the desired stable equilibrium state of the system will disappear.

[0124] See attached document Figure 5 and Figure 6 , Figure 5 Different magnitudes of impulse I imp Schematic diagram of the lower bifurcation point location. Figure 6 Different magnitudes of impulse I imp A schematic diagram of the limiting ring at the lower bifurcation point. The larger the single-pulse impulse of the correction mechanism, the greater the amplitude of the limiting ring will be. For two-dimensional corrected projectiles, arranging the correction mechanism off-center from the center of mass can improve lateral correction capability. Furthermore, under the same conditions, increasing the force can increase the projectile's correction distance, but it also increases the likelihood of flight instability. Therefore, in practical applications, when designing parameters for pulse vector engine-type correction mechanisms, the combined effects of axial eccentricity, the magnitude of the pulse correction mechanism's impulse, and projectile flight stability factors should be considered.

Claims

1. A method for evaluating the stability of a projectile flight based on Hopf bifurcation analysis, characterized in that, The method comprises the following steps: Step (1): establishing a nonlinear angular motion model of the pulse correction projectile; Step (2): based on the nonlinear angular motion model of the pulse correction projectile in step (1), selecting a gas dynamic parameter of the nonlinear free motion stage of the projectile or a control parameter μ of the controlled flight stage as a bifurcation control variable; Step (3): according to the bifurcation control variable in step (2), calculating a system bifurcation point, judging a bifurcation type of the system based on a center manifold theorem and a normal form theory, and calculating an amplitude of a limit cycle of the system according to an approximate equilibrium equation of the system; Step (4): stability evaluation.

2. The method of evaluating the stability of a projectile flight according to claim 1, wherein Step (1) specifically comprises the following steps: Step (11): Let the system state variables be [δ1, δ2, ω η , ω ζ ] T , where δ1 is the high-low attack angle, δ2 is the directional attack angle, and ω η and ω ζ are the pitch and yaw direction angular velocities in the body axis coordinate system, respectively. Step (12): according to the center of mass motion theorem and a differential equation of the swing angle velocity, a nonlinear angular motion equation of the projectile is obtained as follows: Step (13): considering the aerodynamic force and moment of the projectile under a large attack angle, a nonlinear aerodynamic modeling is performed on the aerodynamic force and moment of the projectile; the power series expansion of the aerodynamic coefficient is expressed as follows c a = c a0 + c a2 δ 2 +... (2) where c a0 is the linear coefficient derivative of the aerodynamic force and moment, c a2 is the cubic coefficient derivative; only the total angle of attack δ is considered in the polynomial model, and the relationship cosδ = cosδ2cosδ1 exists. Step (14): Substitute formula (2) into nonlinear angular motion equation (1), ignore gravity term and Magnus force, Coriolis force, and set The nonlinear angular motion model of the pulse correction projectile is obtained as follows: Wherein, the expressions of the aerodynamic force and moment of the projectile in the free motion stage and the controlled flight stage are shown in (4) and (5) respectively:

3. The method of evaluating the stability of a projectile flight according to claim 2, wherein Step (3) specifically comprises: Step (31): calculating a system equilibrium point; Step (32): calculating a system bifurcation point; Step (33): reducing the four-dimensional nonlinear system to a two-dimensional system according to the center manifold theorem; Step (34): calculating an amplitude and a period of a limit cycle of the system.

4. The method of evaluating the stability of a projectile flight according to claim 3, wherein Step (31) specifically comprises: Let the right side of the nonlinear angular motion model of the impulse modified projectile, i.e. equation (3), be zero, and the unique equilibrium point O[0, 0, 0, 0] of the impulse modified projectile dynamics system in the uncontrolled free flight state is obtained T ; and the Jacobian matrix at the equilibrium point is: For the controlled flight stage with multiple equilibrium solutions, the system equilibrium point is obtained by solving the Matcont.

5. The method of evaluating the stability of a projectile flight according to claim 4, wherein Step (32) specifically comprises: According to the Jacobian matrix formula (6) at the equilibrium point, when the control parameter μ changes, the system stability change is judged based on the four-dimensional system of the Hurwitz criterion, the point at which the system stability changes is the bifurcation point μ0, and the characteristic values λ1, λ2, λ3, λ4 and the characteristic vectors ξ1, ξ2, ξ3, ξ4 of the system at the bifurcation point are obtained.

6. The method of evaluating the stability of a projectile flight according to claim 5, wherein Step (33) specifically comprises: Introduce nonsingular linear transformation x = py, where y = [yl, y2, y3, y4] T where p is a square matrix composed of real and imaginary parts of eigenvectors corresponding to the second and fourth eigenvalues of J; let u = [yl, y2] T , v = [y3, y4] T , into the system equation (3) According to the center manifold theorem, v is expressed as v=h(u), and h(0)=h'(0)=0; v=h(u) is substituted into the second term of formula (7), and based on the derivation chain rule, it is obtained that Formula (8) is substituted into the first term of formula (7), and the h(u) differential equation is obtained after arrangement: Based on the required center manifold order, the equation is as follows: Substituting equation (10) into the differential equation (9) for h(u), we obtain the expression for v = h(u) in terms of the coefficients a ijk b ijk Substituting h(u) into the first term of equation (7), we obtain the reduced equation for the flow on the central manifold as follows: After reduction, the equation is written in the following form: According to the normal form theory, the three-order normal form is as follows: wherein The polar coordinate transformation y1=rcosθ, y2=rsinθ is introduced, and the equation is obtained as follows:

7. The method of evaluating the stability of a projectile flight according to claim 6, wherein Step (34) specifically comprises: The approximate equilibrium equation of the system is obtained according to equation (14): cμr + ar 3 = 0; Amplitude of limit cycle Period T = 2π / ω, where ω is the frequency.

8. The method of evaluating the stability of a projectile flight according to claim 7, wherein Step (4) specifically comprises: When a≠0 and c≠0, the system appears Hopf bifurcation at μ=0, and the bifurcation characteristics are as follows: (1) When c>0 and a>0, the equilibrium point is unstable for μ>0 and asymptotically stable for μ<0, and there exists an unstable limit cycle when μ<0; (2) When c>0 and a<0, the equilibrium point is unstable for μ>0 and asymptotically stable for μ<0, and there exists an asymptotically stable limit cycle when μ<0; (3) When c<0 and a>0, the equilibrium point is asymptotically stable for μ>0 and unstable for μ<0, and there exists an unstable limit cycle when μ<0; (4) When c < 0 and a < 0, the equilibrium is asymptotically stable for μ > 0 and unstable for μ < 0, and there exists an asymptotically stable limit cycle for μ < 0.

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