Hyper Catastrophe on 4-Dimensional Canards

When discovering the potential of canards flying in 4-dimensional slow-fast system with a bifurcation parameter, the key notion “symmetry” plays an important role. It is of one parameter on slow vector field. Then, it should be determined to introduce parameters to all slow/fast vectors. It is, however, there might be no way to explore for another potential in this system, because the geometrical structure is quite different from the system with one parameter. Even in this system, the “symmetry” is also useful to obtain the potentials classified by R. Thom. In this paper, via the coordinates changing, the possible way to explore for the potential will be shown. As it is analyzed on “hyper finite time line”, or done by using “non-standard analysis”, it is called “Hyper Catastrophe”. In the slow-fast system which includes a very small parameter , it is difficult to do precise analysis. Thus, it is useful to get the orbits as a singular limit. When trying to do simulations, it is also faced with difficulty due to singularity. Using very small time intervals corresponding small , we shall overcome the difficulty, because the difference equation on the small time interval adopts the standard differential equation. These small intervals are defined on hyper finite number N, which is nonstandard. As and the intervals are linked to use 1/N, the simulation should be done exactly.

Canards Flying on Bifurcation

There exists a property “structural stability” for “4-dimensional canards” which is a singular-limit solution in a slow-fast system with a bifurcation parameter. It means that the system includes the possibility to have some critical values on the bifurcation parameter. Corresponding to these values, the pseudo-singular point, which is a singular point in the time-scaled-reduced system should be changed to another one. Then, the canards may fly to another pseudo-singular point, if possible. Can the canards fly? The structural stability gives the possibility for the canards flying. The precise reasons why happen are described in this paper.

Influence of control strategy on stability of dual-spin projectiles with fixed canards

Existing literature has shown that the control force at the nose could cause dynamic instability for controlled projectiles. To lower the adverse impact on the dual-spin projectile with fixed canards under the premise of meeting guidance system requirements, the influence of control moment provided by a motor on the flight stability is analyzed in this paper. Firstly, the effect of the rolling movement on stability is analyzed based on the stability criterion derived using the Hurwitz stability theory. Secondly, the evaluation parameters combining the features of different control periods that could assess the variation of stability features after the motor torque are obtained. These effective formulas are used to indicate that, to reduce the flight instability risks, the stabilized rolling speed of roll speed keeping period should be as small as possible; the variation trend of motor torque during the rolling speed controlling period and the roll angle of the forward body during roll angle switching period are recommended corresponding to the projectile and trajectory characteristics. Moreover,detailed numerical simulations of 155 mm dual-spin projectile are satisfactory agreement with the theoretical results.

Canards in a rheodynamic model of cardiac pressure pulsations

This paper reports on the canard phenomenon occurring in a rheodynamic model of cardiac pressure pulsations. By singular perturbation techniques the corresponding parameter value at which canards exist is obtained. The physiological significance of canards in this model is given.

Structural Stability in 4-Dimensional Canards

Let us consider higher dimensional canards in a sow-fast system R2+2 with a bifurcation parameter. Then, the slow manifold sometimes shows various aspects due to the bifurcation. Introducing a key notion “symmetry” to the slow-fast system, it becomes clear when the pseudo singular point obtains the structural stability or not. It should be treated with a general case. Then, it will also be given about the sufficient conditions for the existence of the center manifold under being “symmetry”. The higher dimensional canards in the sow-fast system are deeply related to Hilbert’s 16th problem. Furthermore, computer simulations are done for the systems having Brownian motions. As a result, the rigidity for the system is confirmed.

Impact point prediction guidance based on iterative process for dual-spin projectile with fixed canards

The fixed canards configuration of a dual-spin projectile makes it difficult to apply the traditional guidance law. In this study, a modified impact point prediction guidance strategy based on an iterative process was developed for a class of dual-spin projectiles with fixed canards, to reduce the impact point dispersion. The guidance strategy is dependent on the modified projectile linear theory to rapidly predict the flight states and the impact point. For projectiles with control applied to the trajectory, the modified projectile linear theory method is known to achieve poor impact point prediction. To improve the prediction accuracy, improvements were made to the modified projectile linear theory by considering the products of the yaw rate and other small quantities.The guidance strategy is based on the iterative process for the continuous adjustment of the expected output of the roll angle of the course correction fuze, to minimize the direction error between the predicted impact point and target location. Studies were conducted on a model dual-spin projectile configuration to demonstrate the guidance details. The numerical simulations indicate that the proposed guidance strategy can effectively reduce the projectile impact point dispersion.

A review of dual-spin projectile stability

This paper gives a succinct review of dual-spinprojectile stability and some technologies relating to them.It describes how the traditional stability factors from linear projectile theory are modified to better describe a controlled dual-spin projectile.Finally,it reviews works which have investigated how different aspects of a controlled dual-spin design can affect flight stability,primarily airframe structure and canard properties.A conclusion is given,highlighting important guidelines from the enclosed discussions.

On Canard Homoclinic of a Liénard Perturbation System

The classification on the orbits of some Liénard perturbation system with several parameters, which is relation to the example in [1] or [2], is discussed. The conditions for the parameters in order that the system has a unique limit cycle, homoclinic orbits, canards or the unique equilibrium point is globally asymptotic stable are given. The methods in our previous papers are used for the proofs.

Rolling Control Characteristic Experimental Investigation of a Canard Missile with Free-Spinning Tail

To study the rolling control characteristics of a canard-controlled missile, a series of wind tunnel experiment is conducted. The experimental method, the structure features of wind tunnel model and the experimental results are introduced in this paper. The experimental data show that the canard is an inefficient rolling control device for canard-controlled missile with fixed tail fins; but for the free-spinning tail fin configuration, the canard can conduct rolling control of the missile, and even have higher controlling efficiency under larger canard deflection angle.

A Calculation Method of Rolling Characteristics of Canard Missile with Free-Spinning Tails

The steady flow field of a canard missile on different angles of attack and Mach numbers were studied. Based on analysis, a method was proposed to reduce the calculation for the rolling characteristics of the canard missile with free-spinning tails, and was tested to obtain the relations between rolling moment coefficient, Mach number, and angle of attack. All the computed rolling moment coefficients obtained from the proposed method greatly agreed with the experimental results of FD-06 wind tunnel in CAAA, which proved that the method can not only reduce the calculation cost but also keep precision in calculating the rolling characteristics of canard missiles.

Canard Solutions in a Predator-Prey Model

The canard explosion phenomenon in a predator-prey model with Michaelis-Menten functional response is analyzed in this paper by employing the geometric singular perturbation theory. First, some turning points, such as, fold point, transcritical point, pitchfork point, canard point, are identified;then Hopf bifurcation, relaxation oscillation, together with the canard transition from Hopf bifurcation to relaxation oscillation are discussed.

Canard轨迹原理

为了精确的求出Canard轨迹的描述方程,提出了一个分段表达的非线性模型,能直接求出一个非线性二阶自治方程的相轨迹方程的解析解,能定量求出Canard轨迹的描述方程和参数条件,可以用两类方程简单描述所有的Canard轨迹.

单结管二阶自治电路中的Canard现象

在非线性动力学系统中,三阶以上的自治系统才能产生混沌运动,这使得有关非线性方面的研究集于三阶以上的自治系统(二阶以上的非自治系统)。然而,二阶自治系统内也会出现很复杂的动力学现象;文献[1]首先在数学研究中报道了Canard现象,而Braaksma等对细胞膜方程中的Canard现象进行了研究和报道。Canard现象的名称来源于相轨迹,形状像1只鸭子,它的产生过程简述如下:在含有非线性函数f(x)的二阶自治系统中,在控制参数的分岔点附近的一个邻域内,该系统的相图会出现象鸭子形状的极限环,随控制参数的变化,会出现无头鸭极限环(图1:acbda闭合路径);有头鸭极限环(图1:acbdea闭合路径);驰张振荡器极限环(图1:acba闭合路径)。我们用单结管作非线性元件构成一个二阶自治电路,发现了许多与文献[1]和文献[2]相同和不同的特点。

一氧化碳表面催化氧化反应中canard突变与内信号随机双共振

Canard突变是普遍存在于可激发体系内的一种复杂时间行为,其特征表现为极限环振幅与周期在极小的参数范围内急剧变化.分析了在超临界霍普夫分叉附近canard突变的动力学行为及其对噪声的响应特性,发现并说明了Pt(110)/CO+O2表面催化氧化反应体系中的内信号随机双共振,这是一种噪声对不同频率信号的选择性放大作用.

Extending Slow Manifold Near Generic Transcritical Canard Point

We consider the dynamics of planar fast-slow systems near generic transcritical type canard point. By using geometric singular perturbation theory combined with the recently developed blow-up technique, the existence of canard cycles, relaxation oscillations and solutions near the attracting branch of the critical manifold is established. The asymptotic expansion of the parameter for which canard exists is obtained by a version of the Melnikov method.

Flight dynamics characteristics of canard rotor/wing aircraft in helicopter flight mode

The aerodynamic layout of the Canard Rotor/Wing(CRW) aircraft in helicopter flight mode differs significantly from that of conventional helicopters. In order to study the flight dynamics characteristics of CRW aircraft in helicopter mode, first, the aerodynamic model of the main rotor system is established based on the blade element theory and wind tunnel test results. The aerodynamic forces and moments of the canard wing, horizontal tail, vertical tail and fuselage are obtained via theoretical analysis and empirical formula. The flight dynamics model of the CRW aircraft in helicopter mode is developed and validated by flight test data. Next, a method of model trimming using an optimization algorithm is proposed. The flight dynamics characteristics of the CRW are investigated by the method of linearized small perturbations via Simulink. The trim results are consistent with the conventional helicopter characteristics, and the results show that with increasing forward flight speed, the canard wing and horizontal tail can provide considerable lift,which reflects the unique characteristics of the CRW aircraft. Finally, mode analysis is implemented for the linearized CRW in helicopter mode. The results demonstrate that the stability of majority modes increases with increasing flight speed. However, one mode that diverges monotonously,and the reason is that the CRW helicopter mode has a large vertical tail compared to the conventional helicopter. The results of the dynamic analysis provide optimization guidance and reference for the overall design of the CRW aircraft in helicopter mode, and the model developed can be used for control system design.

Closed-loop dynamic control allocation for aircraft with multiple actuators

A closed-loop control allocation method is proposed for a class of aircraft with multiple actuators. Nonlinear dynamic inversion is used to design the baseline attitude controller and derive the desired moment increment. And a feedback loop for the moment increment produced by the deflections of actuators is added to the angular rate loop, then the error between the desired and actual moment increment is the input of the dynamic control allocation. Subsequently, the stability of the closed-loop dynamic control allocation system is analyzed in detail. Especially, the closedloop system stability is also analyzed in the presence of two types of actuator failures： loss of effectiveness and lock-in-place actuator failures, where a fault detection subsystem to identify the actuator failures is absent. Finally, the proposed method is applied to a canard rotor/wing （CRW） aircraft model in fixed-wing mode, which has multiple actuators for flight control. The nonlinear simulation demonstrates that this method can guarantee the stability and tracking performance whether the actuators are healthy or fail.

Canard phenomenon in second-order autonomous unijunction transistor circuit

In the nonlinear dynamic system the chaotic phenomenon can only appear in the autonomous system whose order is higher than three (included), which impels the research work in the nonlinear field to focus on the above-mentioned system. However,very complex dynamic phenomenon may also occur in the second-order autonomous sys-

Aeroservoelastic modeling and analysis of a canard-configured air-breathing hypersonic vehicles

Air-breathing hypersonic vehicles （HSVs） are typically characterized by interactions of elasticity, propulsion and rigid-body flight dynamics, which may result in intractable aeroservoelastic problem. When canard is added, this problem would be even intensified by the introduction of low-frequency canard pivot mode. This paper concerns how the aeroservoelastic stability of a canard-configured HSV is affected by the pivot stiffnesses of all-moveable horizontal tail （HT） and canard. A wing/pivot system model is developed by considering the pivot torsional flexibility, fuselage vibration, and control input. The governing equations of the aeroservoelastic system are established by combining the equations of rigid-body motion, elastic fuselage model, wing/pivot system models and actuator dynamics. An unsteady aerodynamic model is developed by steady Shock-Expansion theory with an unsteady correction using local piston theory. A baseline controller is given to provide approximate inflight characteristics of rigid-body modes. The vehicle is trimmed for equilibrium state, around which the linearized equations are derived for stability analysis. A comparative study of damping ratios, closed-loop poles and responses are conducted with varying controller gains and pivot stiffnesses. Available bandwidth for control design is discussed and feasible region for pivot stiffnesses of HT and canard is given.