## 1 Introduction  
The synchronization of chaotic systems plays a tremendous role in nonlinear system analysis and has been applied in a large variety of physical, chemical, neural network and biological systems [1; 2; 3] since the pioneering work of [PERSON] and [PERSON] [4]. The synchronization of chaotic systems is a process wherein two chaotic systems adjust a certain characteristic of their motion to standard performance because of a coupling or a forcing. Many different synchronization regimes have been studied in the past several decades, for instance, complete (or identical) synchronization (CS) [5; 6], phase synchronization (PS) [7], lag synchronization (LS) [8; 9; 10; 11; 12], generalized synchronization (GS) [13; 14], anticipating synchronization (AS) [15; 16; 17] and so on. Among all types of synchronization, CS is the simplest one, where the states of all oscillators are identical. GS is considered to be the type of chaos synchronization most frequently occurring in natural systems and is characterized by the existence of a functional relation between the state variables of drive-response systems. Different types of synchronization, such as CS, LS and AS, can be viewed as special types of GS with different functional relations. In technology and nature, the time delay is ubiquitous due to finite signal transmission times, switching speeds as well as memory effects. LS, typically appearing as a coincidence of the shifted-in-time states of two systems, has received extensive attention in secure communications [18], laser networks [9] and digital image encryption [19]. AS was firstly proposed by [PERSON] [20; 21], who considered two chaotic systems unidirectionally coupled in a drive-response configuration and showed that the current states of a response system can be synchronized with the future states of a drive system. Two different coupling schemes were suggested by [PERSON]: One is for the case where there exists an internal delay in a drive system that defines the anticipation time of a synchronized state. The other is for the case where a drive system does not possess internal delays, while in a salve system, there exists time-delay self-feedback. The work by [PERSON] has led to the proven possibility of a real-time predictor of chaotic dynamics for systems without delays. Moreover, the scheme allows one to change the anticipation time without altering the dynamics of a drive system. As a possible application, AS can be used to design control methods for avoiding unwanted pulses from a drive system. AS has been reported in many applications in semiconductor lasers [22], chaotic laser diodes and cryptography [23; 24]. Both LS and AS are mostly observed in systems using a time-delay factor. However, approximate LS has been found in bidirectionally and unidirectionally coupled oscillators with parameter mismatch [25]. AS has been also observed in systems without time delay. Such a type of phenomenon was first reported in the classical Voss scheme, where the true time-delay term was replaced with its first-order approximation [26]. Nevertheless, AS is not exact, that is, the advanced trajectory of a drive system only approximately coincides with the current trajectory of a response system. In addition, the anticipation time, in this case, is small. [PERSON] et al. [25] considered a more general case of nonidentical systems without time delay in which the mean frequency of a free response system is greater than the mean frequency of a master system. They showed that AS may occur in the Voss coupling scheme with vanishing time delay. If the above condition is satisfied in neural systems, the chaotic spikes of a presynaptic neuron may be predicted by a postsynaptic neuron. The Voss scheme provides a simple memoryless method to achieve LS and AS between coupled oscillators.