Code:

import 'dart:async';
import 'package:animated_text_kit/animated_text_kit.dart';
import 'package:animations/animations.dart';
import 'package:flutter/material.dart';
import 'package:page_transition/page_transition.dart';

class SplashScreen4 extends StatefulWidget {
  @override
  _SplashScreen4State createState() => _SplashScreen4State();
}

class _SplashScreen4State extends State<SplashScreen4> {
  @override
  Widget build(BuildContext context) {
    return Scaffold(
      body: Center(
        child: Center(
          child: OpenContainer(
            closedBuilder: (_, openContainer) {
              return Container(
                height: 60,
                width: 220,
                child: Center(
                  child: Text(
                    'View Splash Animation',
                    textAlign: TextAlign.center,
                  ),
                ),
              );
            },
            openColor: Colors.indigo,
            closedElevation: 20,
            closedShape:
                RoundedRectangleBorder(borderRadius: BorderRadius.circular(10)),
            transitionDuration: Duration(milliseconds: 700),
            openBuilder: (_, closeContainer) {
              return SplashScreen4Sub();
            },
          ),
        ),
      ),
    );
  }
}

class SplashScreen4Sub extends StatefulWidget {
  @override
  _SplashScreen4SubState createState() => _SplashScreen4SubState();
}

class _SplashScreen4SubState extends State<SplashScreen4Sub>
    with TickerProviderStateMixin {
  AnimationController scaleController;
  Animation<double> scaleAnimation;

  @override
  void initState() {
    super.initState();

    scaleController = AnimationController(
      vsync: this,
      duration: Duration(milliseconds: 600),
    )..addStatusListener(
        (status) {
          if (status == AnimationStatus.completed) {
            Navigator.of(context).pushReplacement(
              PageTransition(
                type: PageTransitionType.bottomToTop,
                child: ThirdPage(),
              ),
            );
            Timer(
              Duration(milliseconds: 300),
              () {
                scaleController.reset();
              },
            );
          }
        },
      );

    scaleAnimation =
        Tween<double>(begin: 0.0, end: 12).animate(scaleController);

    Timer(Duration(seconds: 2), () {
      setState(() {
        scaleController.forward();
      });
    });
  }

  @override
  void dispose() {
    scaleController.dispose();
    super.dispose();
  }

  @override
  Widget build(BuildContext context) {
    return Scaffold(
      backgroundColor: Colors.indigo,
      body: Center(
        child: DefaultTextStyle(
          style: TextStyle(fontSize: 30.0),
          child: AnimatedTextKit(
            animatedTexts: [
              TyperAnimatedText(
                'APP NAME',
                speed: Duration(milliseconds: 150),
              ),
            ],
            isRepeatingAnimation: false,
            repeatForever: false,
            displayFullTextOnTap: false,
          ),
        ),
      ),
    );
  }
}

class ThirdPage extends StatelessWidget {
  @override
  Widget build(BuildContext context) {
    return Scaffold(
      appBar: AppBar(
        title: Text('Go Back'),
        centerTitle: true,
        brightness: Brightness.dark,
        backgroundColor: Colors.indigo,
      ),
      body: Center(
        child: Text(
          "HOME",
          style: TextStyle(
            fontWeight: FontWeight.bold,
            fontSize: 20,
          ),
        ),
      ),
    );
  }
}